Editorial Note:

Polygon is MDC Hialeah's Academic Journal. It is a multi-disciplinary online publication whose purpose is to display the

academic work produced by faculty and staff. In this issue, we find seven articles that celebrate the scholarship of teaching

and learning from different academic disciplines. As we cannot understand a polygon merely by contemplating its sides, our goal

is to present work that represents the campus as a whole. We encourage our colleagues to send in submissions for the next issue of

Polygon. The editorial committee and reviewers would like to thank Dr. Goonen, Dr. Bradley-Hess, Dr. Castro, and Prof. Jofre for

their unwavering support. Also, we would like to thank Mr. Samuel Hidalgo and Mr. John Munoz of Media Services for their

work on the design of the journal. In addition, the committee would like to thank the contributors for making this edition possible.

It is our hope that you, our colleagues, continue to contribute and support the mission of the journal.

Sincerely, The Polygon Editorial Committee The Editorial Committee: Dr. Mohammad Shakil - Editor-in-Chief Dr. Jaime Bestard Prof. Victor Calderin

Patrons: Dr. Norma M. Goonen, Campus President Dr. Ana Maria Bradley-Hess, Academic Dean Dr. Caridad Castro, Chair of Arts and Sciences Prof. Maria Jofre, Chair of EAP and Foreign Languages

Reviewers: Prof. Steve Strizver-Munoz Prof. Joseph Wirtel

Mission of Miami Dade College The mission of the College is to provide accessible, affordable, high-quality education that keeps the learner’s needs at the center of the decision-making process.

Miami Dade College District Board of Trustees Helen Aguirre Ferré, Chair Peter W. Roulhac, Vice Chair Armando J. Bucelo Jr. Marielena A. Villamil Mirta "Mikki" Canton Benjamin León III Robert H. Fernandez Eduardo J. Padrón, College President

Editorial Notes

i

Guidelines for Submission

ii-iii

The MAT0024 Express (8 weeks) pilot at MDC- Hialeah Campus, Spring 2011

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Deafening Footfalls: A Comparative Analysis of the Poetic Motifs of Borges and Heaney

1931

V. Calderin

The Benefits of Foreign Language Study

3235

M. Orro

Monotonicty, Convexity, and Some Inequalities for Entropy 36of Record Value Distributions 80

M. Shakil

A Note on the Definite Integral

M. Shakil

1-15

J. Bestard

Disclaimer: The views and perspectives presented in these articles do not represent those of Miami Dade College. Â

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POLYGON: Many Corners, Many Faces (POMM) A premier professional refereed multi-disciplinary electronic journal of scholarly works, feature articles and papers on descriptions of Innovations at Work, higher education, and discipline related knowledge for the campus, college and service community to improve and increase information dissemination. Published by MDC Hialeah Campus Liberal Arts and Sciences Department (LAS).

Editorial Committee: Dr. Mohammad Shakil (Mathematics) Editor-in-Chief Dr. Jaime Bestard (Mathematics)

Editor

Prof. Victor Calderin (English)

Editor

Manuscript Submission Guidelines: Welcome from the New POLYGON Editorial Team: The Department of Liberal Arts and Sciences at the Miami Dade College–Hialeah Campus and the new members of editorial committee—Dr. Mohammad Shakil, Dr. Jaime Bestard, and Professor Victor Calderin —would like to welcome you and encourage your rigorous, engaging, and thoughtful submissions of scholarly works, feature articles and papers on descriptions of Innovations at Work, higher education, and discipline related knowledge for the campus, college and service community to improve and increase information dissemination. We are pleased to have the opportunity to continue the publication of the POLYGON, which will be bi-anually during the Fall & Spring terms of each academic year. We look forward to hearing from you. General articles and research manuscripts: Potential authors are invited to submit papers for the next issues of the POLYGON. All manuscripts must be submitted electronically (via e-mail) to one of the editors at mshakil@mdc.edu, or jbestard@mdc.edu, or vcalderi@mdc.edu. This system will permit the new editors to keep the submission and review process as efficient as possible. Typing: Acceptable formats for electronic submission are MSWord, and PDF. All text, including title, headings, references, quotations, figure captions, and tables, must be typed, with 1 1/2 line spacing, and one-inch margins all around. Please employ a minimum font size of 11. Please see the attached template for the preparation of the manuscripts.

Length: A manuscript, including all references, tables, and figures, should not exceed 7,800 words (or at most 20 pages). Submissions grossly exceeding this limit may not be accepted for review. Authors should keep tables and figures to a minimum and include them at the end of the text. Style: For writing and editorial style, authors must follow guidelines in the Publication Manual of the American Psychological Association (5th edition, 2001). The editors request that all text

2 pages be numbered. You may also please refer to the attached template for the preparation of the manuscripts. Abstract and keywords: All general and research manuscripts must include an abstract and a few keywords. Abstracts describing the essence of the manuscript must be 150 words or less. The keywords will be used by readers to search for your article after it is published. Book reviews: POLYGON accepts unsolicited reviews of current scholarly books on topics related to research, policy, or practice in higher education, Innovations at Work, and discipline related knowledge for the campus, college and service community to improve and increase information dissemination. Book reviews may be submitted to either themed or open-topic issues of the journal. Book review essays should not exceed 1,900 words. Please include, at the beginning of the text, city, state, publisher, and the year of the book’s publication. An abstract of 150 words or less and keywords are required for book review essays. Notice to Authors of Joint Works (articles with more than one author). This journal uses a transfer of copyright agreement that requires just one author (the Corresponding Author) to sign on behalf of all authors. Please identify the Corresponding Author for your work when submitting your manuscript for review. The Corresponding Author will be responsible for the following: • • •

ensuring that all authors are identified on the copyright agreement, and notifying the editorial office of any changes to the authorship. securing written permission (via email) from each co-author to sign the copyright agreement on the co-author’s behalf. warranting and indemnifying the journal owner and publisher on behalf of all coauthors.

Although such instances are very rare, you should be aware that in the event a co-author has included content in their portion of the article that infringes the copyright of another or is otherwise in violation of any other warranty listed in the agreement, you will be the sole author indemnifying the publisher and the editor of the journal against such violation. Please contact the editorial office if you have any questions or if you prefer to use a copyright agreement for all coauthors to sign.

3 Instructions for the Preparation of Manuscripts for the Polygon (THE TITLE IS HERE) (12 pt, bold, 32 pt above) NAME IS HERE (11 pt16 pt above, 32 pt below)

ABSTRACT. Abstract is here, not exceeding 160 words. It must contain main facts of the work. (11 pt) Key words and phrases: (11 pt)

1. Introduction (11 pt, bold, 24 pt above, 12 pt below) Main Body

REFERENCES (11 pt, 30 pt above, 12 pt below)

[1] [2] [3] [4]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables. Dover, New York, 1970. J. Galambos and I. Simonelli, Products of Random Variables – Applications to Problems of Physics and to Arithmetical Functions, CRC Press, Boca Raton / Atlanta, 2005. S. Momani, Non-perturbative analytical solutions of the space- and time-fractional Burgers equations. Chaos, Solitons & Fractals, 28(4) (2006), 930-937. Z. Odibat, S. Momani, A Application of variational iteration method to nonlinear differential equations of fractional order, Int. J. Nonlin. Sci. Numer. Simulat. 1(7) (2006), 15-27. (11 pt)

XXXX YYYYY. Received his Master’s/Ph. D. Degree in Physics from the University of ZZZZZ (Country) in 1987 under the direction of Dr. M. N. OPQR. Since 1989, he has been at CCCC College in Hawaii, USA. His research interests focus on the Fractals, Solitons, Undergraduate Teaching of Physics, and Curriculum Development. (11 pt) Department of Liberal Arts & Sciences (Physics Program), CCCC College, P. O. Box 7777, Honolulu, Hawaii, USA. e-mail: xxyy@ccc (11 pt)

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“The MAT0024 Express (8 weeks) pilot at MDC- Hialeah Campus, Spring 2011” Dr. Jaime Bestard, Ph.D. Associate Professor of Mathematics, MDC- Hia. Campus, March, 2011

Abstract: The accelerated learning process in some college preparatory courses has become a modality to increase students’ retention via the student commitment and student motivation. The following paper shows the results of a pilot related to a replication of the experiment related at MDCHomestead Campus in MDC Hialeah Campus. The study linked two consecutive courses like MAT0024“Beginning Algebra and MAT1033”Intermediate Algebra” back to back in compressed 8 weeks semesters linked two disciplines in an instructional practice that is intended to reinforce the competencies of the two courses by the application of the intensive delivery. The students practiced in an intensive format according to brief theoretical information and selfinstructional awareness of what is presented by the course competencies explained in the classroom, including a service learning activity in favor of the community and the motivation of the students produced a positive effect in the academic results that in similar courses taught separately was not experiencing and not expected in the typical courses taught independently in a regular full semester schedule. Theme: Educational Research Key Words: Linked Courses, Accelerated Instruction, Motivation, STEM Education

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1) INTRODUCTION There is certain trend to advise students to take the mathematics requirements very late in the curriculum. It is imperative to eliminate this advising trend with the right position of the math courses in the curriculum. Due to the belief that students with poor expectations in quantitative reasoning may perform better in a semester with another further course that expedite content and avoid in-necessary replications of topics or related course to mathematics, this experimental iteration was conducted in campus. The belief that the inclusive practice of the principles of a mathematics course in a same semester, upon curricular arrange with another discipline which uses quantitative reasoning to argument may produce good result, brought the idea to form the learning community of MAC1105 “College Algebra” with GEO2000”Physical Geography” in the MDC -Hialeah Campus fall semester 2008-01. But at the same time, it is not so feasible to produce learning communities when there are principle- building disciplines that needs intensive practice, like in the case of developmental mathematics. This rationale led to the express concept to link across the full semester to sections of mathematics in eight weeks each: MAT0024 “College Preparatory Algebra” and MAT1033”Intermediate Algebra” Some statements across the nation were leading to this decision and the pilot is supported by previous experiments and analyses college –wide in Homestead and in the Hialeah Campuses: The instructors usually teach the courses independently and the match occurred when two AMATYC Institute projects were developed by them. Final Report, March 2008 Foundations for Success National Mathematics Advisory Panel

•

……”Limitations in the ability to keep many things in mind (working-memory) can hinder mathematics performance. - Practice can offset this through automatic recall, which results in less information to keep in mind and frees attention for new aspects of material at hand. - Learning is most effective when practice is combined with instruction on related concepts. - Conceptual understanding promotes transfer of learning to new problems and better long-term retention”…..

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http://www.ed.gov/MathPanel During two years, the Math advisory Panel worked in the nation: Review of 16,000 research studies and related documents. Public testimony gathered from 110 individuals. Review of written commentary from 160 organizations and individuals, 12 public meetings held around the country. Analysis of survey results from 743 Algebra teachers There is enough scientific evidence cumulated that lead to the previous point, just consider the following research that occurred in our country during the last three years: Foundations for Success National Mathematics Advisory Panel.

2) Methods This experimental study is intended to: To increase student engagement in the instructional process by facilitating the association of the learning outcomes in different disciplines reinforcing the common topics in competencies. To increase motivation of students and the acquisition of quantitative reasoning skills, when the solution of an example in class has potential environmental, technical and social extended impact in a real life application. To improve the student success rate as well as the student pass rates in critical courses, by the effect of the multidisciplinary approach to the solution of problems and cooperative discussions. Improve retention and enrollment in critical courses by effective multidisciplinary and college â€“ wide coordinate interaction. The following factors produce interest in the experimental study: Underprepared students in need of intensive instructional techniques, produce a demand for motivational instructional engines. In order to investigate avenues to better serve the students, faculty at MDC research best practices in teaching learning strategies. According to the MDC Mathematics Discipline Annual Report for the academic year 2007-08(Fig 1) the performance of several math courses is affected to levels of the pass rate under 60 %.

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Fig 1. Pass Rate(%) MDC Mathematics Discipline 2007-08 Academic Annual Report There is certain trend to advice students not to take their mathematics requirements with science or other math related courses, due to the belief that students with poor expectations in quantitative reasoning may perform low in a semester with another science or related course to mathematics. The idea to link math courses within a semester using intensive quantitative reasoning became the foundation of this replication from the experiment at Homestead Campus and there was a very special interest at MDC -Hialeah Campus for the Spring 2011. From this perspective, other authors studied the meta-reflective interactions in the problem space, the cognitive demands, as follows: Meta-reflective interactions in the problem space. Cognitive Demands and Second-Language Learners: A Framework for Analyzing Mathematics Instructional Contexts, Campbell, et all. Mathematical Thinking & Learning, 2007, Vol. 9 Issue 1, p3-30, 28p, 1 chart, 3 diagrams p9

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The authors of this study participated in several previous projects related to Math across the Curriculum, sponsored by the AMATYC Summer and Winter Institutes The preparation of the exercises to extend the interaction of the quantitative reasoning competencies to the multicourse semester was outlined and constructed by not overlapping content that let using the learning momentum. The framework to produce the replication was possible with the support of the administration of the department of Liberal Arts in the person of Dr. Caridad Castro to and the collaborative work of the instructor interacting with the supportive staff of the Academic Support Center at MDC-Hia Campus. The delivery of the competencies was granted across the semester by the pace of the course, according to regular instruction

3) Data Analysis 3.1)The study consisted in the learning process for MAT0024â€?Beginning Algebraâ€? ref # 626359 and MAT1033 ref # 626369 as an experimental group of common students and control groups of independent instruction on similar courses.

5

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The Action Plan had an additional tool, due to the complexity of the interaction to make the learning effective by using the binder and check its completion in class and in the lab time at the Academic Support Center, with the assistance of the tutors. The population under study was the experimental group was formed as described in the following figures. It is remarkable the consistency of the targeted population with the MDC-Hialeah Campus typical students as shown in the following charts:

35 30 25 20 Male

15

Female

10 5 0 Sophomore

Junior

Senior

Fig 3. Previous Mathematics Experience in targeted population 70 60 50 40 30 20 10 0

Series1

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Fig 4. Etxpected Grade Structure of the Target Population 3.2)Assessment of the students’ opinions The intentionality of the activity was declared to the students who prepared their experiment log. The conditions of the instructional time of the activity was a key factor The students were assessed by a pre and post survey, showing the logical change of the population across the semester in Fig 5 80 70 60 50 40

Pre

30

Post

20 10 0 Male

Female

Fig 5. Gender structure of the pre- post survey A Pre-Post survey of the participating students’ opinions was applied in compliance with the AMATYC, MAC3 project. The survey includes twenty one questions related to the students’ perception of skills and or learning outcomes. Those questions were responded in the Pre and the Post applications, as well as ten questions related to the understanding and gains or improvements during the course. Also, it is recorded gender, age group, number of children, number of previous attempts, grade in the last attempt, and ethnicity, keeping the privacy but recording their identification number that lead to matching the pre and post surveys. Those fact are shown in the following group ofcharts:

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3.3) Analysis of the grades: The grades are shown in the Table 1 and in chart of fig 6 the bar graph of de Grade Distribution Review and in Fig 7 the side by side boxplots for the data. Observe the lower results of the test 5 involving the topics of factoring, operations with polynomials.

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TABLE 1 Detail of the course grades by activity and by student T1 95 90 86 75 70 95 90 70 85 0 90 80 70 70 75 70 70 75 65 0 70 70

T2 85 75 89 80 65 90 90 80 90 0 95 80 75 65 76 75 85 85 70 0 75 75

T3 90 85 90 80 70 95 95 70 85 0 90 70 80 70 80 75 90 80 60 0 75 75

T4 95 90 90 75 75 95 90 85 90 0 90 80 75 65 85 80 85 75 65 0 80 70

T5 70 55 60 85 35 50 85 75 70 0 25 75 65 60 45 70 0 75 30 0 75 75

HW 85 77 78 95 82 74 98 70 90 0 71 87 66 76 88 88 81 88 82 0 80 84

Lab 30 32 45 36 32 32 33 32 32 0 34 32 34 32 38 35 33 33 38 0 32 34

Exit 80 63 77 80 77 90 73 73 67 0 70 60 70 80 73 70 70 60 43 0 67 70

FG 79 71 77 76 63 77 82 70 76 0 71 71 65 63 70 71 62 71 57 0 69 69

Gr S S S S S S S S S W S S S S S S S S P W S S

Id #

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Fig 6. Descriptive comparative Grade Distribution Review for the experimental group

Fig 7. Side By Side Box Plots for the Tests and the Exit Exam MAT0024 Express Spring 2011 The regression analysis(Fig 8.) shows a significant strong correlation Grades vs Laboratory Time, as appears in the scatter plot, observe how the regression equation is practical since R2 is greater than 40%, significant and predictable, with all the experimental points in the 95 %prediction interval

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Fig. 8. Fitted Line Plot of the Final Grades vs. Laboratory Time

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The Residual plots show a close to normal distribution and randomly distributed with no specific trend, as in Fig 9.

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Fig. 9. Residuals Plots

Correlations: Lab time, FG Pearson correlation of Lab time and FG = 0.825 P-Value = 0.000

Regression Analysis: FG versus Lab time The regression equation is FG = 20.8 + 1.54 Lab time Predictor Constant Lab time

Coef 20.783 1.5442

S = 12.5096

SE Coef 7.154 0.2367

R-Sq = 68.0%

T 2.90 6.52

P 0.009 0.000

R-Sq(adj) = 66.4%

Analysis of Variance Source Regression Residual Error Total

DF 1 20 21

Unusual Observations

SS 6660.0 3129.8 9789.8

MS 6660.0 156.5

F 42.56

P 0.000

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Obs 10 20

Lab time 0.0 0.0

FG 0.00 0.00

Fit 20.78 20.78

15

SE Fit 7.15 7.15

Residual -20.78 -20.78

St Resid -2.03RX -2.03RX

R denotes an observation with a large standardized residual. X denotes an observation whose X value gives it large leverage.

Fig.10. Fitted Line Plot of Final grades vs. Homework assignment s Grade using My Math Lab The Fig 10 Shows a very strong correlation of the final grades with the Grade of the HW assignments using My Math Lab and about 90 % of the experimental observations are determined by the model and within the 95 % Prediction Interval. In fig 11. The Residuals plot show no trend which confirm the assumptions of the model and the significance and predictability of this model.

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Fig. 11. Residuals Plot Regression Analysis: FG versus HW The regression equation is FG = 4.008 + 0.8060 HW S = 7.07902

R-Sq = 89.8%

R-Sq(adj) = 89.3%

Analysis of Variance Source Regression Error Total

DF 1 20 21

SS 8787.57 1002.25 9789.82

4) Concluding Remarks:

MS 8787.57 50.11

F 175.36

P 0.000

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This pilot study shows the influence of the instructional effective interaction over the quantitative reasoning skills, and this effect can be measured from the both sides of the learning process as follows: IMPACT ON STUDENTS Decrease math anxiety Positive learning attitude Recognition of math in daily life Lifelong appreciation of math Positive experience in math courses Perception that math is relevant Development of quantitative reasoning The positive influence of intensive testing and HW in the final grade IMPACT ON INSTRUCTOR Develops a Comprehensive Instructional Synthesis of the courses Enhance of Departmental Campus Vision Improve Teaching Strategies. Generalization of Conclusions

RECOMMENDATIONS: To compare the study with the subsequent MAT1033 link course and to offer in further major terms and involving greater samples sizes to extend the results to other courses association in the discipline of Mathematics( Like MGF1106-STA2023) To continue incorporating the linked express instructional practices to courses with potential risk in motivation of students

REFERENCES: 1. Bestard, Rodriguez: “Course syllabi for MAC1105, GEO2000 MDC-Hialeah Campus, Fall 2008. 2. Bestard, J: “The T-link Project” AMATYC Summer Institute, Leavenworth, WA, Aug, 2006. 3. Diefenderfer, et all: “Interdisciplinary Quantitative Reasoning” Hollins University, VA, 2000.

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4. Rodriguez, Bestard: â€œThe Global Warming awareness and the environmental math instruction at the MDC-Hialeah Campus. AMATYC Winter Institute, Miami Beach, Fl January 2007)

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Deafening Footfalls: A Comparative Analysis of the Poetic Motifs of Jorge Luis Borges and Seamus Heaney

Victor Calderin, MA

There is a fragile, delicate string that connects Seamus Heaney and Jorge Luis Borges. Although both writers come to us from completely different worlds, they both share a few similar opinions on what art, especially verse, is. The germ for this investigation appeared during an accidental discovery of an interview of Borges by Heaney in Dublin on Bloomsday of 1982. The interview sparked an idea that has culminated in this paper. The concept of expression arises in both of their works. The purity and truthfulness of expression is brought to the surface when both writers are compared. The use of inversion is also something that appears. The theme of the self imposed exile, in reference to Joyce, is also apparent in the their works. The final connection will be nature of expression in the poems. It would not be fair to attempt a comparison that does not touch on the serious differences between both individuals. Heaney and Borges both struggle with similar demons in their respective manners. What appears by looking at both writers is an image of the nature of poetry and its formation. The first item that must be observed is the sphere of their lives. Heaney and Borges appear to come from two drastically different worlds. Heaneyâ€™s upbringing in Ireland sets him in a different intellectual space than Borgesâ€™ Argentine

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roots. An important item to consider when discussing Borges’ origins is that Borges’ childhood is filled with exposure to literature from around the world. Borges’ maternal grandmother was English. His father was an avid reader of English poetry. During one of Charles E. Norton lectures which Borges gave at Harvard in 1967 and 1968 he recalls an: already ancient South American evening, and I see my father…I hear his voice saying words that I understood not, but yet I felt. Those words came from Keats, from his “Ode to a Nightingale” (Borges 98). The library in Borges’ home becomes his primary vehicle in the literary world. He reads writers from various languages. And, while remaining thoroughly Latin American, he embraced and created literature that did not necessarily place him in the category of an Argentine author. Like Heaney, Borges is published in various local literary magazines in his native land. He goes on to establish various relationships with other local writers, like Bioy Casares and Victoria Ocampo. The interesting aspect of Borges’ corpus of works is that it is primarily known for the short stories he composed. The bulk of scholastic writing on Borges deals with his narrative technique in his stories. Heaney is not totally clear about which of Borges’ work he has read. There is an implication that it might have been his poetic material; this question will be address later on in the paper. After the Peron dictatorship, Borges is assigned the head librarian position at the Argentine National Library, an honor he was denied by Peron. Borges spends the latter part of his life giving reading and discussing his work. He passed away in 1986 in Geneva, Switzerland.

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The first items that must be discussed is the nature of inversion and how it functions in Heaney’s idea of play and pattern. In an interview with Haffenden titled “Meeting Seamus Heaney,” Heaney mentions Borges structural style. Heaney states that: Hercules represents the possibility of the play of intelligence. Borges isn’t always writing in verse, but it’s that kind of satisfaction, the exhilarations of the intelligence, the play and pattern, and it’s so different from pleasures of a Neruda, who’s more of an Antaeus-figure (Heaney Interview 22-23). This passage illustrates Heaney awareness of Borges’ work. It is even suggestive of some familiarity of Borges’ verse. The use of myth is crucial to both poets. Neil Corcoran notes that Heaney’s poem about the two mythic figures in North is “obviously primarily an allegory of colonization” with Hercules as the “aggressor” and Antaeus as the “native” (Corcoran 100). This is an evident interpretation. Corcoran sites this depiction of “play and pattern” in Hercules that is associated to Borges by Heaney. This relationship demands closer analysis of “Hercules and Antaeus.” Heaney’s depiction of mythological figures in “Hercules and Antaeus” reveals their symbolic significance. Hercules is named at the beginning of the poem. He is the “Skyborn and royal, / snake-choker, dung-heaver,” (HA ll.1-2). The words associated with Hercules are tied to nature. “Sky”, “snake”, and “dung” connect Hercules to an exterior world. This connection is not a harmonious one. Corcoran sees Hercules as the aggressor. This is evident in the fact that his association with nature is a violent one. Hercules chokes snakes and heaves dung. These images suggest two polar activities. The choking of the snakes sent by Hera is an act of constriction. One can see the fists tightening around the snakes. This is held in opposition to the heaving of hung. The act of heaving calls forth an image of expulsion and repulsion. It is interesting how Heaney

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places these two contradictory images next to each other. Antaeus’s description also reveals much about the poem. Antaeus is seen in opposition to Hercules. He is not “sky-born” as is his counterpart. In mythology, Antaeus draws his strength from the earth, and as long as he is touching the earth, which is his mother, he is invincible. It is only when “Antaeus, the mould-hugger, / is weaned at last:” from the earth, that he is defeated (HA ll.8-9). The epithet of “mould-hugger” alludes to the description of Hercules in earlier lines. The act of hugging is held in opposition to the choking and heaving. Antaeus connects to the material and does not hold himself away from the earth. The giant represents a connection to the natural world and the physical. It is only through a direct disconnection with the earth that Antaeus falters. There is something more behind this simple dichotomy. Heaney chooses the word “wean” to signify the separation between Antaeus and the earth. The word has a gentle tone to it and suggests images of the mother. It is obvious why Corcoran would connect Antaeus to Heaney’s motherland. The mother/child relationship is complicated when Carlanda Green states that “without woman, man is cut off from the earth; he is lost, as is Ge’s son, Antaeus, when he dies at the hand of Hercules” (Green 157). There is a dependency present in respect to Antaeus that is not a component of the character of Hercules. Hercules is “sky-born” and aerial. He is free to move about as he pleases. And it is freedom, which is connected to the intelligence, that brings about the defeat of Antaeus. The conflict manifests itself with the word “wean.” Antaeus’s separation is not gentle, but a severe event that renders him powerless. This inter-conflict which the poem invokes becomes a central theme in the poem.

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The conflict of the contrast between earth and sky is important because it deconstructs our expectation of their roles. This inversion of expectation is a Borgesean technique and is what Heaney refers to as “the play and pattern” in the interview with Haffenden. This play of meanings is evident in the manner in which Antaeus is defeated. Defeat is typically signified by a physical collapse into powerlessness. In Antaeus’ case: a fall was a renewal but now he is raised upthe challenger’s intelligence is a spur of light, (HA ll.10-13). The lifting of Antaeus destroys him. It is through the inversion of positions that Hercules is able to defeat an entity that would normally be invincible. The inversion is a crucial factor. One final observation on inversion complicates the meaning of the poem. The positions of Hercules and Antaeus pose another problem in the reading of inversion in the poem. Hercules and his names are placed at the top of the poem, while Antaeus and his final function as “pap of the dispossessed” are at the bottom (HA l.32). If a reader were to assign spatial directions to the poem, the top could represent the sky and the bottom as the ground. This reverses the position of Hercules and Antaeus and, in turn, the defeat of the giant. There is a suggestion that both these figures are representations of each other. Heaney uses this play to inverse the roles of the two figures and complicate them to the point where there is no true reading of their positions. Henry Hart reveals some insight into this problem when he states that: Defeated Antaeus, paradoxically, will live on as an icon of martyrdom, providing “pap for the dispossessed.” So will Hercules, as a reminder that even the most invincible empires, rationalisms, and technologies are doomed (Hart 9798). There are examples of the play of patterns through the works of Borges. One example is from the poem titled “The Labyrinth.”

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The labyrinth is one of Borges’ most common tropes. He uses it to set up numerous narratives, like Death and a Compass, The Garden of the Forking Paths, The Library of Babel. The labyrinth becomes a symbol for the world in the works of Borges. Its main function is to illustrate the problem with reality and fiction. The real and fictional become inverted and confused in the labyrinth that Borges creates. The poem titled after the maze illustrates this point and illuminates the kind of play of patterns mentioned by Heaney in conversation with Haffenden. “The Labyrinth” also uses mythology to illustrate the theme of inversion. The myth of Theseus and the Minotaur is invoked in this poem. Borges places a nameless individual in the labyrinth. The narrator states that “Zeus himself could not undo the web / of stone closing around me” (L ll.1-2). The reference to Zeus is a subtle connection to Hercules, but primarily, it is a statement of the severity and difficulty of the labyrinth. The physical nature of the labyrinth is also importance. There are “severe galleries which curve in secret circles to the end of the years” (L ll.5-7). This suggests an infinite component to the labyrinth both temporally, as seen in the words “to the end of years”, and spatially in its “secret circles.” The labyrinth becomes the world and is endless. Besides this infinite component, there is the question of who is the narrator. In the traditional myth there are 4 main components: Theseus the hunter, Ariadne the weaver, the Minotaur the murderer, and the 14 virginal victims. Theseus and the Minotaur are the principal characters and are locked in an eternal and archetypal battle just as Hercules and Antaeus are. The representation of these personas is also the same. Theseus can be compared to Hercules in that both function under the hero’s role and use reason and cunning in order to succeed. The Minotaur and Antaeus are symbolically united because

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both are derived from nature, although the Minotaur’s birth is not completely natural but a subversion of the natural process. The inversion that is present in Heaney’s poem is also present in “The Labyrinth.” The identity of the narrator is never truly revealed in Borges’ poem. This not only alludes to the theme of inversion but also questions the act of reading itself. This result is directly connected to Heaney’s observation on play and pattern in Borges. The core to the dilemma in the poem is presented when the narrator reveals that s/he: knows that there is an Other in the shadows, whose fate is to wear out the long solitudes which weave and unweave this Hades and long for [his/her] blood and devour [his/her] death (L ll13-16). The narrator of the poem might be the either the Minotaur or Theseus. Both figures are respectively hunted by the “Other.” The weaving and unweaving allude to Ariadne but also connect the mould-hugging Antaeus and dung-heaving Hercules. This paradox of a world that is constrictive but at the same time liberating is played out in this poem. The main difference between the poems is the imagery connected with the events. Heaney creates a vibrant image in “Hercules and Antaeus” while Borges’ “Labyrinth” is a dark place. Heaney’s poem is filled with natural images like “river-veins,” “secret gullies,” and “hatching grounds.” These images are used to contrast Antaeus and Hercules. Hercules’ description is created through his actions and his thoughts. His mind is “big with golden apples, / his future hung with trophies,” (HA ll.3-4). The aggressor has visions of grandeur in both poems. The Other longs for the demise of the narrator in “the Labyrinth.” The imagery in the Borges’ poem is dark and nondescript. This is one of the major differences between the two poets. Ramona Lagos notes that in Borges’ mature work “the labyrinth acquires the connotation of a jail, horror, and a true

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inferno” (Lagos 144). The bleak maze filled with “pale dust” and frightening “signs” support this idea of the labyrinth as a prison. The most horrific and fantastic aspect of this poem is the ambiguity of the narrator. It states that “each of us seeks the other” (L l.17). This line inverts the reader’s belief about the roles in this poem. Theseus seeks the Minotaur as the Minotaur seeks Theseus. This mutual pursuit opens the door of interpretation regarding who is speaking in the poem. This is a different inversion that the type that Heaney uses in his poem. The play of patterns in Heaney and Borges, while possessing significant differences, is revealed through the inversion of role and characteristics. Heaney presents a world full and action and color in “Hercules and Antaeus.” Borges creates and schizophrenic environment where one does not know who is speaking. Both poets create situation where the victim is the aggressor and the aggressor is the victim. Hercules/Theseus is a distorted reflection of Antaeus/Minotaur. This inversion is evident in the form and content of these poems. And, while not reflective of the poets’ respective works as a whole, it does suggest that there might have been some connection between both poets. There is another avenue of comparison. Both poets incorporate James Joyce in their work. While Joyce will not figure prominently in this paper, it might be worthwhile to observe this function in the works of Heaney and Borges. In June 16th 1982, Seamus Heaney and Richard Kearney interviewed Jorge Luis Borges in Dublin. The first interesting item to note is that when Borges is discussing his acclaim as a writer, Heaney asks “perhaps it is Borges rather than you who writes your works?” (Borges Interview 73). This is direct reference to “Borges and I.” This short prose poem is characteristic of the ephemeral

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nature of identity addressed in “The Labyrinth.” The narrator is discussing the other Borges, the public persona, as the opposite of the narrator’s quiet, private nature. At the end of the piece, the narrator states: Thus my life is an escape. I will lose everything, and everything will belong to oblivion, or to the other. I don’t know which of us wrote this poem (Borges, Selected Poems 93). This poem addresses the self and the difficulties of being controlled by one’s creations. It is interesting how Heaney alludes to this piece in the interview. The authors’ opinion on Joyce is interesting, but what is more interesting is the use of Joyce in their work. Heaney uses Joyce as a guiding voice at the end of “Station Island” while Borges uses Joyce in as a model for the labyrinth-maker. In the final section of “Station Island,” Joyce confronts Heaney and addresses the poet’s preoccupations. The Dubliner states: The main thing is to write for the joy of it. Cultivate a work-lust that imagines your haven like your hands at night dreaming the sun in the sunspot of a breast (SI XII ll.22-25) Henry Hart notes that Joyce “bestows boons in the form of advice regarding the artist’s vocation and badgers Heaney about his fastidious devotions” (Hart 173). Joyce becomes the voice that assists Heaney in understanding that expression is the core of poetry and writing. Heaney dilemma concerning his allegiances is addressed in this section. Heaney like Joyce, and to some extent Borges, is an exile. Heaney discusses this idea of “inner émigré” in his interview with Haffenden when he states that he “had no security, no rails to run on except the ones I invented for myself” (Heaney Interview 24). Borges also sees Joyce in this view of an exile.

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Borges sees Joyce’s writing as works of art that capture the eternal and essential. He admires Joyce’s “plurality of styles” (Borges Interview 73). In “Invocation to Joyce” the speaker says to Joyce: You, meanwhile, forged in the cities of exile in that exile which was your loathed and chosen instrument, the weapon of your art, (IJ ll.19-24). This self-imposed exile of Joyce is parallel to Heaney’s. Their art comes from this stress between home and elsewhere. It only when Hercules lifts Antaeus “out of his element” that the titan enters “into the dream of loss / and origins” (HA ll.15, 16-17). This issue of displacement is what fuels the creative process at various points for Heaney, who mentions “Exposure” as the culmination of this sentiment to Haffenden, and for Borges, who viewed himself as “a European writer in exile. Neither Hispanic nor American, nor Hispanic-American, but an expatriate European” (Borges Interview 74). This disconnection can allow on to freely express oneself without restriction. In “Station Island,” Joyce tells the poets “Let go, let fly, forget. / You’ve listened long enough. Now strike your note” (SI XII ll.29-30). The poet must freely express what they feel. Heaney uses the conflicting situations in both “Hercules and Antaeus” and “Station Island.” Borges uses the paradoxical nature of the labyrinth to express a metaphor for the universe and existence. Personal expression lies at the core of their poetry. The final section of this paper will focus on how expression works between both poets. Borges and Heaney have different beliefs about what poetry and art is. Borges’ view of art and poetry was initially pure expression, but as he mature he saw art as allusion (Borges 117). Borges states that “I do not try, as I tried once, to be a ‘South American

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writer.’ I merely try to convey what dream is” (Borges 119). This idea of dream as impression is important for Borges. During Heaney’s interview with Borges, Heaney shows some interest in the use of dreams to create art. Heaney asks, “Is it then the mode rather than the actual material of dreams that primarily inspires and influences your work?” (Borges Interview 76). Borges responds that dreams “have left their imprint on my fiction in one form or another. The symbols often differ, but the patterns and structures remain the same” (Borges Interview 76). This is crucial to understanding the work of Borges. Expression is defined by these “patterns.” This mixture of disorder, the dream-like, and order, the labyrinthine, becomes the make-up of the corpus of Borges’ work. Borges is completely dependent on the pattern and play. Heaney has a different approach to art. During Heaney’s conversations with Haffenden, the issue of art is brought up. Concerning writing, Heaney says that he “only writes when [he’s] in the trance…. it is a mystery of sorts, if you are possessed by a subject, if you have a subject in you…” (Heaney Interview 13). This is a crucial statement. The driving force before poetry becomes this internal manifestation which “possesses” the poet. Heaney, like Borges, connects art to this internal artistic spirit that much be expressed. Heaney makes another statement about the construction of imagery that flows the same vein of thoughts: I think there’s some kind of psychic energy that cries out for a house, and you have to build the house with the elements of your poetry, with the elements of your imagery, which have to have a breath of life in them (Heaney Interview 18). Heaney believes that the imagery has to possess life. The plastic nature of Heaney’s poetry is evident and confirms this statement. The poetry we read not only has to be inspired, but it must have “a breath of life” in its images. The expression of Borges is a

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mixture of the rational and the irrational. Borges inverts a reader’s expectation in order to make an important point about reality and fiction. Heaney presents a different kind of expression in the form of living, organic imagery in his poetry. His use of inversion is mainly to express contrasts and conflicts present in his poems. Both poets tap into something primal to express their poetry. Borges touches into the archetypal, while Heaney reaches into the dirt and “digs” for something that is universal. Seamus Heaney and Jorge Luis Borges are poets who create poetry that, while at time pertaining to regional concerns, accesses the depth of human experience. Although both are extremely different and come from different environment, they both share certain beliefs in poetics. It is difficult to say how much influence is truly present. It is evident that Heaney read some Borges and that they met and spoke at least once. The few references that do exist are quite fascinating. Heaney and Borges, in their works, reveal that poetry is something that contains the chaotic and the ordered, existing on that delicate border between fiction and reality. Abbreviations: HA = “Hercules and Antaeus” by Seamus Heaney L = “The Labyrinth” by Jorge Luis Borges SI = “Station Island” by Seamus Heaney IJ = “Invocation to Joyce” by Jorge Luis Borges Works Cited: Borges, Jorge Luis. “Borges and the World of Fiction.” An interview by Seamus Heaney and Richard Kearney. The Crane Bag. Vol. 6 No.2 Pgs.71-78 1982. Borges, Jorge Luis. Selected Poems. Ed. Alexander Coleman. New York: Penguin Books, 1999

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Borges, Jorge Luis. This Craft of Verse. Cambridge: Harvard University Press, 2000. Corcoran, Neil. Seamus Heaney. London: Faber and Faber, 1986. Green, Carlanda. “The Feminine Principle in Seamus Heaney’s Poetry” Critical Essays on Seamus Heaney. Ed. Robert F. Garratt. New York: G.K. Hall & Co., 1995. 151-158. Hart, Henry. Seamus Heaney: Poet of Contrary Progression. Syracuse: Syracuse University Press, 1992. Heaney, Seamus. “Meeting Seamus Heaney.” An Interview by John Haffenden. London Magazine. n.s. Vol. 19 No. 3 pgs. 5-28. June, 1979. Heaney, Seamus. Opened Ground. New York: Farrar, Straus and Giroux, 1998. Lagos, Ramona. “The Return of the Repressed: Objects in Borges’ Literature.” Borges the Poet. Ed. Carlos Cortínez. Fayetteville: The University of Arkansas Press, 1986. 142-184

Margarita Orro, Ph.D.* Associate Professor** ABSTRACT. In an increasingly independent but at the same time very much connected world, foreign language proficiency has become more significant than ever. Success depends in large measure on the ability of an individual to function as a member of a global village whose inhabitants speak an ample variety of languages. Learning another language thus, is no longer a pastime; it is a necessity that when fulfilled, can produce numerous cognitive, professional, cultural, and social benefits. This paper presents some of the multiple benefits that can be obtained through the study of foreign languages. At Miami Dade College, students are currently offered the opportunity to master not just one, but two, three, or more foreign languages, by choosing from some of the most widely spoken languages in the world such as: Chinese, French, German, Italian, Portuguese, and Spanish. Any foreign language course(s) taken would also satisfy several of the Learning Outcomes of MDC, most notably #1 [Communicate effectively using listening, speaking, reading, and writing skills], #5 [Demonstrate knowledge of diverse cultures, including global and historical perspectives], and to a lesser degree #3 [Solve problems using critical and creative thinking and scientific reasoning]

KEY WORDS: Foreign languages, benefits, globalization, communication skills, thinking critically, culture, job opportunities.

*Margarita Orro obtained her Ph.D. degree in Spanish and Luso-Brazilian Literatures from the City University of New York in 1993. Since 1989 she has been teaching Spanish language courses at Miami Dade College.

**Department of ESL and Foreign Languages Miami Dade College/ North Campus 11380 NW 27th Avenue Miami, Florida 33167-3495 e-mail: morro1@mdc.edu.

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The Benefits of Foreign Language Study Studying a foreign language is regarded as a positive thing, but what are the real benefits behind learning another language? Well, it turns out there are plenty of them! There are indeed several benefits to learning other languages. Speaking even just one foreign language can open many doors to a person’s future, because it is something that can be used in each and every corner of the globe nowadays. Among other things, the study of a foreign language helps to expand one’s view of the world, it encourages critical reflection on the relation between language and culture, it develops the intellect, and it also aids in building practical skills that may be used in other disciplines. In today’s global economy, there is a rising need for individuals with advanced skills in foreign languages. This is the effect of an increased activity in international business, as well as an awareness of the need to conduct business and diplomatic relations in the language of the host country. Foreign language study does not just look well on transcripts - it looks great on resumes, too! Knowing a foreign language might give the edge when job hunting. As technology makes our world shrink, and the economy becomes more and more “globalized”, many businesses are experiencing a growing need to employ people who can effectively communicate beyond their native tongue; therefore, a second language has become an essential part of the basic preparation in several careers these days. Even cases in which knowing a second language may not be required for a particular job, many find that their foreign language skills would often enhance (and even improve) their opportunities for mobility and promotions within their field. Only a few of the most important job opportunities that a foreign language can open up these days may include: The U.S. Government The United States Government employs citizens with foreign language skills on a regular basis. For example, several agencies and departments such as the IRS, CIA, the FBI, the State Department, the DEA, and the US Armed Services, just to name a few, make substantial use of people with foreign language skills. American and International Businesses

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Knowing a foreign language is a much sought after asset for many foreign companies doing business in the US, as well as in US-based international companies. Business leaders know that the ability to communicate with others in their own language is fundamental in marketing American products abroad. Also, with nearly every industry targeting fast-growing foreign markets, more companies are requiring foreign experience for top management and other important positions. Social and Public Services The social and public services have to deal with so many diverse groups of people, that not knowing foreign languages can, and in many cases does, hamper their ability to serve the public adequately. Health Care The same thing can be said for Health Care. The ability to communicate with non-English speaking people can often make a difference between life and death. ___________________________________________________________________________ Besides a field of specialization, or any technical skills that college students may opt to develop, they should definitely aim to take classes that would help them build on their communication skills, and knowing a second language can certainly give them tangible advantages in the current job market, since foreign language courses precisely focus on developing communication skills, both written and oral (MDCâ€™s Learning Outcome #1). Also, having oneâ€™s attention engaged in another language requires thinking critically (MDCâ€™s Learning Outcome #3) about something that was essentially taken for granted before, namely, the structure of language and its use in day-to-day communications, at the same time that it increases the ability to seek alternative problem-solving techniques. One of the larger, although somewhat less direct benefits of studying a foreign language, is the immersion in a foreign culture. This experience helps us see the world as others do, and that is a very necessary skill in the modern world. As mentioned above, as a result of modern globalization, international business opportunities have multiplied. Thus, both mutual understanding and effective communication between nations have rapidly gained an increased relevance these days, and consequently, there should be a valid urgency for more foreign language study as a means to attaining this goal, since the study of another language can provide the most effective tool for penetrating the barrier of a language and its culture. Furthermore, experiencing another culture would allow for achieving a considerably deeper 34

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understanding of one’s own culture. (MDC’s Learning Outcome #5) In conclusion, picking up a new language has many obvious benefits. If one is intent on learning a new language, it would be a superb idea that will never be regretted! ___________________________________________________________________________ References DeGalan Julie and Stephen Lambert. Great Jobs for Foreign Language Majors. Lincolnwood: VGM Career Horizons, 1994. "Foreign Language Education." The International Encyclopedia of Education. 1985 Ed. Fradd. "Bilingualism, Cognitive Growth, and Divergent Thinking Skills." EDUCATIONAL FORUM 46 (1982): 469-474. Hamayan, Else. "The Need for Foreign Language Competence in the United States." Eric Digest November 1986. Seelye, H. Ned., and Laurence J. Day. Careers for foreign language aficionados & other multilingual types. Lincolnwood: VGM Career Horizons, 1992. Weatherford, H. Jarold. "Personal Benefits of Foreign Language Study." Eric Digest October 1986.

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MONOTONICITY, CONVEXITY AND SOME INEQUALITIES FOR ENTROPY OF RECORD VALUE DISTRIBUTIONS â€“ TECHNICAL REPORT M. Shakil Department of Mathematics Miami Dade College, Hialeah Campus Hialeah, FL 33012, USA, E-mail: mshakil@mdc.edu Abstract In this paper, monotonicity and convexity of entropy of record value distributions obtained from some commonly used continuous probability models are investigated. Some entropy inequalities are obtained. The results show some interesting entropy or information properties of record value distributions. Key words: Entropy, record value distributions, monotonicity, convexity, entropy inequality, digamma functions, error functions. AMS Subject Classifications: 94A15, 94A17, 33B15, 33B20, 62B10, 54C70, 62G07, 62G30, and 62H10.

1. INTRODUCTION: Entropy arises from an important concept known as information (or uncertainty), which is also, sometimes, called self-information. The scale of uncertainty in the outcome of a

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probability experiment with a discrete and finite number of outcomes was first formulated by Shannon (1948) in his studies of communication engineering. 1.1. LITERATURE REVIEW Since the seminal work of Shannon, entropy appears as an important tool with applications in various disciplines of science and engineering. The notion of entropy as developed and studied by Shannon can be used to provide some powerful descriptive and inferential statistical methods. In particular, entropy provides an excellent tool to quantify the amount of information (or uncertainty) contained in a random observation regarding its parent distribution (population). A large value of entropy implies greater uncertainty in the data. In other words, the larger will be the entropy, the smaller will be the information. This follows from the fact that maximizing the entropy is equivalent to minimizing its negative. Further development of Shannonâ€™s concept of entropy, its generalizations, applications, and other entropy measures continued with the contributions of Kullback and Leibler (1951), Lindley (1956), Jaynes (1957a, b), Renyi (1962), Good (1968), Kapur (1989), Soofi and Gokhale (1991), Maasoumi (1993), Soofi (1994), Soofi, Ebrahimi and Habibullah (1995), Mazzuchi, Soofi and Soyer (2000), and Soofi and Retzer (2002), among others. The entropy of various commonly used continuous distributions have been tabulated by many authors, see, for example, Johnson and Kotz (1970), Verdugo Lazo and Rathie (1978), and Ebrahimi, Maasoumi and Soofi (1999). Formulas for RĂŠnyi information and related measures for some standard continuous distributions have been derived by Nadarajah and Zografos (2003). Ebrahimi, Habibullah and Soofi (1992) have developed a test of fit for exponentiality using the estimated Kullback-Leibler discrimination information function. Alwan,

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Ebrahimi and Soofi (1998) have developed an information theoretic process control (ITPC) algorithm for monitoring of processes moments based on Shannon’s entropy and Kullback-Leibler discrimination information function. For entropy-based parameter estimation in hydrology, see, for example, Singh (1998). The use of entropy measures for comparative analyses of educational systems has also been considered, see, for example, Mueller (2000). The entropy of record value distributions from some commonly used continuous probability models have been studied by Shakil (2003). The properties of entropy, Kullback-Leibler information, and mutual information for order statistics have also been explored, see, for example, Ebrahimi, Soofi and Zahedi (2004). 1.2. ENTROPY OF DISCRETE AND CONTINUOUS DISTRIBUTIONS It is defined as follows. Def. 1. ENTROPY OF A DISCRETE PROBABILITY DISTRIBUTION For a discrete random variable associated with n (countable) possible outcomes Ei’s, where P(Ei) = pi, and P = (p1, p2, …, pn), the entropy, Hn(P), is defined as n Hn(P) = – Σ pi ln pi . i=1 It can be easily verified that the entropy, Hn(P), satisfies the following conditions: (i) Hn(P) is maximum when p1 = p2 = …= pn = 1 / n. (ii) Hn(P) is minimum when pi = 1, pj = 0, j ≠ i, i = 1, 2, … , n, that is, Hn(P) is minimum, when one of the probabilities is unity and all others are zero. (iii) From (i) and (ii), it follows that, for the discrete case 0 ≤ Hn(P) ≤ ln(n).

(2.3.1)

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Def. 2. ENTROPY OF A CONTINUOUS PROBABILITY DISTRIBUTION Entropy of an absolutely continuous random variable X, with a probability density function f (x), is defined as follows +∞ HX [ f(x) ] = – ∫ f(x) logf(x) dx, –∞

(2.4.1)

where log is taken at base e ( i.e., natural log). 1.3. ENTROPIC AND INFORMATION INEQUALITIES The subject of entropic and information inequalities also appears as an important and interesting area of study. For example, Wyner and Ziv (1969) have shown that, given E|X|k, we have HX[f(x)] ≤ 1 [ ln { 2k e Γk(1/k)E|X|k } ], k > 0, k kk - 1

(2.4.3)

where |X| denotes the absolute value of X. The equality in (2.4.3) is attained by the maximum entropy distribution with density given by f*(x) = C(η) exp(–η|x|k),

(2.4.4)

where the model parameter η is obtained as the Lagrange multiplier for satisfying the constraint E|X|k ≤ θ < ∞, and C(η) is the normalizing constant. When k = 2, the inequality (2.4.3) can be simplified to the following inequality: [exp(2HX{f(x)})]/( 2πe) ≤ Var(X)

(2.4.5)

The ratio in the inequality (2.4.5) is the entropy power fraction proposed by Shanon (1948) for comparison of continuous random variables. The equality in (2.4.5) holds if and only if X has a normal distribution. Several other inequalities related to entropy and information divergence for discrete and continuous probability distributions and their applications have also been proposed and studied by various authors and researchers, see,

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for example, Stam (1959), Blachman (1965), Witsenhausen (1974), Jones (1979), Costa (1985), Allouche, France and Tenebaum (1988), Kapur (1989), Cover and Thomas (1991), Dembo and Cover (1991), Zamir (1993), Jardas, Pecaric, Roki and Sarapa (1999), Topsoe (2000), Bercher and Vignat (2002), Harremoes and Vignat (2003), Cerone, Dragomir and Osterreicher (2003), and Dragomir (2004), among others. 1.4. SOME REMARKS Note that entropy, HX[p(x)], is always non-negative in the case of a discrete random variable X. Also, when X is discrete, HX[p(x)] is invariant under any one-to-one transformation of X. However, when X is a continuous random variable, the entropy is not necessarily invariant under a one-to-one transformation, and it can take any values in (– ∞, + ∞). Note that E(X2) < ∞ implies HX[f(x)] < ∞, but the converse is not necessarily true. In this paper, monotonicity and convexity of entropy of record value distributions obtained from some commonly used continuous probability models are studied. Some entropy inequalities are obtained. The results show some interesting entropy or information properties of record value distributions. The organization of this paper is as follows. Some basic definitions and concepts of record value distributions are reviewed in section 2. A general expression for entropy of a record value distribution and the entropies of record value distributions associated with the uniform, exponential, Pareto, Weibull, and normal distributions are presented in section 3. Monotonicity, convexity and some inequalities of entropy of record value distributions are discussed in section 4. Some concluding remarks are presented in section 5. Entropies of some well known continuous distributions are given in Appendix I.

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2. RECORD VALUE DISTRIBUTIONS An observation is called a record if its value is greater than (or analogously, less than) all the preceding observations. In the classical record model, the underlying sample from which record values are observed is considered to consist of independent and identically distributed observations from a continuous probability distribution. There are four main variables of interest in the study of records, namely, record times, number of records, inter-record times, and record values. 2.1. LITERATURE REVIEW The probability distributions and related properties of these variables, and also the applications of statistics to the study of record values, have been extensively studied by many researchers. The development of the general theory of statistical analysis of record values began with the work of Chandler (1952), who studied the stochastic behavior of random values arising from the â€œclassical record model,â€? and established that for the random record sequence the waiting times between records have infinite expectations. Further development continued with the contributions of Foster and Stuart (1954), Dwass (1960), Renyi (1962), Gupta (1984), Lin (1988), Qasem (1996), Arnold, Balakrishnan and Nagraja (1998), Gulati and Padgett (2003), Ahsanullah (1995, 2004), among others. The entropy of record value distributions associated with some common continuous probability models, and their properties are investigated, both analytically and numerically, in Shakil (2003). 2.2. Distributions of Record Values Let {Xn : n = 1, 2, â€Ś } be a sequence of independent and identically distributed ( i. i. d.) random variables from an absolutely continuous distribution function F(x), with a

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probability density function f(x). Then a record occurs at index time j, if Xj > Xi ,

(2.2.1)

∀ 1 ≤ i ≤ j – 1, that is, Xj is larger than each of the previous values X1, X2, …, Xj – 1. Thus, the first observation is always a record, and if a record occurs at time t, then Xt will be called a record value. Let R(j) denote the time (index) at which the jth record value is observed. Since the first observation is always a record value, we have R(1) = 1, …, R(j + 1) = min {i: Xi > XR (j)},

(2.2.2)

where we define R(0) = 0. Let Δ(j) = R(j + 1) – R(j)

(2.2.3)

denote the inter-record time between the jth record value and (j + 1)st record value, and let the jth record value XR(j) be denoted by X(j) for simplicity. Then the joint probability density function of the record values X(1), …, X(j) is given by g X ( 1 ), …, X ( j ) ( x1, …, x j ) = [ f ( x j ) ]

j–1 Π [ f ( x r ) / { 1 – F ( x r ) } ], r=1

(2.2.4)

for x1 < x2 < … < x j, and the marginal probability density function (pdf) of the jth record value X(j) is given by, see, for example, Qasem (1996). g X ( j ) ( x ) = { – ln ( 1 – F ( x ) ) } ( j – 1 ) f ( x ). (j – 1)!

(2.2.5)

3. ENTROPY OF A RECORD VALUE DISTRIBUTION 3.1. DERIVATION OF ENTROPY EXPRESSION Let X1, X2, … be i. i. d. observations from an absolutely continuous distribution function

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F(x), with a probability density function f(x). Let R(j) denote the time (index) at which the jth record value is observed, and let the jth record value XR(j) be denoted by X(j). Let be the entropy of the jth record value X(j). Then, using (1.2) for the pdf of X(j) in (1.4), and applying the integrals (2.7), (2.9) and (2.10), the entropy of the jth record value, X(j), is given by

, (3.2) where 1, 2, 3, …, and is the digamma function, see, for example, Shakil (2003, 2006, 2007), Shakil et al. (2008), and Zahedi and Shakil (2007). It is easy to see that for 1, (3.2) easily reduces to the entropy of the parent distribution given by

. Hence, since the first observation from the parent distribution is always considered a record value, the entropy of the first non-trivial record value is obtained when j ≥ 2. 3.2. Entropy Differential Def. Let the entropy of the jth record value, X(j), and the entropy of the (j + 1)st record value, X(j + 1), be denoted by H(j) and H(j + 1), respectively. The jth entropy differential, Δ(j), is defined as Δ(j) = H(j + 1) – H(j), j = 1, 2, 3, … . Note that Δ(j) represents the change in entropy in observing the record value from the jth to the (j + 1)st record value. 3.3. Relative Differential Entropy Index

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Def. Parallel to the concept of relative information index, developed in Ebrahimi and Soofi (1990), the relative differential entropy index, η(r, s), between the rth and sth (r < s) record values, is defined as follows. Let the entropy of the rth record value X(r), and the entropy of the sth record value X(s) be denoted by H(r), and H(s), respectively, r < s. Then η(r, s) = (H(s) – H(r)) / (H(s) – H(1)) ≡ ΔH(r, s) ΔH(1, s)

(2.4.2)

where ΔH(r, s) = H(s) – H(r), and ΔH(1, s) = H(s) – H(1). 3.4. ENTROPY OF A RECORD VALUE OBTAINED FROM SOME COMMONLY USED CONTINUOUS DISTRIBUTIONS Entropy of a record value distribution, associated with the uniform, exponential, Pareto, Weibull and normal distributions, is provided in TABLE 3.4.1 below, see, for example, Shakil (2003, 2006, 2007), Shakil et al. (2008), and Zahedi and Shakil (2007).

TABLE 3.4.1 ENTROPY OF A RECORD VALUE DISTRIBUTION FROM THE UNIFORM, EXPONENTIAL, WEIBULL, PARETO AND NORMAL DISTRIBUTIONS

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Distribution UNIFORM

PDF of the jth record value gX(j)(x) = {– ln(1 – (x/θ))}j – 1 (1/θ)I(0, )(x), Γ(j) where θ > 0, j ≥ 1 is an integer, and 0 ≤ x ≤ θ.

Entropy ln(θ)+ ln[Γ(j)] – (j – 1)ψ(j)

θ

EXPONENTIAL

gX(j )(x) = θ(– j ) (x)j – 1[e – (x / )]I(0, )(x), Γ(j) where θ > 0, j ≥ 1 is an integer, and 0 ≤ x < ∞.

ln(θ)+ ln[Γ(j)] – (j – 1)ψ(j) + j

WEIBULL

gX(j)(x) = (λ)(j)(β) [x(j – 1){exp(– λx )}]I(0, )(x), Γ(j) where λ > 0, β > 0, j ≥ 1 is an integer, and 0 ≤ x < ∞.

ln[Γ(j)] – ln(β) – [(ln(λ))/β] – {j – 1/β}ψ(j) + j

PARETO

gX(j)(x) = {(θ)ln(x)}(j – 1) (θ)[x – ( + 1)]I(1, )(x), Γ(j) where θ > 0, j ≥ 1 is an integer, and 1 ≤ x < ∞.

ln[Γ(j)] – ln(θ) – (j – 1)ψ(j) + [{1 + (1/θ)} × Γ(j + 1)]

NORMAL

_ gX(j)(x) = exp(–x2/2)[ln{2/{1 – erf(x/√2 )}}]j – 1 , √(2π)Γ(j) where j = 1, 2, 3, ..., and – ∞ < x < + ∞.

θ

∞

β

β

θ

∞

∞

4. MONOTONICITY, CONVEXITY AND SOME INEQUALITIES FOR ENTROPY OF A RECORD VALUE DISTRIBUTION In this section, the monotonicity and convexity properties for entropy of a record value distribution, when the parent distributions are uniform, exponential, Pareto, Weibull and normal, are discussed. Some entropy inequalities are also given. 4.1. UNIFORM DISTRIBUTION Def. A continuous random variable X is said to have a uniform distribution over interval (α, β), X ~ uniform (α, β), if its pdf f (x) and cdf F(x) = P(X ≤ x) are, respectively, given by f(x) = [{1/(β – α)] I( , )(x); α

and

β

(3.1.1)

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F(x) = 0, x < α = (x – α)/(β – α), α < x < β

(3.1.2)

= 1, x ≥ β; where I ( , )(x) represents the indicator function defined over the interval (α, β), and α

β

– ∞ < α < β < + ∞. In what follows, without loss of generality, for simplicity of computations, we only consider the entropy for uniform (0, θ), from which the entropy of any general uniform (α, β) can easily be obtained by a simple transformation of the parameter θ. Let X1, X2, … be i. i. d. observations from a uniform (0, θ) parent distribution. The entropy, H(j), of the jth record value, X(j), is given by H(j) = ln(θ) + ln(Γ(j)) – (j – 1)ψ(j), j = 1, 2, 3, …, and θ > 0,

(3.1.6)

where ψ(j) denotes the digamma function. Theorem 4.1.1. Let the entropies of the jth record value, X(j), and the (j + 1)st record value, X(j + 1), be denoted by H(j) and H(j + 1), respectively. Then H(j + 1) = H(j) + ln(j) – ψ(j) – 1, j = 1, 2, 3, ….

(3.2.3)

Proof From (3.1.6), H(j) and H(j + 1), respectively, are given by H(j) = ln(θ) + ln(Γ(j)) – (j – 1) ψ(j),

(3.2.4)

H(j + 1) = ln(θ) + ln(Γ(j + 1)) – (j) ψ(j + 1).

(3.2.5)

and

Subtracting (3.2.4) from (3.2.5), we have H(j + 1) – H(j) = ln(θ) + ln(Γ(j + 1)) – (j)ψ(j + 1) – ln(θ) – ln(Γ(j)) + (j – 1)ψ(j) = ln(Γ(j + 1)) – (j)ψ(j + 1) – ln(Γ(j)) + (j – 1)ψ(j)

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= ln[ Γ(j + 1) ] – (j) [ψ(j + 1) – ψ(j)] – ψ(j) Γ(j) = ln(j) – (j)(1/j) – ψ(j) = ln(j) – ψ(j) – 1, from which, we obtain H(j + 1) = H(j) + ln(j) – ψ(j) – 1, j = 1, 2, 3, … .

(3.2.6)

This proves the theorem. Theorem 4.1.2. Let the entropies of the jth record value, X(j), and the (j + 1)st record value, X(j + 1), be denoted by H(j) and H(j + 1), respectively. Let Δ(j) ≡ H(j + 1) – H(j), j = 1, 2, 3, …, denote the jth entropy differential. Then (a) (1/2j) – 1 ≤ Δ(j) ≤ (1/j) – 1, j = 1, 2, 3… . (b) The sequence {H(j)} of the entropies of the jth record values is monotone decreasing in j, for each θ > 0. (c) lim Δ(j) = lim(H(j + 1) – H(j)) = – 1. j→∞ j→∞ (d) Δ(j) is monotone decreasing in j ≥ 1, and that Δ(j) + ln(2) – 1 ≤ Δ(j + 1) ≤ Δ(j), ∀ j ≥ 1, Proof From (3.2.6), we have Δ(j) = H(j + 1) – H(j) = ln(j) – ψ(j) – 1, j = 1, 2, 3, … . As noted earlier, the digamma function ψ(j) satisfies the following inequality

(3.2.7)

Polygon 2011 Vol. V (1/2j) ≤ ln(j) – ψ(j) ≤ (1/j), j = 1, 2, 3, …. .

48 (3.2.8)

Incorporating the inequality (3.2.8) in (3.2.7), we get (1/2j) – 1 ≤ Δ(j) ≤ (1/j) – 1, j = 1, 2, 3, …,

(3.2.9)

which proves part (a). Also (3.2.9) implies that Δ(j) = H(j + 1) – H(j) ≤ 0, since both lower limit and upper limit ≤ 0, ∀ integer j ≥ 1. This completes the proof of part (b). Proof of (c) follows by taking the limit of (a) as j → ∞. Proof of part (d) follows from the fact that the function D(j) ≡ Δ (j + 1) – Δ(j) = ln[(j + 1)/j] – [ψ(j + 1) – ψ(j)] = ln[(j + 1)/j] – (1/j) It is easy to check that D(j) is a non-positive, and increasing function of j, and hence. min D(j) = D(1) = ln(2) – 1 ≈ –0.3068528… j and max D(j) = limD(j) = 0. j j→∞ This completes the proof of theorem 3.2. Theorem 4.1.3. Let {H(j)} be the sequence of the entropies of the jth record values. Then, for each j, H(j) is an increasing concave function of θ. Proof This easily follows from (3.1.6), by differentiating it twice, w. r. t. θ. 4.1.4. Relative Differential Entropy Index As defined in (2.4.2), the relative differential entropy index, η(r, s), between the rth and sth (r < s) record values, is given by

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η(r, s) = (H(s) – H(r)) / (H(s) – H(1)) ≡ ΔH(r, s) ΔH(1, s)

(3.2.10)

where ΔH(r, s) = H(s) – H(r), and ΔH(1, s) = H(s) – H(1). From (3.1.6), we have ΔH(r, s) = H(s) – H(r) = [ln(θ) + ln(Γ(s)) – (s – 1) ψ(s)] – [ln(θ) + ln(Γ(r)) – (r – 1) ψ(r)] = [ln(Γ(s)) – ln(Γ(r))] + (r – 1) ψ(r) – (s – 1) ψ(s) = ln[Γ(s)] + (r – 1) ψ(r) – (s – 1) ψ(s), Γ(r)

(3.2.11)

and ΔH(1, s) = H(s) – H(1) = [ln(θ) + ln(Γ(s)) – (s – 1) ψ(s)] – [ln(θ) + ln(Γ(1)) – (1 – 1) ψ(1)] = ln(Γ(s)) – (s – 1) ψ(s).

(3.2.12)

Incorporating (3.2.11) and (3.2.12) in (3.2.10), the relative differential entropy index, η(r, s), is obtained as follows η(r, s) = ln[Γ(s)/ Γ(r)] + (r – 1) ψ(r) – (s – 1) ψ(s). [ln(Γ(s)) – (s – 1) ψ(s)]

(3.2.13)

The numerical computations of η(r, s) indicate that, when s is fixed, the relative differential entropy index is decreasing in r. As an illustration, using S-Plus, the graphical and tabular representations of the relative differential entropy index, η(r, s), when s = 2, 5, 10, 15, are provided in Table 4.1.4.1 and Graph 4.1.4.1 below. Table 4.1.4.1

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Relative Differential Entropy Index, η(r, s), when s = 2, 5, 10, and 15, respectively. r 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

s=2

s=5

s = 10

s = 15

1.000000 0.000000

1.0000000 0.8514679 0.5951326 0.3055854 0.0000000

1.0000000 0.9433565 0.8456016 0.7351812 0.6186445 0.4985097 0.3760063 0.2518262 0.1263965 0.0000000

1.00000000 0.96548596 0.90592195 0.83864058 0.76763246 0.69443192 0.61978819 0.54412272 0.46769594 0.39068000 0.31319461 0.23532637 0.15713994 0.07868487 0.00000000

Figure 4.1.4.1 Graph of the Relative Differential Entropy Index, η(r, s), when s = 2, 5, 10, and 15, respectively. 4.2. EXPONENTIAL DISTRIBUTION Def. A continuous positive random variable X is said to have an exponential distribution, with mean θ > 0, if its pdf f (x) and cdf F(x) = P(X ≤ x) are, respectively, given by

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f (x; θ) = (1/θ) e–x/ I(0, )(x),

(4.1.1)

F (x; θ) = (1 – e–x/ )I(0, )(x),

(4.1.2)

θ

∞

and θ

∞

where I(0, )(x) represents the indicator function defined on (0, ∞). Let X1, X2, … be i. i. d. observations from an exponential (θ) parent distribution. The ∞

entropy, H(j), of the jth record value, X(j), is given by H(j) = ln(θ) + ln(Γ(j)) – (j – 1) ψ(j) + j, θ > 0, j = 1, 2, 3, … .

(4.1.4)

where ψ(j) denote the digamma function. Theorem 4.2.1. Let the entropies of the jth record value, X(j), and the (j + 1)st record value, X(j + 1), be denoted by H(j), and H(j + 1), respectively. Then H(j + 1) = H(j) + ln(j) – ψ(j), j = 1, 2, 3, … .

(4.2.3)

Proof From (4.1.2), H(j) and H(j + 1), respectively, are given by H(j) = ln(θ) + ln(Γ(j)) – (j – 1) ψ(j) + j,

(4.2.4)

H(j + 1) = ln(θ) + ln(Γ(j + 1)) – (j) ψ(j + 1) + (j + 1).

(4.2.5)

and

Subtracting (4.2.4) from (4.2.5), we have H(j + 1) – H(j) = ln(θ) + ln(Γ(j + 1)) – (j) ψ(j + 1) + (j + 1) – ln(θ) – ln(Γ(j)) + (j – 1) ψ(j) – j = ln(Γ(j + 1)) – (j) ψ(j + 1) – ln(Γ(j)) + (j – 1) ψ(j) + 1 = ln[ Γ(j + 1) ] – (j) [ψ(j + 1) – ψ(j)] – ψ(j) + 1 Γ(j)

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= ln(j) – ψ(j), from which, we get H(j + 1) = H(j) + ln(j) – ψ(j), j = 1, 2, 3, … .

(4.2.6)

This completes the proof of theorem 4.1. Theorem 4.2.2. Let H(j) be the entropy of the jth record value X(j). Let Δ(j) ≡ H(j + 1) – H(j) denote the jth entropy differential. Then (a) the sequence {H(j)} of the entropies of the jth record values, is monotone increasing in j, for each θ > 0; (b) lim Δ(j) = lim[H(j + 1) – H(j)] = 0. j→∞ j→∞ Proof From (4.2.6), we have H(j + 1) – H(j) = ln(j) – ψ(j), j = 1, 2, 3, … . As noted earlier, since ψ(j) satisfies the inequality 0 < 1/(2j) ≤ ln(j) – ψ(j) ≤ 1/(j), j = 1, 2, 3…, we have H(j + 1) ≥ H(j), j = 1, 2, 3… . This completes the proof of part (a). Proof of part (b) follows by noting that, since j–1 ψ(j) = – γ + Σ (k–1), ∀ integer j ≥ 2, where γ denotes the Euler’s constant, k=1 we have j–1 H(j + 1) – H(j) = ln(j) + γ – Σ (k–1). k=1

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That is, j–1 H(j + 1) – H(j) = γ – [ Σ (k–1) – ln(j)]. k=1

(4.2.7)

j–1 γ = lim[ [ Σ (k–1) – ln(j)], j→∞ k = 1

(4.2.8)

But

by the definition of Euler’s constant. Hence, incorporating (4.2.8) in (4.2.7), we have lim Δ(j) = lim(H(j + 1) – H(j)) = 0. j→∞ j→∞ Theorem 4.2.3. Let {H(j)} be the sequence of the entropies of the jth record values. Then, for each j, H(j) is a monotone increasing concave function of θ. Proof This easily follows from (4.1.4), by differentiating it twice, w. r. t. θ. 4.3. PARETO DISTRIBUTION Def. A continuous random variable X is said to have a Pareto distribution in (1, ∞), with the shape parameter θ > 0, if its pdf f (x) and cdf F(x) = P(X ≤ x) are, respectively, given by f (x; θ) = θ x–( and

θ

+ 1)

I(1, )(x), ∞

F (x; θ) = (1 – x– ), θ

where I(1, )(x) represents the indicator function defined on (1, ∞). ∞

Let X1, X2, … be i. i. d. observations from a Pareto(θ) distribution in (1, ∞). The

(5.1.1) (5.1.2)

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entropy, H(j), of the jth record value, X(j), is given by H(j) = ln(Γ(j)) – ln(θ) – (j – 1)ψ(j) + [1 + (1/θ )] Γ(j + 1), j = 1, 2, 3, …, θ > 0, 5.1.5) where ψ(j) is the digamma function. Theorem 4.3.1. Let the entropies of the jth record value, X(j), and the (j + 1)st record value, X(j + 1), be denoted by H(j) and H(j + 1), respectively. Then H(j + 1) = H(j) + ln(j) – ψ(j) + [1 + (1/θ )](j)Γ(j + 1) – 1, j = 1, 2, 3, …, θ > 0,

(5.2.3)

where ψ(j) is the digamma function. Proof From (5.1.5), H(j) and H(j + 1), respectively, are given by H(j) = ln(Γ(j)) – ln(θ) – (j – 1)ψ(j) + [1 + (1/θ)]Γ(j + 1),

(5.2.4)

and H(j + 1) = ln(Γ(j + 1)) – ln(θ) – (j)ψ(j + 1) + [(1 + (1/θ)]Γ(j + 2).

(5.2.5)

Subtracting (5.2.4) from (5.2.5), we have H(j + 1) – H(j) = ln[Γ(j + 1)] – (j)[ψ(j + 1) – ψ(j)] + [1 + (1/θ)][Γ(j + 2) – Γ(j + 1)] – ψ(j) Γ(j) = ln(j) – ψ(j) + [1 + (1/θ)](j)Γ(j + 1) – 1.

(5.2.6)

Hence, from (5.2.6), we get H(j + 1) = H(j) + ln(j) – ψ(j) + (1 + (1/θ)](j)Γ(j + 1) – 1, j = 1, 2, 3, …, θ > 0,

(5.2.7)

where ψ(j) denotes the digamma function. This completes the proof of theorem 5.1. Theorem 4.3.2. The sequence {H(j)} of the entropies of the jth record values is monotone increasing in j, for all θ > 0.

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Proof From (5.2.7), we have H(j + 1) – H(j) = ln(j) – ψ(j) + [{1 + (1/θ)}(j)Γ(j + 1)] – 1. As noted earlier, since ψ(j) satisfies the following inequality (1/2j) ≤ ln(j) – ψ(j) ≤ (1/j), j = 1, 2, 3, …, we have (1/2j) + [{1 + (1/θ)}(j)Γ(j + 1)] – 1 ≤ H(j + 1) – H(j) ≤ (1/j) + [{1 + (1/θ)}(j)Γ(j + 1)] – 1, θ > 0, j = 1, 2, 3, … .

(5.2.8)

(5.2.8) implies that H(j + 1) – H(j) ≥ 0, since both lower limit and upper limit ≥ 0, ∀ integer j ≥ 1, and θ > 0. Hence H(j + 1) ≥ H(j), j = 1, 2, 3, …, ∀ θ > 0 . This completes the proof of theorem 5.2. Theorem 4.3.3. Let {H(j)} be the sequence of the entropies of the jth record values. Then, for each j, H(j) is a decreasing convex function of θ. Proof This easily follows from (5.1.8), by differentiating it twice, w. r. t. θ. 4.4. WEIBULL DISTRIBUTION Def. A continuous positive random variable X is said to have a Weibull (λ, β) distribution with parameters β > 0 and λ > 0 , if its pdf f (x) and cdf F(x) = P(X ≤ x) are, respectively, given by

Polygon 2011 Vol. V f (x; λ, β) = [λβx(

56 β

– 1)

]exp(–λx )I(0, )(x), β

∞

(6.1.1)

and F (x; λ, β) = [1 – exp(–λx )]I(0, )(x),

(6.1.2)

β

∞

where β > 0 is a shape parameter, and I(0, )(x) represents the indicator function defined ∞

on (0, ∞). Let X1, X2, … be i. i. d. observations from a Weibull (λ, β) parent distribution. The entropy, H(j), of the jth record value, X(j), is given by H(j) = ln[Γ(j)] – ln(β) – [{ln(λ)}/β] – [j – (1/β)]ψ(j) + j, j = 1, 2, 3, … ,

(6.1.5)

where λ > 0, β > 0, and ψ(j) is the digamma function. Theorem 4.4.1. Let the entropies of the jth record value, X(j), and the (j + 1)st record value, X(j + 1), be denoted by H(j) and H(j + 1), respectively. Then H(j + 1) = H(j) + ln(j) – ψ(j) – [(β – 1)/βj], β > 0, j = 1, 2, 3, … ,

(6.2.3)

where ψ(j) is the digamma function. Proof From (6.1.5), we have H(j + 1) – H(j) = ln[Γ(j + 1)] – ln(β) – [{ln(λ)}/β] – [(j + 1) – (1/β)]ψ(j + 1) + (j + 1) – ln[Γ(j)] + ln(β) + [{ln(λ)}/β] + [j – (1/β)]ψ(j) – j = ln[ Γ(j + 1) ] – [j – (1/β)][ψ(j + 1) – ψ(j)] – ψ(j + 1) + 1 Γ(j) = ln(j) – ψ(j) – [(β – 1)/βj], β > 0, j = 1, 2, 3, … , where ψ(j) is the digamma function. (6.2.3) easily follows from (6.2.4). This completes the proof of theorem 6.1. Theorem 4.4.2.

(6.2.4)

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Let {H(j)} be the sequence of the entropies for the record values from of a Weibull (λ, β) parent distribution. Then {H(j)} is monotone increasing in j, for 0 < β ≤ 2, ∀ λ > 0. Proof As noted earlier, since (1/2j) ≤ ln(j) – ψ(j) ≤ (1/j), j = 1, 2, 3, …, from (6.2.4), we have (1/2j) – [(β – 1)/βj] ≤ H(j + 1) – H(j) ≤ (1/j) – [(β – 1)/βj]. That is, [(2 – β)/βj] ≤ H(j + 1) – H(j) ≤ (1/βj). (6.2.5) implies that H(j + 1) – H(j) ≥ 0, since both lower limit and upper limit > 0, for 0 < β ≤ 2, ∀ integer j ≥ 1. This completes the proof. 4.4.3. Remark Our numerical evaluation of H(j) for some selected values of β suggest that H(j) is decreasing in j, for large β > 2. See Figures 6.1 and 6.2 for illustrations.

(6.2.5)

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Figure 6.1: Illustration of monotonic property when 0 < β ≤ 2.

Figure 6.2: Illustration of monotonic property when β > 2. Theorem 4.4.4. Let {H(j)} be the sequence of the entropies for the record values from a Weibull (λ, β) parent distribution. (a) For j ≥ 2, {H(j)} is monotone decreasing in β, if λ > 1. (b) For j = 1, {H(j)} is monotone decreasing in β, if λ ≥ e ≈ 1.781, γ

where γ ≈ 0.5772166…, is the Euler’s constant.

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Proof Note that ψ(1) = –γ ≈ –0.577216649 < 0, and that ψ(j) ≥ 0 for j ≥ 2, where γ is the Euler’s constant. Differentiating (6.1.5), w. r. t. β, we get dH(j)/dβ = –1/β – [ln(λ)/β2] – [ψ(j)/β2].

(6.2.6)

Note that since β > 0, ln(λ) > 0 for λ > 1, and ψ(j) ≥ 0 for j ≥ 2, it follows that the derivative in (6.2.6) is less than zero for λ > 1 and j ≥ 2. This proves part (a). For (b), note that –β – ln(λ) – ψ(1) = –β – ln(λ) – (–γ) = –β – ln(λ) + γ ≤ – ln(λ) + γ ≤ 0, if λ ≥ e = e0.577216649 ≈ 1.781, γ

which completes the proof of part (b). 4.4.5. Remark Our numerical evaluation of H(j) for some selected values of λ and j suggest that H(j) is decreasing in β. See Figures 6.3 - 6.9 for some illustrations of the behavior of H(j).

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Figure 6.3: Illustration of monotonic property for β = 1, 2, 3, … , 10, when λ = 0.5, and j = 1, 2, 5, and 10 respectively.

Figure 6.4: Illustration of monotonic property for β = 1, 2, 3, … , 10, when λ = 1, and j = 1, 2, 5, and 10 respectively.

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Figure 6.5: Illustration of monotonic property for β = 1, 2, 3, … , 10, when 2 < λ < 2.5, and j = 1, 2, 5, and 10 respectively.

Figure 6.6: Illustration of monotonic property for β = 1, 2, 3, … , 10, when λ = 2.5, and j = 1, 2, 5, and 10 respectively.

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Figure 6.7: Illustration of monotonic property for β = 1, 2, 3, … , 10, when λ > 2.5, and j = 1, 2, 5, and 10 respectively.

Figure 6.8: Illustration of monotonic property for β = 1, 2, 3, … , 10, when λ = 5, and j = 1, 2, 5, and 10 respectively.

62

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Figure 6.9: Illustration of monotonic property for β = 1, 2, 3, … , 10, when λ = 10, and j = 1, 2, 5, and 10 respectively. Theorem 4.4.6. Let {H(j)} be the sequence of the entropies for the record values from a Weibull (λ, β) parent distribution. Then H(j) is a monotone decreasing convex function of λ, ∀ β > 0, ∀ integer j ≥ 1. Proof This easily follows from (6.1.5), by differentiating it twice, w. r. t. λ. 4.4.7. Remark Our numerical evaluation of H(j) for some selected values of β and j suggest that H(j) is decreasing in λ. See Figures 6.10 - 6.12 for some illustrations of the behavior of H(j).

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Figure 6.10: Illustration of monotonic property for λ = 1, 2, 3, … , 10, when β = 0.25, and j = 1, 2, 5, and 10 respectively.

Figure 6.11: Illustration of monotonic property for λ = 1, 2, 3, … , 10, when β = 1, and j = 1, 2, 5, and 10 respectively.

64

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Figure 6.12: Illustration of monotonic property for λ = 1, 2, 3, … , 10, when β = 5, and j = 1, 2, 5, and 10 respectively. 4.5. NORMAL DISTRIBUTION Def. A continuous random variable X is said to have a normal distribution with mean µ and variance σ2, X ~ N(µ, σ2), if its pdf f (x) and cdf F(x) = P(X ≤ x) are, respectively, given by __ f(x) = [1/{σ (√2π)}] exp[–{(x – µ)/σ}2}/2], – ∞ < x < + ∞, and __ x F(x) = [1/{σ (√2π)}] ∫ exp[–{(u – µ)/σ}2}/2] du, – ∞ < x < + ∞ , –∞ where – ∞ < µ < + ∞ and σ > 0 are location and scale parameters, respectively. A normal distribution with µ = 0 and σ = 1 is called a standard normal distribution. Note that if Z ~ N(0, 1) and X = µ + σ Z, then X ~ N(µ, σ2), and conversely if X ~ N(µ, σ2) and Z = [(X – µ)/σ], then Z ~ N(0, 1). In what follows, without loss of generality, for simplicity of computations, we only consider the entropy for N(0, 1), from which the

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entropy of any general N(µ, σ2) can easily be obtained by a simple location and scale transformation. Let X1, X2, … be i. i. d. observations from a N(0, 1) parent distribution, with pdf fX and cdf FX, respectively, given by __ fX(x) = [1/(√2π )] exp[–x2/2]

(7.1.1)

and x FX(x) = Φ(x) = ∫ f(u)du –∞ __ x = [1/(√2π)] ∫ exp[–u2/2 ]du. –∞

(7.1.2)

(7.1.2) may be expressed in terms of the error function, erf(x), as follows _ F(x) = Φ(x) = 1 + erf{x/(√2 )}, 2

(7.1.3)

_ x erf(x) = {2/(√π)}∫ exp[–u2]du = P{|Y| ≤ x}, 0

(7.1.4)

where

Y ~ N(µ = 0, σ2 = 1/2), and __ 0 [1/(√2π)] ∫ exp[–u2/2]du = 1/2. –∞

(7.1.5)

Let X(j) denote the jth record value. The entropy, H(j), of the jth record value, X(j), is given by

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___ H(j) = ln[√(2π) Γ(j)] – (j – 1)ψ(j) +∞ _ 2 2 + ∫ x exp(–x /2)[ln{2/{1 – erf(x/√2)}}]j – 1 dx, j = 1, 2, 3, ... , –∞ 2√(2π)Γ(j)

(7.1.9)

where ψ(j) denotes the digamma function. Note that the integral in the expression (4.5.3) for H(j), j > 1, is too complex and does not have a tractable closed form. 4.5.1. Entropy of the 1st record value Taking j = 1 in (7.1.9), and noting +∞ __ 2 2 ∫ x exp(–x /2)dx = √2π, –∞ the entropy of the first record value, X(1), is given by __ H(1) = ln(√2π) + (1/2) = [ln(2πe)]/2 ≈ 1.4189385,

(7.2.1)

which represents the entropy of a standard normal parent distribution. 4.5.2. Entropy of the 2nd (first non-trivial) record value Taking j = 2 in (7.1.15), the entropy of the second record value, X(2), (which represents the first nontrivial record), is given by __ +∞ +∞ _ 2 2 2 2 H(2) = ln(√2π) – ψ(2) + ∫ ln(2)[x exp(–x /2)] dx – ∫ x exp(–x /2)ln{1 – erf( x/√2 )}dx. –∞ 2√(2π) –∞ 2√(2π) _ +∞ _ 2 2 = ln(2√π) + γ – 1 – ∫ x exp(–x /2)ln{1 – erf(x/√2 )}dx, –∞ 2√(2π)

(7.2.2)

_ where γ is the Euler’s constant, and erf(x/√2 ) denotes the error function. Note that the integral in the expression (4.5.3) for H(2), is too complex and does not have a tractable

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closed form. Theorem 4.5.3. Let the entropy of the jth record value, X(j), and the entropy of the (j + 1)st record value, X(j + 1), be denoted by H(j), and H(j + 1), respectively. Then H(j + 1) = H(j) + ln(j) – ψ(j) – 1 +∞ _ _ 2 2 j + ∫ x exp(–x /2) [1{ln{2/(1 – erf(x/√2 ))}} – {ln{2/(1 – erf(x/√2 ))}} j – 1]dx, – ∞ 2√(2π)Γ(j) j j = 1, 2, 3, … . Proof From (7.1.9), we have H(j + 1) – H(j) = ln[ Γ(j + 1) ] – (j) [ψ(j + 1) – ψ(j)] – ψ(j) Γ(j) +∞ _ 2 2 + ∫ x exp(–x /2)[ln{2/{1 – erf(x/√2)}}]j dx, –∞ 2√(2π)Γ(j + 1) +∞ _ – ∫ x2exp(–x2/2)[ln{2/{1 – erf(x/√2)}}]j – 1 dx. –∞ 2√(2π)Γ(j) (7.2.3) implies that

(7.2.3)

H(j + 1) = H(j) + ln(j) – ψ(j) – 1 +

+∞ _ _ ∫ x2exp(–x2/2) [1{ln{2/(1 – erf(x/√2 ))}} j – {ln{2/(1 – erf(x/√2 ))}} j – 1]dx, – ∞ 2√(2π)Γ(j) j

j = 1, 2, 3, … . This completes the proof.

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4.5.4. Remark (i) Our numerical calculations of H(j)’s strongly suggest that the entropies of record values from a standard normal parent distribution is a decreasing function of j. As an illustration, using S-Plus, graphical and tabular representations of entropy of the jth record value, X(j), are provided in Table 7.1 and Figure 7.1, for j = 1, …, 10, below. (ii) Assuming these indications can be proved analytically, then, it is easy to show that 0 ≤ H(j) ≤ [ln(2πe)] /2 ≈ 1.4189385, ∀ j ≥ 1. Table 7.1 Entropy of the jth record value from a standard normal parent distribution j 1 2 3 4 5 6 7 8 9 10

H(j) 1.418891 1.293929 1.241067 1.210799 1.190767 1.176178 1.164494 1.153812 1.142155 1.127193

Figure 7.1: Graph of entropy of the jth record value, for j = 1, …, 10.

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5. Concluding Remarks In this research project, some properties of the entropies for the record value distributions associated with the uniform, exponential, Weibull, and Pareto distributions, which had tractable closed forms, were investigated both analytically (when feasible) and numerically. The entropies of the record value distributions were computed using Shannon’s entropy. The entropy of a record value distribution associated with the normal parent distribution did not have a tractable closed form and needs further investigations. Some general conclusions are drawn as follows. (a) When the parent distribution is uniform (0, θ) (i) The sequence {H(j)} of the entropy of the jth record value is monotone decreasing in j, for each θ > 0. (ii) H(j) is an increasing concave function of θ, for each integer j ≥ 1. (iii) Our numerical results (for those cases that we considered) indicate that η(r, s), the relative differential entropy index, η(r, s), when s is fixed, is decreasing in r. (b) When the parent distribution is exponential (θ) (i) The sequence {H(j)} of the entropies of the jth record values, is monotone increasing in j, for each θ > 0. (ii) H(j) is a monotone increasing concave function of θ, for each integer j ≥ 1. (c) When the parent distribution is Pareto (θ) in (1, ∞) distribution (i) The sequence {H(j)} of the entropies of the jth record values is monotone increasing in j, for all θ > 0. (ii) H(j) is a monotone decreasing convex function of θ, for each integer j ≥ 1.

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(d) When the parent distribution is Weibull (λ, β) distribution (i) The sequence {H(j)} is monotone increasing in j, for 0 < β ≤ 2, ∀ λ > 0. (ii) For j ≥ 2, {H(j)} is monotone decreasing in β, if λ > 1. (iii) For j = 1, {H(j)} is monotone decreasing in β, if λ ≥ e ≈ 1.781, where γ ≈ γ

0.5772166… is the Euler’s constant. (iv) H(j) is a monotone decreasing convex function of λ, ∀ β > 0, ∀ integer j ≥ 1. (e) When the parent distribution is a standard normal N(0, 1) parent distribution (i) Our numerical results for those cases of entropies of record values associated with the normal parent distribution suggest that H(j) is a decreasing function of j. (ii) Assuming these indications can be proved analytically, then, it is easy to show that 0 ≤ H(j) ≤ [ln(2πe)] /2 ≈ 1.4189385, ∀ j ≥ 1. These results will be useful in the applications of Shannon’s entropy in the statistical data analysis and characterization of record value distributions. It will be useful in quantifying information contained in observing each record value. For future work, one can consider to develop inferential procedures for the parameters of the parent distributions based on the entropies of their corresponding record value distributions. REFERENCES 1. Abramowitz, M., and Stegun, I. A. (1970), Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables, Dover, New York. 2. Ahsanullah, M. (1995), Record Statistics, Nova Science Publishers, Commack, New York. 3. Alzer, H. (1997), “On some inequalities for the gamma and psi functions,” Math. Comp., 66, 373 – 389. 4. Arnold, B. C., Balakrishnan, N., and Nagaraja, H. N. (1998), Records, John Wiley & Sons, N. Y.

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5. Chandler, K. M. (1952), “The Distribution and Frequency of Record Values,” Journal of the Royal Statistical Society, Series B 14, 220 – 228. 6. Chaudhry, M. A., and Zubair, S. M. (2002), On a class of Incomplete Gamma Functions with Applications, Chapman & Hall/CRC, Boca Raton. 7. Dwass, M. (1960), “Some k – Sample Rank Order Tests,” in Contributions to Probability and Statistics, 198 – 202, Stanford University Press, Stanford, California. 8. Ebrahimi, N., Maasoumi, E., and Soofi, E. S. (1999), “Ordering Univariate Distributions by Entropy and Variance,” Journal of Econometrics, 90, 317 – 336. 9. Ebrahimi, N., Soofi, E. S. (1990), “Relative Information Loss Under Type II Censored Experimental Data,” Biometrika, 77, 2, 429 – 435. 10. Ebrahimi, N., Soofi, E. S., and Zahedi, H. (2003), “Information Properties of Order Statistics and Spacings,” Technical Report (to appear). 11. Foster, F. C., and Stuart, A. (1954), “Distribution – Free Test in Time Series Based on the Breaking of Records,” Journal of the Royal Statistical Society, Series B 16, 1 – 22. 12. Good, I. J. (1968), “Utility of a Distribution,” Nature 219 (5161). 13. Gradshteyn, I., and Ryzhik, I. (1980), Tables of Integrals, Series and Products, Academic Press, New York. 14. Gulati, S., and Padgett, W. J. (2003), Parametric and Nonparametric Inference from Record – Breaking Data, Springer – Verlag, New York. 15. Gupta, R. C. (1984), “Relationships between order statistics and record values and some characterization results,” Journal of Applied Probability, 21, 425 – 430. 16. Jaynes, E. T. (1957a), “Information Theory and Statistical Mechanics,” Physical Review, 106, 620 – 630. 17. Jaynes, E. T. (1957b), “Information Theory and Statistical Mechanics,” Physical Review, 171 – 197. 18. Jeffrey, A. (1995), Mathematical Formulas and Integrals, Academic Press, San Diego. 19. Johnson, N. L., and Kotz, S. (1970), Distributions in Statistics: Continuous Univariate Distributions, Vol. I - II, John Wiley & Sons, New York. 20. Kapur, J. N. (1989), Maximum-Entropy Models in Science and Engineering, Wiley-

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Eastern, New Delhi. 21. Kullback, S., and Leibler, R. A. (1951), “On Information and Sufficiency, Annals of Mathematical Statistics,” 22, 79 – 86. 22. Lin, G. D. (1988), “Characterizations of Uniform Distributions and of Exponential Distributions,” Sankhya, Ser. A, 50, 64 – 69. 23. Lindley, D. V. (1956), “On a Measure of Information Provided by an Experiment,” Annals of Mathematical Statistics, 27, 986 – 1005. 24. Maasoumi, E., (1993), “A Compendium to Information Theory in Economics and Econometrics,” Econometric Reviews, 12(2), 137 – 181. 25. Mazzuchi, T. A., Soofi, E. S., and Soyer, R. (2000), “Computation of Maximum Entropy Dirichlet for Modeling Lifetime Data,” Computational Statistics & Data Analysis, 32, 361 – 378. 26. Mitrinovic, D. S. (1970), Analytic Inequalities, Springer, New York. 27. El-Qasem, A. A. (1996), “Estimation via Record Values,” Journal of Information & Optimization Sciences, Vol. 17, No. 3, pp. 541 – 548. 28. Renyi, A. (1962), “Theorie des Elements Saillants d’une Suite d’Observations, with Summary in English,” Colloquium on Combinatorial Methods in Probability Theory, 104 – 117, Mathematisk Institut, Aarhus Univeritet, Denmark. 29. Shakil, M. (2003), “Entropy Study of Record Value Distributions obtained from Some Commonly Used Continuous Probability Models,” Masters’ Thesis, Florida International University, Florida, USA. 30. Shakil, M. (2005). “Entropies of Record Values Obtained From The Normal Distribution and Some of their Properties.” Journal of Statistical Theory and Applications, Vol. 4, No. 4, 371-386. 31. Shakil, M. (2006). “Entropy and Information Properties of Record Values from a Weibull Distribution.” Journal of Statistical Theory and Applications, Vol. 5, No. 2, 105-126. 32. Shakil, M., Singh, J.N., and Kibria, B.M.G. (2008). A note on Kullback-Leibler discrimination information properties of record values. Journal of Statistical Research, Vol. 42, No. 2, pp. 89-98. 33. Shannon, C. E. (1948), “A Mathematical Theory of Communication,” Bell System

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Tech. Journal, 27, 379 – 423, 623 – 656. 34. Soofi, E. S. (1994), “Capturing the Intangible Concept of Information,” Journal of the American Statistical Association, Vol. 89, No. 428, 1243 – 1254. 35. Soofi, E. S., Ebrahimi, N., and Habibullah, M. (1995), “Information Distinguishability with Application to Analysis of Failure Data,” Journal of the American Statistical Association, Vol. 90, No. 430, 657 – 668. 36. Soofi, E. S., and Gokhale, D. V. (1991), “Minimum Discrimination Information Estimator of the Mean with known Coefficient of Variation,” Computational Statistics & Data Analysis, 11, 165 – 177. 37. Soofi, E. S., and Retzer, J. J. (2002), “Information Indices: Unification and Applications,” Journal of Econometrics, 107, 17 – 40. 38. Suhir, E. (1997), Applied Probability for Engineers and Scientists, McGraw-Hill, New York. 39. Verdugo Lazo, A. C. G., and Rathie, P. N. (1978), “On the Entropy of Continuous Probability Distributions,” IEEE Transactions of Information Theory, Vol. IT-24, No. 1. 40. Wyner, A. D., and Ziv, J. (1969), “On communication of Analog Data from Bounded Source Space,” Bell System Technical Journal, 48, 3139 – 3172. 41. Zahedi, H. and Shakil, M. (2006). “Properties of Entropies of Record Values in Reliability and Life Testing Context.” Communications in Statistics, 35: 997–1010.

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75 APPENDIX I TABLE 1

ENTROPY OF SOME IMPORTANT LOCATION – SCALE FAMILY OF UNIVARIATE CONTINUOUS PROBABILITY DISTRIBUTIONS* ( α = location parameter; β = scale parameter; α, β > 0; – ∞ < x < + ∞ ) Distributions Gaussian (Normal)

PDF __ f(x; α, β2) = [1/(β√2 π)]exp[–(x – α)2/(2β2)], where – ∞ < x < + ∞, α > 0, β > 0.

Extreme Value (Gumbel)

ln(β) + 1 + γ f(x; α, β) = (1/β)exp[–(x – α)/β – exp{–(x – α)/β}], , where – ∞ < x < + ∞, α > 0, β > 0. where γ = 0.5772… is Euler’s constant ln(β) + ln(2e) f(x; α, β) = (1/2β)exp{–|x – α|/β}, where – ∞ < x < + ∞, α > 0, β > 0.

Laplace ( Double Exponential )

Logistic

–2

Entropy (1/2)ln(2πeβ2 )

ln(β) + 2

f(x; α, β) = (1/β)[{1 + exp{–(x – α)/β}} × exp{–(x – α)/β}], where – ∞ < x < + ∞, α > 0, β > 0. Uniform

f(x; α, β) = (1/β), where α – (β/2) < x < α + (β/2), β > 0.

* Source: Ebrahimi, N., Maasoumi, E., and Soofi, E. S. (1999).

ln(β)

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76 TABLE 2

ENTROPY OF SOME IMPORTANT SHAPE – SCALE FAMILY OF UNIVARIATE CONTINUOUS PROBABILITY DISTRIBUTIONS* ( α = shape parameter; β = scale parameter; α, β > 0; 0 < x < + ∞ ) Distributions PDF Gamma; Exponential: α = 1; f(x; α, β) = [1/{βΓ(α)}](x/β) – 1e– x/ , Erlang: α = 1, 2, …; where 0 < x < + ∞, α > 0, β > 0. Chi-square: α = 1/2, 2/2, … , β=2

Entropy ln(β) + ln(Γ(α)) + (1 – α)ψ(α) + α

Inverse Gamma; Inverse Chi-square: α = 1/2, 2/2, … , β = 1/2.

ln(β) + ln(Γ(α)) – (1 + α)ψ(α) + α

α

β

f(x; α, β) = [1/{βΓ(α)}](x/β)– – 1 exp{–(x/β)2}, where 0 < x < + ∞, α > 0, β > 0. α

Generalized Normal; f(x; α, β) = [1/{2βΓ(α/2)}](x/β) Half-Normal: α = 1; where 0 < x < + ∞, α > 0, β > 0. Rayleigh: α = 2; MaxwellBoltzmann: α = 3; Chi: α = 1, 2, … .

α

Inverse Generalized - Normal; Inverse Chi: _ β = √2, α = 1, 2, …

Log - Normal

Pareto

f(x; α, β) = [1/{2βΓ(α/2)}] (x/β) where 0 < x < + ∞, α > 0, β > 0.

–1

exp{–(x/β)2},

–α–1

2

exp{–(x/β) },

__ f(x; α, β) = [1/(αx√2π)]exp[{–(lnx – lnβ)2}/(2 α2)], where 0 < x < + ∞, α > 0, β > 0. f(x; α, β ) = (α/β)(x/β)– where 0 < β < x, α > 0

α

–1

,

ln(β) + ln[(Γ(α/2))/2 ] + {(1 – α)/2}ψ(α/2) + (α/2) ln(β) + ln[(Γ(α/2))/2 ] – (α/2)ψ(α/2) + (α/2) ln(β) + ln(α) + (1/2)ln(2πe) ln(β) – ln(α) + (1/α) + 1

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77 TABLE 2 (CONTINUED)

Distributions Triangular

PDF f(x; α, β ) = [2/(αβ)](x/β), 0 ≤ x ≤ (αβ),

Entropy ln(β) – ln(2) + (1/2)

= [2/{(1 – α)β}][1 – (x/β)], (αβ) ≤ x ≤ β, where α > 0, β > 0. Weibull

ln(β) – ln(α) + [{γ(α – 1)}/α] + 1, where γ = 0.5772… is Euler’s constant

f(x; α, β) = (α)(x/β) – 1exp[–(x/β)2], where 0 < β ≤ x; and α > 0 α

* Source: Ebrahimi, N., Maasoumi, E., and Soofi, E. S. (1999).

TABLE 3 ENTROPY OF STUDENT – t, F, AND BETA FAMILIES OF PROBABILITY DISTRIBUTIONS* Distributions Student – t: α = 1, 2, …; Cauchy: α = 1

F: α = 1, 2, … , β = 1, 2, … .

Entropy ln[(α) B(1/2, α/2)] + ((α + 1)/2)[ψ{(α + 1)/2} – ψ(α/2)] 1/2

f(x; α) = [1/{(α)1/2B(1/2, α/2)}] × {1 + (x2/α)}– ( + 1)/2, where – ∞ < x < + ∞; and α > 0. α

f(x; α, β) = [(α) /2(β) /2x( /2) – 1] ÷ [{B(α/2, β/2)}(β + αx)( + where x > 0; and α, β > 0 are integers. α

β

α

α

Beta

–1

f(x; α, β) = [1/{B(α, β)}][x (1 – x ) where 0 ≤ x ≤ 1; and α, β > 0. α

* Source: Ebrahimi, N., Maasoumi, E., and Soofi, E. S. (1999).

β

β

)/2

–1

],

],

ln[{αB(α/2, β/2)}/β] – {(α – 2)/2}ψ(α/2) – {(β + 2)/2}ψ(β/2) + {(α + β)/2}ψ{( α + β)/2} ln[B(α, β)] – (α – 1)[ψ( α ) – ψ (α + β)] – (β – 1)[ψ(β) –ψ(α+β)]

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REVIEW OF SOME USEFUL MATHEMATICAL RESULTS The following definitions and mathematical results are useful in the computation of entropy of record value distribution, see, for example, Abramowitz and Stegun (1970), and Gradshteyn and Ryzhik (1980). 4.1. Gamma Function Def. Let α > 0. The integral ∞ ∫ t –1e–tdt ≡ Γ(α) 0

(2.1)

α

is called a (complete) gamma function. 4.2. Digamma Function Def. A digamma function, denoted by ψ(z) and also called a psi function, is defined as (2.2) 4.3 Exponential Integral Def. The exponential integral is defined as follows =

where

,

(2.3)

and it assumed that the path of integration excludes the origin and does

not cross the negative real axis. 4.4 Error Function Def. (i) The error function, denoted by

, is defined as

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(2.4)

where

.

(ii)

(2.5a)

(iii)

(2.5b)

(iv)

(2.6a)

(v) x = – ∞ ⇒

(– ∞) = –

(∞)= –1; x = + ∞ ⇒

(+ ∞) =1.

(2.6b)

4.3. Lemma (a) Let Γ(α) be a gamma function defined in (2.1). Then (i) Γ(1) = 1. (ii) Γ(α) = (α – 1) Γ(α – 1). (iii) Γ(n) = (n – 1)!, if n is a positive integer. (iv) Γ(1/2) = (v)

0, 1, 2, 3, … .

∞ (vi) ∫ t –1e–t/ dt = λ Γ(α), for any 0 < λ < ∞. 0 α

λ

α

(2.7)

(b) Let ψ(z) be a digamma function defined in (2.2). Then ψ(z) satisfies the following properties. (i) ψ(z + 1) = ψ(z) + (1/z).

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(ii) ψ(1 – z) = ψ(z) + πcot(πz). (iii) ψ(1) = – γ, where γ = lim [{(1 + 1/2 + 1/3 + … + 1/(j – 1)} – ln(j – 1)] ≈ 0.57721566 j→∞ is the Euler’s constant. (iv) ψ(2) = – γ + 1. n–1 (v) ψ(n) = – γ + Σ (k–1), ∀ integer n ≥ 2. k=1

(2.8)

(vi) ψ(1/2) = – γ – 2 ln2. (vii) 1/2z < ln(z) – ψ(z) < 1/z , (z > 0), see, for example, Alzer (1997). (viii) 1/z < ψ(z) < 1/(z – 1), (z > 1), see, for example, Mitrinovic (1970). (c)

∞ ∫ tj – 1 e–t ln(t)dt = Γ(j)ψ(j), j ≥ 1 is an integer. 0

(d)

∞ ∫ tj–1 0

ln(t)dt = θjΓ(j)[ψ(j) + ln(θ)], j ≥ 1 is an integer, θ > 0.

(2.9)

(2.10)

Proof For (a) and (b), see, for example, Abramowitz and Stegun (1970), Gradshteyn and Ryzhik (1980), Jeffrey (1995), and Chaudhry and Zubair (2002). (c) and (d) are based on some more complex mathematical results, see, for example, Shakil (2003, 2006).

t j 1e t ln(t )dt

A NOTE ON THE DEFINITE INTEGRAL

0

M. Shakil Department of Mathematics Miami Dade College, Hialeah Campus Miami, FL 33012, USA E-mail: mshakil@mdc.edu

ABSTRACT

t j 1e t ln(t )dt , which involves logarithmic, exponential and power

The definite integral

0

functions, occurs frequently in many problems of applied sciences, such as physics, engineering, probability, reliability theories, etc. The purpose of this paper is to give analytical proof of this

t j 1e t ln(t )dt , with some historical background and applications in probability

integral

0

theory.

Key words: Incomplete integral, gamma function, digamma function, Macdonald function, Mellin transform, entropy, probability density function.

MSC 2000: 01A05, 26A36, 28A25, 33B15, 62G30, 94A15

2

1. INTRODUCTION The definite integral

t j 1e t ln(t )dt , where 0 and j 1 is an integer,

(1.1)

0

occurs frequently in many problems of applied sciences, such as physics, engineering, probability, reliability theories, etc. Recently, we came across this integral in some problems on entropies of record values. For example, the computation of Shannon’s entropy of an absolutely continuous random variable X with a probability density function f (x ) , that is,

H x f ( x) f ( x) ln[ f ( x)]dx ,

(1.2)

can be reduced to the evaluation of the integral (1.1) and similar integrals (see, for example, Shakil (2003), Zahedi and Shakil (2006), among others). After a thorough search, no analytical evaluation or any historical evidence of the integral (1.1) was found in the available literature. Only some references appear, for example, in Gradshteyn and Ryzhik (1980), Bierens de Haan (1867), and Erdelyi (1954, Vol. 1). In this article, an analytical evaluation of the integral (1.1) is presented, which is based on some complex mathematical results. The organization of this paper is as follows. Section 2 contains historical background of the integral (1.1). The analytical evaluation of the integral (1.1) is presented in Section 3. Applications of the integral in the computations of entropies of record values corresponding to some continuous probability distributions are discussed in Section 4. Some concluding remarks are given in Section 5.

2. SOME HISTORICAL REMARKS As pointed above, the integral (1.1) occurs frequently in the computations of entropies of record values corresponding to continuous probability distributions. A formula for the integral can be found in Gradshteyn and Ryzhik (1980, page 576), as given below 1 x

x

e

ln xdx

1 ln , where Re 0, Re 0.

0

No analytical proof of the integral is given in Gradshteyn and Ryzhik (1980), except the references of Bierens de Haan (1867) and Erdelyi (1954, Vol. 1). Our search continued. We were able to find a 1957 reprint of Bierens de Haan's 1867 "Nouvelles Tables D'Integrales Definies," published by Hafner Publishing Company, New York and London. The above integral appears in Bierens de Haan (1957, page 496, Table 353, Formula Number 1) as follows

e qx lx.x p 1 dx

1 p Z / p lq, qp

0

/

where Z

p is used for gamma function and

lq denotes the natural logarithm of q. No analytical

3

proof of the integral is given in Bierens de Haan's 1867 edition, except a reference to his own 1862 publication of Tables D'Integrales Definies, published as Volume VIII of the Memoirs of the Royal Academy of Sciences of Amsterdam. It is interesting to note that David Bierens de Haan (1822 - 1895) was a professor of mathematics at Leiden University. He is known for compiling tables of integrals. Following the nomination by D. Bierens de Haan and H.G. van de SandeBakhuyzen, a doctorate in mathematics and astronomy, honoris causa, was conferred upon Thomas Joannes Stieltjes by Leiden University in June, 1884. For more on the life and work of D. Bierens de Haan, the interested reader is referred to Sheldon (1912), Schrek (1955), and Talvila (2001). In Erdelyi (1954, Vol. I, page 315), the above integral appears as the Mellin

x

transform of f ( x ) x e

ln x , Re( ) 0 , given by

[x

e x ln x ]x s 1dx s s s ln , Res Re ,

0

where s denotes a complex number. But no derivation or proof of the integral is given in Erdelyi (1954, Vol. I).

t j 1e t ln(t )dt

3. ANALYTICAL EVALUATION OF THE DEFINITE INTEGRAL

0

In this section, an analytical evaluation of the above integral is provided. For this, we consider a more general class of an incomplete integral given by

M , t; a, b x

b ax ( 1) x

e

[ln x] dx,

(3.1)

t

where t 0, a 0, b 0, , and 0 is an integer. The integral (3.1) is used

in the study of the generalized inverse Gaussian distribution, whose probability density function is given by at 1

f (t )

b t

t e , where t 0, a 0, b 0, . M 0 ; a, b

For details on (3.2), see, for example, Chaudhry and Zubair (2002), and references therein. In

(3.2)

4

(3.2), M 0 ; a, b is given by

M 0 ; a, b t

b at 1 t

2

b dt 2 K 2 ab , a

e

(3.3)

0

where

a 0, b 0, and K is Macdonald function, (see, for example, Gradshteyn and Ryzhik

(1980), and Chaudhry and Zubair (2002), among others). The chi-square, exponential, Erlang, gamma, log-normal, Raleigh, and Weibull probability densities are the special cases of the probability density function (3.2), and can be easily derived by simple transformations of the variable t or by taking special values of the parameters

a, b, and . A closed form solution of

the incomplete integral M , t ; a, b as given in (3.1) above is not known. However, a solution

of M , t ; a, b , when t 0 and 0 , is given in Gradshteyn and Ryzhik (1980, page 340). A solution of an incomplete integral of the form

b

b ax x ( 1) t x a x x 2 e [ln x] dx, where

(3.4)

t 0, a 0, b 0, and , can be found in Gradshteyn and Ryzhik

(1980), for

1 , where 0, 1, 2, 3, . A solution for t 0 and is given in 2

Chaudhry and Ahmad (1992). The incomplete integral M , t ; a, b as given in (3.1) satisfies the following functional recurrence relation

aM 1, t; a, b M , t; a, b bM 1, t; a, b b at t

M 1 , t ; a, b t e

[ln t ] ,

(3.5)

where t 0, a 0, b 0, and , (see, for example, Chaudhry (1994), among

others). If t 0 , the above recurrence relation (3.5) holds true a 0, b 0, and

. However, for t 0 , the following functional recurrence relation holds

5

aM 1, 0; a, b M , 0; a, b bM 1, 0; a, b M 1 , 0; a, b ,

(3.6)

where a 0, b 0, and , (see, for example, Chaudhry (1994), among others).

t j 1e t ln( t )dt , where 0 and j 1 is an integer, note

In order to evaluate the integral

0

that, since t 0 , substituting j , a , b 0, and 1 in (3.6), we obtain the following

recurrence relation

M 1 j 1, 0; , 0 jM 1 j , 0; , 0 M 0 j, 0; , 0 ,

(3.7)

where

t j e t ln(t )dt .

M 1 j 1, 0; , 0

(3.8)

0

From (3.7) and (3.8), we have

j

t

( ) t e

j 1 t

ln(t )dt ( j ) t

0

e

ln(t )dt t j 1e t [ln(t )] 0 dt

0

0

( j ) t j 1e t ln(t )dt t j 1e t dt. 0

(3.9)

0

Note that the second integral on the right-side of (3.9) represents the celebrated gamma function

t j 1e t dt

given by

1 ( j ) . Hence (3.9) becomes j

0

j

t

( ) t e 0

where

ln(t )dt ( j ) t j 1e t ln(t )dt

1 ( j ) , j

(3.10)

0

j 1 is an integer and 0 . Combining (3.7), (3.8) and (3.10), it follows that 1 j M 1 j 1, 0; , 0 M 1 j , 0; , 0 j 1 ( j ) ,

which implies that

(3.11)

6

1 j 1 M 1 j, 0; , 0 M 1 j 1, 0; , 0 j ( j 1) ,

(3.12)

From (3.11) and (3.12), the following recurrence relation follows

1 1 j ( j 1) M 1 j 1, 0; , 0 M 1 j 1, 0; , 0 j 1 ( j 1) j 1 ( j ) . 2 Proceeding in this manner, we have

j ( j 1) 1 j ( j 1) 2 j ( j 1) 3 M 1 j 1, 0; , 0 M 1 1, 0; , 0 (1) j j 1 ( 2) j 1

j ( j 1) j 1

j 1 ( j 2) j 1 ( j 1) j 1 ( j )

1 (1) (2) ( j 1) ( j ) j! j M 1 1, 0; , 0 , 2! ( j 1)! j! 1!

(3.13)

But

M 1 1, 0; , 0 e t ln(t ) dt 1 ln( ),

(3.14)

0

where lim 1 m

1 1 1 ln(m) 0.577216 is Euler’s constant, (see, for 2 3 m

example, Gradshteyn and Ryzhik (1980, p. 573), among others). Applying (3.14) in (3.13), and simplifying, we have

j! 1 1 1 M 1 ( j 1, 0; , 0) j 1 ln( ) 1 2 3

1 j

1 1 1 1 j! j 1 ln( ) 1 . j 1 j 2 3 Noting the following properties of digamma function

1 1 1 (1) and ( j ) 1 , j 2 is an integer, j 1 2 3 and applying in (3.15), we have

(3.15)

7

j! M 1 ( j 1, 0; , 0) j 1 ( j ) ln( )

1 . j

(3.16)

Combining (3.8) and (3.16), it follows that

t j e t ln( t )dt

M 1 j 1, 0; , 0

0

1 j! j 1 ( j ) ln( ) . j

(3.17)

Consequently, in (3.17), replacing j by ( j 1) , we obtain

1 ( j 1)! . t j 1e t ln(t )dt = ( j 1) ln( ) j ( j 1)

M 1 j , 0; , 0

(3.18)

0

Noting the following properties of digamma and gamma functions

( j 1) ( j )

1 and ( j 1)! ( j ) , j

and applying in (3.18), the required expression for the integral (1.1) is obtained as follows

t j 1e t ln(t )dt

M 1 j , 0; , 0

0

( j )ln( ) ( j ) j ,

(3.19)

where j 1 is an integer, 0 , and ( j ) and ( j ) denote gamma and digamma functions.

Substituting 1 in (3.19), we get, in particular, the following integral

t j 1e t ln( t ) dt ( j) ( j ) ,

M 1 j , 0; 1, 0

0

where j 1 is an integer. Note that the integral on the left-side of (3.20) can be deduced by

differentiating both sides of the following celebrated Eulerian integral

( ) t 1e t dt , 0 0

with respect to the parameter so that

(3.20)

8

/ ( ) t 1e t ln(t )dt , 0 . 0

The derivative of ln ( ) is called digamma function or psi function, denoted by ( ) , and is

given by

( )

/ t t d ln ( ) ( ) e et dt , where is Euler’s constant. d ( ) 1 e 0

There exists a vast literature on gamma and psi functions. Historically, the gamma function was first introduced by Euler (1707 – 1783) in his studies of generalizing the factorial ! for any noninteger values of . Legendre (1752 – 1833) was first to introduce the notation ( ) and name

it as gamma function, while Gauss (1777 – 1855) used the notation ( ) , which represents

( 1) . For details on history, analysis, properties and applications of gamma and psi functions, see, for example, Sibagaki (1952), Davis (1959), Artin (1964), Srivastava and Manocha (1984), Titchmarsh (1986), Nikiforov and Uvarov (1988), Gautschi (1998), Gourdon and Sebah (2002), Chaudhry and Zubair (2002), and references therein.

4. SOME APPLICATIONS IN THE COMPUTATION OF ENTROPY In this section, applications of the integrals (3.19) and (3.20) in the computations of entropies of record values, corresponding to continuous probability distributions, namely, uniform and exponential distributions, are presented. For entropies of record values, corresponding to Weibull, Pareto, normal, gamma, beta, and Cauchy distributions, and their properties, see, for example, Shakil (2003), and Zahedi and Shakil (2006), among others.

Let X 1 , X 2 , be i. i. d. observations from an absolutely continuous distribution function F (x ) , with a probability density function f (x ) . Let R( j ) denote the time (index) at which the jth

record value is observed, and let the jth record value X R ( j ) be denoted by X ( j ) . A record

value occurs at time

j , j 1, 2 , , where j n , if X j X i , 1 i j , that is, X j is

larger than each of the previous values X 1 , X 2 , X

. The first observation is always a j 1

record value, and if a record value occurs at time t , then X t will be called a record value. The

9

joint probability density function of the record values X 1 , X 2 , X j

is given by

j 1 f (xr ) g X (1) , X ( 2) ,, X ( j ) ( x1 , x 2 , x j ) f ( x j ) , r 1 {1 F ( x )} r

(4.1)

and the marginal probability density function (pdf) of the

jth record value X ( j ) is given by

ln{1 F ( x)}j 1 f ( x) g X ( j ) ( x)

,

(4.2)

( j ) For details on (4.1) and (4.2), see, for example, Ahsanullah (2004), among others.

4.1. UNIFORM DISTRIBUTION Definition. A continuous random variable X is said to have a uniform distribution over an interval ( , ) , that is, X ~ uniform ( , ) , if its pdf f (x ) and cdf F ( x ) P ( X x ) are, respectively, given by

f ( x)

1 I ( , ) ( x) ; ( )

and

0 , x x F ( x) , x ; 1, x where I ( , ) ( x ) represents the indicator function defined over the interval ( , ) , and . In what follows, without loss of generality, for simplicity of computations, we only consider the entropy of a record value corresponding to a uniform (0 , ) , from which the

entropy of a record value corresponding to any general uniform ( , ) can easily be obtained by

a simple transformation of the parameter . Let X 1 , X 2 , be a sequence of i. i. d. random variables from a uniform parent distribution in

given by

(0 , ) with the cdf FX and pdf f X , respectively,

10

1 f X ( x ; ) I ( 0 , ) ( x),

(4.1.1)

x FX ( x ; ) I ( 0 , ) ( x ) ,

(4.1.2)

and

where I ( 0 , ) ( x) represents the indicator function defined over the interval (0 , ) .

Let

X ( j ) denote the jth observed record value. Using (4.1.1) and (4.1.2) for f (x) and F (x)

respectively in (4.2), g X ( j ) ( X ) , the pdf of

X ( j ) , is given by

j 1

x ln1 g X ( j) ( X ) ( j )

I ( 0 , ) ( x) ,

(4.1.3)

where 0 , j 1 is an integer, and 0 x .

Let the entropy of the jth record value X ( j ) from the uniform (0 , ) parent distribution be

denoted by H X ( j)

, or simply H ( j ) . Then, using (4.1.3) in the expression (1.2) for Shannon’s

entropy, we have

H ( j ) E ln g ( j ) ( x ) g ( j ) ( x ) ln g ( j ) ( x ) dx

j 1 x ln1 j 1 x ln1 ln ( j ) dx . ( j )

(4.1.4)

0

Substituting ln1

x t t , which gives dx ( e )dt , t 0 as x 0 , and t as

x , the above expression (4.1.4) for the entropy of the jth record value X ( j ) , reduces to

11

H ( j)

t j 1 t t j 1 ln e ( j ) dt ( j )

0

1 j 1 t t e ( j 1) ln(t ) ln{ ( j )}dt ( j ) 0 j 1 j 1 t ln{ ( j )} j 1 t t e ln( t ) dt t e dt , ( j ) 0 ( j) 0 from which, using Eulerian integral and the integral (3.20), the entropy of the jth record value

X ( j ) from the uniform (0 , ) parent distribution is obtained as follows H ( j ) ln( ) ln( j ) ( j 1) ( j ) , where j 1 is an integer, 0 , and ( j ) is digamma function.

4.2. EXPONENTIAL DISTRIBUTION Definition. A continuous positive random variable X is said to have an exponential distribution, with mean 0 , if its pdf f (x ) and cdf F ( x ) P ( X x ) are, respectively, given by

1 f X ( x ; ) e x / I ( 0 , ) ( x),

(4.2.1)

FX ( x ; ) 1 e x / I ( 0 , ) ( x) ,

(4.2.2)

and

where I ( 0 , ) ( x ) represents the indicator function defined on (0 , ) .

Let X 1 , X 2 , be a sequence of i. i. d. observations from an exponential ( ) parent distribution. Let X ( j ) denote the jth observed record value. Using (4.1.1) and (4.1.2) for

f (x) and F (x) respectively in (4.2), g X ( j ) ( X ) , the pdf of X ( j ) , is given by

g ( j ) ( x)

1 ln(1 FX ( x))j 1 f X ( x) ( j )

12

j 1

=

1 ln{1 (1 e x / )} ( j )

=

1 x j 1 e x / I ( 0 , ) ( x) , j 1, 2, 3, ..., 0 , 0 x . ( j )

e x / I ( 0 , ) ( x)

(4.2.3)

j

Note that

X ( j ) ~ Gamma ( shape parameter j , scale parameter ) . The entropy of a

gamma distribution is well-known, but, for the sake of completeness and to show the applications of integrals (3.19) and (3.20), we derive the entropy here. Let the entropy of the jth record

value X ( j ) from the exponential ( ) parent distribution be denoted by H X ( j)

, or simply H ( j ) .

Then, using (4.2.3) in the expression (1.2) for Shannon’s entropy, we have

H ( j)

E lng ( j)

( x) g

( x) lng ( j ) ( x) dx

( j)

x j 1 e x / x j 1 e x / ln j ( j ) j ( j )

dx

0

j ln( ) ln{( j )} j 1 x / j x / 1 dx j 1 dx x e x e j ( j ) 0 ( j ) 0 j 1 j 1 x / j ln( x) dx . x e ( j ) 0

(4.2.4)

Thus, using Eulerian integral and the integral (3.19) in (4.2.4), the entropy of the jth record

value X ( j ) from the exponential ( ) parent distribution is given by

H ( j ) ln( ) ln( j) ( j 1) ( j) j , where

j 1 is an integer, 0 , and ( j ) is digamma function.

5. CONCLUDING REMARKS In this paper, we have attempted to give a new analytical proof, with some historical background, of the definite integral

13

t j 1 e t ln(t ) dt ( j )ln( ) ( j ) j ,

(6.1)

0

where j 1 is an integer, 0 , and ( j ) and ( j ) denote gamma and digamma

functions. This integral appears to be well-known. It is useful in the studies of probability theory, entropies of record values, order statistics, numerical analysis, physics, and theory of special functions. However, even after an extensive search, we were not able to find any proof or derivation of this formula in any book or journal, or, at least, the name of the discoverer of this formula. One must not forget to acknowledge the contribution of the mathematician, who two hundred years ago, proved or came up with such a great formula, which now has such tremendous applications in entropy, information, probability and reliability theories. The accomplishments of the past and present mathematicians can serve as pathfinders to their contemporary and future colleagues. The achievements of many mathematicians, and their contributions, both small and large, have been overlooked when chronicling the history of mathematics. By describing the academic history of these personalities within mathematical sciences, we can see how the efforts of individuals have advanced human understanding in the world around us. History bears testimony to their achievements, abilities and accomplishments. It should be the responsibility of the present mathematical world to highlight the achievements of the past mathematicians behind such great formulas and theories. At this point it is clear to pose the following open problem. Though the integral (6.1) appears to be well-known, it will be interesting if we were able to find the name of its discoverer and other proofs of the integral.

REFERENCES 1. Abramowitz, M., and Stegun, I. A. (1970), Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables, Dover, New York. 2. Ahsanullah, M. (2004). Record Values – Theory and Applications, University Press of America, Lanham, MD. 3. Artin, E., The Gamma Function (1964), New York, Holt, Rinehart and Winston. 4. Bierens de Haan, D. (1862), "Expos´e de la th´eorie, des propri´et´es, des formulas de transformation et des m´ethodes d'´evaluation des int´egrales d´efinies," C.G. Van der Post,

14

Amsterdam. 5. Bierens de Haan, D. (1867), Nouvelles tables d'int´egrales d´efinies, P. Engels, Leiden. 6. Bierens de Haan, D. (1957), Nouvelles tables d'int´egrales d´efinies, Hafner, New York. 7.Chaudhry, M. A., and Zubair, S. M. (2002), On a class of Incomplete Gamma Functions with Applications, Chapman & Hall/CRC, Boca Raton. 8.Chaudhry, M. A. (1994). On a Family of Logarithmic and Exponential Integrals occurring in Probability and Reliability Theory, J. Austral. Math. Soc., Ser. B, 35, pp. 469 – 478. 9.Chaudhry, M. A., and Ahmad, M. (1992). On some infinite integrals involving logarithmic, exponential and powers, Proceedings of the Royal Society of Edinburgh, 122A, 11 – 15. 10.Davis, P. J. (1959), “Leonhard Euler’s Integral: A historical profile of the gamma function,” Amer. Math. Monthly, 66, 849 - 869. 11.Erd´elyi, A.(1954), Tables of Integral Transforms, Vol. 1, McGraw-Hill, New York. 12.Gautschi, W. (1998), “The incomplete gamma function since Tricomi,” in: Tricomi’s Ideas and Contemporary Applied Mathematics, Atti Convegni Lincei, 147, Accad. Naz. Lincei, Rome, 207 - 237. 13.Gourdon, X., and Sebah, P. (2002), Introduction to the Gamma Function, World Wide Web site at the address: http:// numbers.computation.free.fr/Constants/constants.html . 14.Gradshteyn, I., and Ryzhik, I. (1980), Tables of Integrals, Series and Products, Academic Press, N. Y. 15.Nikiforov, A. F., and Uvarov, V. B. (1988), Special Functions of Mathematical Physics, Birkhauser Verlag, Basel, Germany. 16.Schrek, D.J.E. (1955), “David Bierens de Haan,” Scripta Math., 21, pp. 31 – 41. 17. Shakil, M. (2003), “Entropy Study of Record Value Distributions obtained from Some Commonly Used Continuous Probability Models,” Masters’ Thesis, Florida International University, Florida, USA. 18.Sheldon, E.W. (1912), “Critical revision of de Haan's Tables of Definite Integrals,” Amer. J. Math. 34, pp. 88 - 114. 19.Sibagaki, W. (1952), Theory and applications of the gamma function, Iwanami Syoten, Tokyo, Japan. 20.Srivastava, H. M., and Manocha, H. L. (1984), A Treatise on Generating Functions, John

15

Wiley and Sons, New York. 21.Talvila, E. (2001), "Some Divergent Trigonometric Integrals," Amer. Math. Monthly 108, No. 5, 432 - 436. 22.Titchmarsh, E. C. (1986), The theory of the Riemann Zeta-function, Oxford Science publications, second edition, revised by D.R. Heath-Brown. 23. Zahedi, H. and Shakil, M. (2006). “Properties of Entropies of Record Values in Reliability and Life Testing Context.” Communications in Statistics, 35: 997–1010.

Polygon 2011

Published on Apr 8, 2014

Polygon is Hialeah Campus’ multi-disciplinary online journal, featuring the academic work of our distinguished faculty and staff.

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