1

Problems

1.1

Events. Probability Space

1. Indicate which of the following statements is true and defend your opinions. (a) A relative frequency is not necessarily a probability. (b) Two sample points (basic outcomes) may or may not be mutually exclusive. (c) Two events may or may not be mutually exclusive. (d) For two events, A and B, with P (B) > 0, the conditional probability of A given B is defined as P (A/B) = P P(A∩B) . (A) 2. Indicate which of the following statements is true and defend your opinions. (a) P (E1 /E2 ) is always equal to P (E1 ). (b) P (E1 /E2 ) is never equal to P (E1 ). (c) P (A/B) is always equal to P (B/A). (d) P (A)P (B/A) is always equal to P (B)P (A/B). 3. Numbers from 1 to n are randomly arranged. What is the probability that numbers 1, 2, 3 come up in this order? What is the probability that numbers k, k+1, k+2 will come in this order (k is a fixed natural number from 1 to n − 2)? 4. (The Matching Problem) A box contains n balls labeled from 1 to n. Successive extractions are done without replacement, until the box is empty. Find the probability to obtain at least one match - that is, the ball labeled i is obtained at extraction i. 5. A box contains 100 numbered balls: 1, 2, ..., 100. 10 balls are drawn without replacement from the box. What is the probability that at least one ball has an even number? 6. There are five married couples in a room. Suppose that for one dance, we randomly pair up each of the men with one of the women. What is the probability that every husband and wife will be matched correctly? What is the probability that at least one of the pairs will be an actual husband-wife pair? 7. An electronic assembly consists of two subsystems, say A and B. Suppose we have the following information: P (B f ails) = 0.5, P (A and B f ail) = 0.3 and P (A f ails, but B doesn0 t f ail) = 0.1. Find the probability that B fails, given that A fails. 8. Let (Ω, K, P ) a probability field and A, B ∈ K such that P (A ∩ B) = 0.01, P (A ∩ B) = 0.03, P (A ∩ B) = 0.05. Calculate P (A) + P (A ∪ B) + P (B/A).

9. Let (Ω, K, P ) a probability field and A, B, C ∈ K such that P (A) = 31 , P (B) = 41 , 1 1 P (C) = 15 , P (A ∩ B) = 16 , P (A ∩ C) = 18 , P (B ∩ C) = 10 and P (A ∩ B ∩ C) = 12 . Calculate P (C/A ∩ B). 10. Consider (Ω, K, P ) a probability field and A1 , A2 , A3 events in K such that A1 ∩ A3 = Ø and P (A1 ∩ A2 ) = P (A2 ∩ A3 ) = 0.2, P (A1 ∪ A2 ∪ A3 ) = 0.8, P (A1 ) = P (A3 ) = 0.3. Find P (A2 ∩ A1 ∪ A3 ) and P (A2 /A3 ). Are A1 , A2 independent events? Justify your answer. Are the events A2 and A3 mutually exclusive? Why? 11. The same problem as the previous, with: P (A1 ∩ A2 ) = P (A2 ∩ A3 ) = 0.1, P (A1 ∪ A2 ∪ A3 ) = 0.9, P (A1 ) = P (A3 ) = 0.2. 12. Let {Ω, K,P } be a probability field and A, B ∈ K, with 0 < P (A) < 1, 0 < 1 P (B) < 1. Find P (A) and P (B) if P (A/B) = 25 , P (A/B) = 10 and P (B/A) = 53 . 13. How many times one needs to roll a die, to get the face ’6’ at least once with a probability greater than 0.7? 14. The letters m, m, a, a, e, t, h, h, o, r, r are written each one on separate small sheets of paper. Six sheets of paper are randomly extracted, one by one. Find the probability to obtain the word 0 mother0 . 15. The Sebastien Cabinet Company, manufacturer of high-priced wood cabinets, employs 50 skilled cabinetmakers, 26 of whom have received formal trade school and apprenticeship training and 24 of whom are self-taught. The production manager categorized these craftsman according to type of training received and according to quality of work, 1 being the best possible rating. The bivariate frequency distribution is shown in the table below. Table 1: Quality Rating of Work T ype of training 1 (B1 ) 2 (B2 ) 3 (B3 ) 4 (B4 ) Total T rade school and apprenticeship (A1 ) 6 11 7 2 26 Self − taugh (A2 ) 3 6 10 5 24 T otal 9 17 17 7 50

What is the probability that a cabinet maker picked at random will (a) Be self-taught? (b) Do work classified as quality-level 1? (c) Do work classified as quality-level 1 given that he or she is self-taught? (d) Do work classified as quality-level 3 or 4 given that he or she is self-taught? (e) Be self-taught given he or she achieved a quality level rating of 2? (f) Be self-taught or do work classified as quality-level 3? 2

(g) Do work classified as quality-level 1 and be self-taught? (h) Be formally trained and do work classified as either quality-level 2 or qualitylevel 3? (i) Do work classified as quality-level 3 given that he or she is either formally trained or self-taught? 16. Refer to table above. (a) Does P (A1 /B1 ) = P (B1 /A1 ) ? (b) Does P (A1 âˆŠ B1 ) taken directly from the table equal P (A1 )P (B1 ) ? What does your answer convey about the independence, or lack of it, of events A1 and B1 ? (c) Are events A1 and B2 independent? Explain. (d) Are events A1 and B3 independent? Explain. (e) Are events A2 and B2 independent? Explain. 17. In a particular town, 20 percent of people buy the morning newspaper, 30 percent buy the evening newspaper, and 10 percent buy both. What is the probability that a person from this town buys at least one of the two newspapers? 18. A recent customer taste test of the top three soft drinks yielded the following results: Category P ref erence (percentage) Coca Cola 25 Pepsi 25 7UP 8 Coca Cola and Pepsi 12 Coca Cola and 7UP 5 Pepsi and 7UP 4 Coca Cola and Pepsi and 7UP 3 What is the probability of a participant selected at random preferring Coca Cola or Pepsi or 7UP? 19. Consider an experiment of flipping a fair coin three times. (a) What is the probability of obtaining two heads? (b) What is the probability of obtaining no heads?

3

20. Two students take a Mathematics exam. The probabilities to pass the exam are: 0.6 for the first student and 0.8 for the second one. Calculate the probabilities of the following events: (a) at least one student passes the exam; (b) only the first student passes the exam; (c) only one student passes the exam; (d) no student passes the exam. 21. Consider three boxes A, B, C containing black and white balls. The probabilities to extract one white ball from these boxes are pA = 0.7, pB = 0.9, pC = 0.6. One box is chosen at random and a ball is drawn. (a) Determine the probability that the extracted ball is white. (b) Find the probability to extract a black ball under the hypothesis that the considered boxes have the following probabilities to be chosen: 0.5, 0.4, and 0.1. (c) In case a), find the probability that the extracted ball comes from box A, given that the drawn ball is white. 22. Consider two boxes A and B as follows: A - 3 red and 3 black balls, B - 4 red and 6 black balls. A box is selected at random and a ball is extracted, also at random. If the selected ball is red find the probability that it came from box A. 23. Boxes of sweets contain toffees and chocolates. Box A contains 6 toffees and 4 chocolates, box B contains 5 toffees and 3 chocolates and box C contains 3 toffees and 7 chocolates. One of the boxes is chosen at random and two sweets are picked and eaten. (a) Find the probability that they are both toffees. (b) Given that they are both toffees, find the probability that they both came from box A. 24. Two balls are extracted with replacement out of a box with 2 white and 3 black balls. What are the chances that the extracted balls are of different colors? 25. Two balls are extracted without replacement out of a box with 2 white and 3 black balls. What are the chances to get balls of the same color? 26. 6% of Type A spark plugs are defective, 4% of Type B spark plugs are defective, and 2% of Type C spark plugs are defective. A spark plug is selected at random from a batch of spark plugs containing 50 Type A plugs, 30 Type B plugs, and 20 Type C plugs. The selected plug is found to be defective. What is the probability that the selected plug was of Type A?

4

27. When Don plays tennis, 65% of his first serves are correct. If the first serve is correct, his chances of winning the point are 90%. If his first serve is not correct, Don is allowed a second serve, and of these, 80% are good. If the second serve is a good one, his chances of winning the point are 60%. If neither serve is correct, Don loses the point. (a) Find the probability that Don loses the point. (b) Find the conditional probability that Don’s first serve was correct, given that he lost the point. 28. The chances of the Los Angeles Lakers winning at home are 70%, whereas the chances of winning on the road are 50%. The Lakers are scheduled to play two home games followed by two road games in the coming weeks. (a) What is the probability of the Lakers winning all four games? (b) What is the probability of the Lakers winning three out of four games? (c) What are the chances of the Lakers losing all four games? 29. 25 percent of students at a large university smoke cigarettes. Consider a group of 12 randomly selected students. (a) What is the probability that exactly 5 will smoke? (b) What is the probability that 5 or less will smoke? (c) What is the probability that either 4 or 5 will smoke? 30. 15 percent of students on a course are not satisfied with the textbook used. Consider a group of 10 randomly selected students. (a) What is the probability that exactly 3 will not be satisfied with the textbook? (b) What is the probability that 3 or more will not be satisfied? (c) What is the probability that 3 or less will not be satisfied? 31. The table below summarizes 60 responses to a survey question: ’Do you favor a no-smoking rule on airplanes?’ Yes No

Male 24 16

Female 14 6

(a) What is the probability that a randomly selected person answers ’yes’ ? (b) What is the probability of selecting someone from the group who answered ’yes’ given that the selected person is a male? (c) Are events of being a male and answering ’yes’ independent?

5

32. Three boxes contain black and white balls, as follows. Box 1:3 white and 5 black, Box 2: 5 white and 3 black, Box 3: 2 white and 3 black. One ball is extracted from each box, without replacement. (a) What is the probability to obtain one white ball? (b) What is the probability to obtain only black balls? (c) Suppose that after extracting one ball from each box, the three balls are reintroduced in their correspondent boxes and this operation is repeated 5 times. What is the probability to obtain exactly 3 times the following combination (2 white balls, 1 black ball)? 33. Three boxes contain black and white balls, as follows. Box 1: 3 white and 4 black, Box 2: 5 white and 10 black, Box 3: 7 white and 2 black. One ball is extracted 2 times with replacement from the first box, while from the second and third one are extracted two balls, without replacement. What is the probability to obtain the following combination: (1 white ball, 1 black ball) either from the first box or from the second and third boxes, or both? 34. Consider 3 boxes with white and black balls as follows: B1:5 white and 5 black, B2: 4 white and 6 black, B3: 4 white and 5 black. We extract, with replacement, 5 balls from each box. Find the probability to obtain the combination (2 white and 3 black) from 2 boxes and from the third, any other combination. 35. Consider 3 boxes with white and black balls as follows: B1: 5 white and 5 black, B2: 4 white and 6 black, B3: 4 white and 5 black. We extract, without replacement, 5 balls from each box. Find the probability to obtain the combination (2 white and 3 black) from 2 boxes and from the third, any other combination.

1.2

Random Variables

36. Consider the following two discrete, independent random variables: µ ¶ µ ¶ −1 0 1 −1 0 1 X: , Y : 1 a + 61 b + 13 13 2a − b 12a2 3 (a) Write the distribution table of the random variable 2XY . (b) Determine k ∈ Z such as P (X + Y = k) > 29 . 37. Consider a box containing 6 white balls and 4 black balls. Three balls are extracted at random and let X be the random variable representing the number of white balls extracted. Write the distribution table of the random variable X, assuming that the extractions are done with replacement. 38. Consider a box containing 3 while balls and 2 black balls. Three balls are extracted at random and let X be the random variable representing the number of white balls extracted. Write the distribution table of random variable X, assuming that the extractions are done without replacement. 6

39. Consider X, Y independent random variables, µ ¶ µ ¶ −1 0 1 0 1 2 3 X: , Y : 3p 2p 5p 3q 2q q 4q and FXY the cumulative distribution function of random variable XY . Compute FXY (3) − FXY (1). 40. Let

µ X:

¶ µ ¶ 2 3 5 1 4 6 , Y : 0.2 0.3 0.5 0.6 0.2 0.2

two independent discrete r.v. on the same probability field. Determine the distribution tables for X + Y , X − Y , XY and calculate E(X), E(Y ), V ar(X), V ar(3X − 5), V ar(7XY ). 41. Consider the function F : R → R, a kx2 F (x) = b

with if x ≤ 0 if 0 < x ≤ 1 , a, b, k ∈ R if x > 1

(a) Determine a, b, k ∈ R such that F is a cumulative distribution function. (b) Determine P ( 14 ≤ X ≤ 43 ). 42. Calculate the expectation and variance of (a) the Binomial random variable with parameters n and p; (b) the Geometric random variable with parameter p; (c) the Poisson random variable with parameter λ. 43. Let (X, Y ) be the bivariate discrete random vector with distribution table XY 2 3

-2 0 0.2 0.25 0.1 0.25

2 0.15 0.05

(a) Determine the marginal distributions of the random variables X, Y . Are the r.v. X, Y independent? (b) Determine the distribution of the random variable X +Y and the conditional distribution of (X + Y |X = 3). (c) Compute F(X,Y ) ( 52 , 12 ) where F(X,Y ) (x, y) is the bivariate c.d.f of the r.v (X, Y ). 44. Consider the independent r.v. X and Y with E(X) = −1, E(Y ) = 1, V ar(X) = V ar(Y ) = σ 2 . Determine V ar(XY ). 7

45. Consider the discrete r.v. X, µ ¶ n X: pn n∈N ∗ with pn = e−λ (1 − e−λ )n−1 . Calculate E(X) and V ar(X). 46. Two fair dice are rolled. Let the random variable X be the smaller of the two scores if the dice show different faces, or the common score if the dice show the same face. Write the distribution table of X and calculate E(X). 47. Consider two discrete random variables X, Y with the following distribution tables: µ ¶ µ ¶ −1 1 −1 2 X: 1 1 , Y : 2 1 2

2

3

3

and let P (X = −1, Y = −1) = λ, where λ is a real parameter. (a) Determine the distribution table of the random vector Z = (X, Y ) (depending on λ); (b) Find the correlation coefficient of X and Y ; (c) Find the value of λ for which X and Y are uncorrelated. For this value of λ, are the random variables X and Y independent? 48. Consider the discrete random variable (X,Y) with the distribution table XY -1 -2

-2 0 0.1 0.1 0.2 0.2

1 0.3 0.1

Calculate E(X), E(Y ), V ar(X + Y ) and ρ(X, Y ). Are the random variables X and Y independent? Justify your answer. 49. Consider the discrete random variables (X, Y ) with distribution table below. Calculate E(X), E(Y ), V ar(X + Y ) and ρ(X, Y ). Are the random variables X and Y independent? Justify your answer. XY -1 1

-2 0.125 0.125

8

1 5 0.25 0.125 0.25 0.125

50. Consider the discrete bivariate random variable Z = (X, Y ) given by the following table. Find x, y, a, b, c, d such that E(X) = 31 , E(Y ) = 14 and compute F(X,Y ) (0, 2) and ρ(X, Y ). XY x 1

0 a

y

7 12 3 4

c d

1 6

b 2 3

51. Consider a box containing 1 white, 2 black and 3 blue balls. We extract, without replacement, two balls and denote by X and Y the random variables representing the number of white and, respectively black balls obtained from the box. Compute ρ(X, Y ), V ar(2X + 3Y ), P (0.5 ≤ min{X, Y } < 1.5) and P (− 21 ≤ min{X, Y } < 52 ). 52. Consider a box containing 1 white and 2 black balls. We extract, with replacement, two balls and denote by X the random variable representing the number of white balls obtained. Determine the distribution of random variable X, compute ρ(X, X 2 ) and the probability that the extracted balls have the same color. 53. Consider three boxes containing white and black balls as follows: U1 : 1 white and 2 black, U2 : 2 white and 3 black and U3 : 1 white and 2 black balls. We extract without replacement, 2 balls from U1 and one ball from U2 and U3 (one from each). Denote by X the random variable representing the number of white balls obtained from U1 and by Y , the random variable representing the number of white balls obtained from U2 and U3. Find the distribution of Z = (X, Y ) and ρ(X, Y ). 54. Consider a box containing 25% white, 50% black and 25% blue balls. We extract with replacement 2 balls. Denote by X and Y the random variables representing the number of white and, respectively black balls obtained. Find the distribution of XY and ρ(X, Y ). 55. Consider two boxes, B1, B2 containing white and black balls as follows: B1: 1 white and 2 black, B2: 2 white and 2 black. We extract one ball from each box and denote by X the random variable representing the number of white balls obtained. Determine the distribution of X, compute ρ(X, X 2 ) and the probability that the extracted balls have the same color. 56. Let X be a random variable with the probability density function (pdf) f (x) = x cxe− 3 , x > 0. Find the parameter c, the cumulative distribution function F , E(X) and P (X ≤ 2/X > 0). 57. Let X a random variable having the probability density function f (x) = e−2|x| , x ∈ R. Calculate P (|X| < n). 9

58. Let X and Y two random variables defined on the same probability field. (a) If X and Y are independent random variables with V ar(X) = 3, V ar(Y ) = 7, find V ar(2X − 3Y ). (b) Repeat your calculations dropping the assumptions of independence and using instead the information that cov(X, Y ) = 1. µ ¶ n , 59. Suppose N is a discrete r.v. with Geometric distribution: N ∼ p · q n−1 where p, q ≥ 0, p + q = 1, n ∈ N∗ . Show that P (N > t + n|N > t) = P (N > n), ∀n > 0, t > 0. ½ αx(1 − x) if 0 ≤ x ≤ 1; 60. A continuous random variable X has pdf f (x) = 0 otherwise. (a) Find α, E(X) and V ar(X). (b) Find the cdf F (·) of X and calculate the probability P (1/2 < X ≤ 3/4). 61. Let X be a discrete r.v. with the following probability mass function: P (X = n) =

αn , (1+α)1+n

n ∈ N, α > 0. Find E(X) and V ar(X). ½ −λx λe if x > 0; 62. Let X ∼ Exp(λ) with f (x) = 0 otherwise. Find the cdf F (·) of X and show that P (X > s + t|X > t) = P (X > s), ∀s, t > 0. µ ¶ n 63. Suppose N is a discrete r.v. with Geometric distribution: N ∼ , p · q n−1 where p, q ≥ 0, p + q = 1, n ∈ N∗ . What is the probability that N is an even number? ½ 1 if −2 ≤ x ≤ 2; 4 64. A uniform random variable X has pdf f (x) = 0 otherwise. (a) Find the cdf F (·) of X, E(X) and V ar(X). (b) Find the pdf of Y = eX . 65. The quality of an electronic device is given by two characteristics U and V having the following probabilistic models: U = 2X + 3Y and V = 4X − Y where X, Y are independent random variables, X ∼ N (3, 2), Y ∼ Beta(10, 0.9). Find the correlation coefficient ρ(U, V ).

10