Fundamentals of probability, with stochastic processes saeed ghahramani 2024 scribd download
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Engineering Fundamentals: An Introduction to Engineering, SI Edition Saeed Moaveni
Stochastic Global Optimization Methods and Applications to Chemical, Biochemical, Pharmaceutical and Environmental Processes 1st Edition Ch. Venkateswarlu Satya Eswari Jujjavarapu
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Ihavetriedtomaintainanapproachthatismathematicallyrigorousand,atthesame time,closelymatchesthehistoricaldevelopmentofprobability.Wheneverappropriate,I includehistoricalremarks,andalsoincludediscussionsofanumberofprobabilityproblemspublishedinrecentyearsinjournalssuchas MathematicsMagazine and American MathematicalMonthly. Theseareinterestingandinstructiveproblemsthatdeservediscussioninclassrooms.
experiment,theeventof headsinthe first flipofthecoin is E = {HT, HH},andtheevent of anoddoutcome whenthedieistossedis F = {T1, T3, T5}
Example1.3 Considermeasuringthelifetimeofalightbulb.Sinceanynonnegative realnumbercanbeconsideredasthelifetimeofthelightbulb(inhours),thesamplespace is S = {x : x ≥ 0}.Inthisexperiment, E = {x : x ≥ 100} istheeventthat thelightbulb lastsatleast100hours, F = {x : x ≤ 1000} istheeventthat itlastsatmost1000hours, and G = {505.5} istheeventthat itlastsexactly505.5hours.
S = b,g,bg,gb,bb,gg,bbb,bgb,bbg,bgg,ggg,gbg,ggb,gbb .
Heretheoutcome b meansthatthechildisaboy,and g meansthatitisagirl.Theevents F = {b,bg,bb,bbb,bgb,bbg,bgg } and G = {gg,bgg,gbg,ggb} representfamilieswhere theeldestchildisaboyandfamilieswithexactlytwogirls,respectively.
Example1.5 Abuswithacapacityof34passengersstopsatastationsometimebetween11:00 A.M. and11:40 A.M. everyday.Thesamplespaceoftheexperiment,consisting ofcountingthenumberofpassengersonthebusandmeasuringthearrivaltimeofthebus, is
where i representsthenumberofpassengersand t thearrivaltimeofthebusinhoursand fractionsofhours.Thesubsetof S definedby F = (27,t):11 1 3 <t< 11 2 3 istheevent thatthebusarrivesbetween11:20 A.M. and11:40 A.M. with27passengers.
Remark1.1 Differentmanifestationsofoutcomesofanexperimentmightleadtodifferentrepresentationsforthesamplespaceofthesameexperiment.Forinstance,inExample 1.5,theoutcomethatthe busarrivesat t with i passengers isrepresentedby (i,t),where t isexpressedinhoursandfractionsofhours.Bythisrepresentation,(1.1)isthesample spaceoftheexperiment.Nowifthesameoutcomeisdenotedby (i,t),where t isthe numberofminutesafter11 A.M. thatthebusarrives,thenthesamplespacetakestheform S1 = (i,t):0 ≤ i ≤ 34, 0 ≤ t ≤ 40 .
Totheoutcomethatthe busarrivesat 11:20 A.M. with 31 passengers,in S thecorresponding pointis 31, 11 1 3 ,whilein S1 itis (31, 20).
wherewehaveassumedthatforanyintegerdollaramount a,Jayrounds a.50 to a +1.The eventofroundingoffatmost3centsinarandomchargeisgivenby
0, 0 01, 0 02, 0 03, 0 01, 0 02, 0 03
Iftheoutcomeofanexperimentbelongstoanevent E ,wesaythattheevent E has occurred.Forexample,ifwedrawtwocardsfromanordinarydeckof52cardsandobserve thatoneisaspadeandtheotheraheart,alloftheevents {sh}, {sh,dd}, {cc,dh,sh}, {hc,sh,ss,hh}, and {cc,hh,sh,dd} haveoccurredbecause sh,theoutcomeoftheexperiment,belongstoallofthem.However,noneoftheevents {dh,sc}, {dd}, {ss,hh,cc}, and {hd,hc,dc,sc,sd} hasoccurredbecause sh doesnotbelongtoanyofthem.
Inthestudyofprobabilitytheorytherelationsbetweendifferenteventsofanexperimentplayacentralrole.Intheremainderofthissectionwestudytheserelations.Inallof thefollowingdefinitionstheeventsbelongtoa fixedsamplespace S .
Subset
Equality
Intersection
Anevent E issaidtobea subset oftheevent F if,whenever E occurs, F alsooccurs.Thismeansthatallofthesamplepointsof E arecontained in F .Henceconsidering E and F solelyastwosets, E isasubsetof F intheusualset-theoreticsense:thatis, E ⊆ F
Events E and F aresaidtobe equal iftheoccurrenceof E impliesthe occurrenceof F ,andviceversa;thatis,if E ⊆ F and F ⊆ E ,hence E = F .
Aneventiscalledthe intersection oftwoevents E and F ifitoccurs onlywhenever E and F occursimultaneously.Inthelanguageofsets thiseventisdenotedby EF or E ∩ F becauseitisthesetcontaining exactlythecommonpointsof E and F .
Union
Complement
Difference
Aneventiscalledthe union oftwoevents E and F ifitoccurswhenever atleastoneofthemoccurs.Thiseventis E ∪ F sinceallofitspointsare in E or F orboth.
Aneventiscalledthe complement oftheevent E ifitonlyoccurswhenever E doesnotoccur.Thecomplementof E isdenotedby E c .
Aneventiscalledthe difference oftwoevents E and F ifitoccurswhenever E occursbut F doesnot.Thedifferenceoftheevents E and F is denotedby E F .Itisclearthat E c = S E and E F = E ∩ F c .
Certainty
Impossibility
Aneventiscalled certain ifitsoccurrenceisinevitable.Thusthesample spaceisacertainevent.
Aneventiscalled impossible ifthereiscertaintyinitsnonoccurrence. Therefore,theemptyset ∅,whichis S c ,isanimpossibleevent.
MutuallyExclusiveness
Ifthejointoccurrenceoftwoevents E and F isimpossible, wesaythat E and F are mutuallyexclusive.Thus E and F aremutually exclusiveiftheoccurrenceof E precludestheoccurrenceof F ,andvice versa.Sincetheeventrepresentingthejointoccurrenceof E and F is
EF ,theirintersection, E and F ,aremutuallyexclusiveif EF = ∅. Asetofevents {E1 ,E2,...} iscalled mutuallyexclusive ifthejoint occurrenceofanytwoofthemisimpossible,thatis,if ∀i = j , E i E j = ∅ Thus {E 1 ,E2 ,...} ismutuallyexclusiveifandonlyifeverypairofthem ismutuallyexclusive.
Theevents n i=1 E i , n i=1 E i , ∞ i=1 E i ,and ∞ i=1 E i aredefinedinawaysimilarto E 1 ∪ E 2 and E 1 ∩ E 2 .Forexample,if {E 1 ,E2,...,En } isasetofevents,by n i=1 E i we meantheeventinwhichatleastoneoftheevents E i , 1 ≤ i ≤ n,occurs.By n i=1 E i we meananeventthatoccursonlywhenalloftheevents E i , 1 ≤ i ≤ n,occur.
SometimesVenndiagramsareusedtorepresenttherelationsamongeventsofasample space.Thesamplespace S oftheexperimentisusuallyshownasalargerectangleand, inside S ,circlesorothergeometricalobjectsaredrawntoindicatetheeventsofinterest. Figure1.1presentsVenndiagramsfor EF , E ∪ F , E c ,and (E c G) ∪ F .Theshadedregions aretheindicatedevents.
Figure1.1 Venndiagramsoftheeventsspecified.
first-servedbasis.Let
E = thereareatleast fiveplaneswaitingtoland, F = thereareatmostthreeplaneswaitingtoland, H = thereareexactlytwoplaneswaitingtoland.
Then
1. E c istheeventthatatmostfourplanesarewaitingtoland.
2. F c istheeventthatatleastfourplanesarewaitingtoland.
3. E isasubsetof F c ;thatis,if E occurs,then F c occurs.Therefore, EF c = E.
4. H isasubsetof F ;thatis,if H occurs,then F occurs.Therefore, FH = H .
5. E and F aremutuallyexclusive;thatis, EF = ∅ E and H arealsomutually exclusivesince EH = ∅.
6. FH c istheeventthatthenumberofplaneswaitingtolandiszero,one,orthree.
Example1.8 ProveDeMorgan’s firstlaw:For E and F ,twoeventsofasamplespace S , (E ∪ F )c = E c F c .
Proof: Firstweshowthat (E ∪ F )c ⊆ E c F c ;thenweprovethereverseinclusion E c F c ⊆ (E ∪ F )c .Toshowthat (E ∪ F )c ⊆ E c F c ,let x beanoutcomethatbelongsto (E ∪ F )c . Then x doesnotbelongto E ∪ F ,meaningthat x isneitherin E norin F .So x belongs toboth E c and F c andhenceto E c F c .Toprovethereverseinclusion,let x ∈ E c F c .Then x ∈ E c and x ∈ F c ,implyingthat x ∈ E and x ∈ F .Therefore, x ∈ E ∪ F andthus x ∈ (E ∪ F )c
NotethatVenndiagramsareanexcellentwaytogiveintuitivejustificationforthe validityofrelationsortocreatecounterexamplesandshowinvalidityofrelations.However, theyarenotappropriatetoproverelations.Thisisbecauseofthelargenumberofcases thatmustbeconsidered(particularlyifmorethantwoeventsareinvolved).Forexample, supposethatbymeansofVenndiagrams,wewanttoprovetheidentity (EF )c = E c ∪ F c . Firstwemustdrawappropriaterepresentationsforallpossiblewaysthat E and F canbe related:casessuchas EF = ∅, EF = ∅, E = F , E = ∅, F = S ,andsoon.Thenineach particularcaseweshould findtheregionsthatrepresent (EF )c and E c ∪ F c andobserve thattheyarethesame.Evenifthesetwosetshavedifferentrepresentationsinonlyone case,theidentitywouldbefalse.
EXERCISES
A1. Adeckofsixcardsconsistsofthreeblackcardsnumbered1,2,3,andthreeredcards numbered1,2,3.First,Vanndrawsacardatrandomandwithout replacement.Then Pauldrawsacardatrandomandwithoutreplacementfromtheremainingcards.Let A betheeventthatPaul’scardhasalargernumberthanVann’scard.Let B bethe eventthatVann’scardhasalargernumberthanPaul’scard.
11. Atacertainuniversity,everyyeareightto12professorsaregrantedUniversityMerit Awards.ThisyearamongthenominatedfacultyareDrs.Jones,Smith,andBrown. Let A, B ,and C denotetheevents,respectively,thattheseprofessorswillbegiven awards.Intermsof A, B ,and C , findanexpressionfortheeventthattheawardgoes to(a)onlyDr.Jones;(b)atleastoneofthethree;(c)noneofthethree;(d)exactly twoofthem;(e)exactlyoneofthem;(f)Drs.JonesorSmithbutnotboth.
12. Provethattheevent B isimpossibleifandonlyifforeveryevent A, A =(B ∩ Ac ) ∪ (B c ∩ A).
13. Let E,F ,and G bethreeevents.Determinewhichofthefollowingstatementsare
correctandwhichareincorrect.Justifyyouranswers.
(a) (E EF ) ∪ F = E ∪ F .
(b) F c G ∪ E c G = G(F ∪ E )c
(c) (E ∪ F )c G = E c F c G.
(d) EF ∪ EG ∪ FG ⊂ E ∪ F ∪ G.
14. Inanexperiment,cardsaredrawn,one byone,atrandomandsuccessivelyfroman ordinarydeckof52cards.Let An betheeventthatnofacecardoraceappearsonthe first n 1 drawings,andthe nthdrawisanace.Intermsof An ’s, findanexpression fortheeventthatanaceappearsbeforeafacecard,(a)ifthecardsaredrawnwith replacement;(b)iftheyare drawnwithoutreplacement.
B15. ProveDeMorgan’ssecondlaw, (AB )c = Ac ∪ B c , (a)byelementwiseproof;(b)by applyingDeMorgan’s firstlawto Ac and B c
16. Let A and B betwoevents.Provethefollowingrelationsbytheelementwise method.
(a) (A AB ) ∪ B = A ∪ B .
(b) (A ∪ B ) AB = AB c ∪ Ac B
17. Let {An }∞ n=1 beasequenceofevents.Provethatforeveryevent B , (a) B ∞ i=1 Ai = ∞ i=1 BAi (b) B ∞ i=1 Ai = ∞ i=1 (B ∪ Ai ).
19. Let {A1 ,A2 ,A3 ,...} beasequenceofevents.Findanexpressionfortheeventthat infinitelymanyofthe Ai ’soccur.
20. Let {A1 ,A2 ,A3 ,...} beasequenceofeventsofasamplespace S .Findasequence {B1 ,B2 ,B3 ,...} ofmutuallyexclusiveeventssuchthatforall n ≥ 1, n i=1 Ai = n i=1 Bi
Definition(ProbabilityAxioms) Let S bethesamplespaceofarandomphenomenon.Supposethattoeachevent A of S ,anumberdenotedby P (A) isassociated with A.If P satisfiesthefollowingaxioms,thenitiscalleda probability andthenumber P (A) issaidtobethe probabilityof A
Axiom1 P (A) ≥ 0
Axiom2 P (S )=1.
Axiom3 If {A1 ,A2 ,A3 ,...} isasequenceofmutuallyexclusiveevents(i.e.,the jointoccurrenceofeverypairofthemisimpossible: Ai Aj = ∅ when i = j ),then
Notethattheaxiomsofprobabilityareasetofrulesthatmustbesatisfiedbefore S and P canbeconsideredaprobabilitymodel.
