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# Concepts of Probability STAT 134

Floy Kub

GPA 3.64

J. Pitman

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COURSE
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J. Pitman
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Class Notes
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123
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## Popular in Statistics

This 123 page Class Notes was uploaded by Floy Kub on Thursday October 22, 2015. The Class Notes belongs to STAT 134 at University of California - Berkeley taught by J. Pitman in Fall. Since its upload, it has received 39 views. For similar materials see /class/226726/stat-134-university-of-california-berkeley in Statistics at University of California - Berkeley.

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Date Created: 10/22/15
In rr39oduc rion To probobili ry S ra r 134 Spring 2009 JIM PITMAN Follows Jim PiTmcm39s book ProbabiliTy SecTions 1415 LecTur39es Prepared by Emilia Huer ToSanchez Many Slides ar39e borrowed from Elchanan Mossel and Yelena ShveTs Named Distributions Uniform ab Dl39srriburion for a lt x lt y lt b Ppoim in xy y xba Hw 1 113 118 136 138 1315 1 46 148 1 55 Three draws from a magic ha r Note Draws are without replacemen39r Space of possible 393 draws39 from The hot IFJill In IE1 Willa WhaT is The chance ThaT on The 2quotd draw if we goT a E on The 1Sf draw we geT an Space of possible 393 draws39 from the hat El l I IE1 IE1 Ell The event I is we get an ipod on 2nd draw and R is the first draw is a rabbit Hal PIIR12 WhaT is The chance ThaT we geT an on The 15 draw if we goT a E on The 2quotd draw Space of possible 393 draws39 from The hot Eil l I lay I y liquot The event I is we get an ipod in 1st draw and R is the 2nd draw is a rabbit an PIR12 Coum ing formula for P A B 39 For39 a fini re se r S2 of equally likely ou rcomes under39 The uniform dis rr39ibu rion and A and B r39epr39esen red as subse rs of S2 The condifiona probabiify of A given B is PA I B ABB The pr39opor39 on of ou rcomes in B Tha r are also in A Her39e AB is The in rer39sec rion of A and B The Frequency In Terpre Taron If PA approximaTes The relaTive frequency of A in a long series of Trials Then PAIB approximaTes The relaTive frequency of Trials producing A among Those Trials which happen To resuIT in B OR In a long sequence of Trials among Those which belong To B The proporTion of Those ThaT also belong To A should be abouT PAIB Le r39s make 3 draws from The magic ha r many Times I Ell LEJI I I ABB 47 z 12 For39 a uniform measure we have A B B A B PAB 9 PAB B m 3 Example Two sided cards A haT ConTains 3 cards One card is black on boTh sides One card is whiTe on boTh sides One card is black on one side and whiTe on The oTher39 The cards are mixed up in The haT Then a single card is drawn and placed on The Table If The visible side 0d The card is black whaT is The chance ThaT The oTher39 side is whiTe Conditional probability in general The condi rional probabili ry of A given B is deno red by PA I B If is given by PA I B PAB PB Example Rich amp Famous In a cer39fain Town 10 of The inhabifanfs are rich 5 are famous and 3 are rich and famous If a Town39s person is chosen af random and she is rich whaf is The pr39obabilify she is famous PR 01 Example RICh 6 Famous PR amp F 003 PF I R 003O1 In a cer39fain Town 10 of g3 The i nhabifanfs are rich 39 39 5 are famous and 3 are rich and famous Thar If a Town39s person is chosen af random and she is rich whaf is The probabilify she is famous 39 39 o 39 39 I n u u I I I I I I I I u u n n n n n n n I n n n n n n n n n n n I I I I n I n n n I I I I 39 39 39 l l Example Rela rive areas A poin r is picked uniformly at random from The big r39ec rangle whose area is 1 Suppose rha r we are Told rha r rhe poin r is in B wha r is The chance Tha r if is in A Example Rela rive areas In o rher39 wor39ds Given Tha r The poin r is in B wha r is The condi rional probabili ry Tha r if is in A Consider The following exper39imenT we firsT pick one of The Two boxes and nexT we pick a ball from The boxed ThoT we picked WhoT39s The chance of geTTing o 4 Tr39ee Diagrams Podd box 1 P AV n box 12 0 O C C O l P2Iodd box Y I even box 12 12 0 C C C 0 P4 P4 and even box Peven box P4Ieven box 12 14 Consider The par39Ti rion BIBZBnQ AB1 AB2 ABn A PABI PABZPABn PA Rule of Average Conditional Probabilities If 81Bn is a disjoint Partition of Q then PA PABI PABZ PABn PA31PB1 PM BzPBzPA BnPBn The overall probability PA is the Weighted average of the conditional probabilities PAIBi with weights PB Consider The following exper39imenT we fir39sT pick one of The Two boxes and nexT we pick a ball from The boxed ThoT we picked WhoT39s The chance of geTTing a number39 less Thon 41 Independence When The probabili ry for39 A does no r depend on Whe rher39 or39 no r B occur39s we say Tha r A and B are independen r In o rher39 words A and B are independen r if PABPAIBC p No re rha r if A and B are i ndependen r Then PM PABPB PAIBCPBC PPpN90 P The above implies Tha r PABPA Independence If A is independen r of B Then A is also independen r of BC Ques rion If A is independen r of B is B independen r of A Multiplication Rule for39 Independen r evenTs PABPABPB PA PB Independence example Consider a ball picked uniformly Then Color39 RedGreen and Number39 1 or39 2 are INDEPENDENT PRedl112PRedl2 PGr39eenl112PGreenl2 Also Consider a ball picked uniformly Then Color39 RedGreen and Number39 1 or39 2 are INDEPENDENT P2 IGr39een 13P2 I Red P1IGreen23P1IRed Independence Consider a ball picked uniformly Then Color39 RedGreen and Number39 1 or39 2 are NOT INDEPENDENT P2IGr39een 13 a P2Red23 Independence Poin rs from a figure have coordino res X and Y If a poin r is picked of uniformly from a rec rongle Then The even rs X gt a and Y gt b are 9 V Independence PXgta amp Ygtb PXgta PYgtb la1b yl NIH Mu V O NIH lw H Pick a Box Then a ball O O O 0 CC 0 O Box 2 Box 3 Box 4 If a ball is drawn from a randomly picked box comes ou r To be green which box would you guess if come from and who r is The chance Tho r you are r39igh r Pick a Box Then a ball PBox 213 PBox 313 PBox 4 13 12 12 0 C 23 13 34 14 By The Rule of Average Conditional Probabilities PG Boll PG Ball I Box 2 PBox 2 PG Bolll Box 3 PBox 3 PG Ball Box 4 PBox 4 13 1323 13 2336 Given That The ball picked is green what is The Probability That it came from box i where i123 1312 2336 1323 2336 1334 2336 Pbox 2 I G Ball Pbox 2 G BaPG Ball Pbox 3 I G Ball Pbox 3 G BaPG Ball Pbox 4 I G Ball Pbox 4 G BaPG Ball BAYES39 RULE For39 a par ri rion Bl l3n of all possible ou rcomes PBi IA PABiPBi PAB1PB1 PM BzPBzPA BnPBn In rr39oduc rion To probabili ry S ra r 134 Spring 2009 Follows Jim PiTman39s book ProbabiliTy SecTions 21 LecTur39es Prepared by Emilia Huer39TaScmchez Many Slides ar39e borrowed from Elchanan Mossel and Yelena ShveTs JIM PITMAN T055 1 coin 100 Times wha r39s The chance of 60 Heads Assump rions n independence probabilities are fixed Each Time we pull an iTem ouT of The haT iT magically reappears sampling wiTh replacemenT WhaT39s The chance of drawing 3 I pods in 10 Trials l lL j i l 7 c l WI M l 3 l A 39l H ea x m 7 gr m H m l H h 2 I 7 39 39 law I39mk f A y v f 7 4n w l in 39 f in w39 quotH 39 w39 quotH 39 X v Am i f l r QM I A 339 X QM iv z x 1 W V Aquot 1 l d 1327 P3 in 10 deS of sequences wi rh 3 A E g Suppose we roll a die 4 Times Wha r39s The chance of k I 9 Le r l mll kO Suppose we roll a die 4 Times Wha r39s The chance of k I 9 Le r l mll k1 Hill at nil Elli IIII Suppose we roll a die 4 Times Wha r39s The chance of k I 9 Le r EEII k2 un nn all hill In In 11 Ill I Suppose we roll a die 4 Times Wha r39s The chance of k I 9 Le r EEII k3 Inna all II I quot IIII Suppose we roll a die 4 Times Wha r39s The chance of k I 9 Le r l mll k4 Suppose we roll a die 4 Times Wha r39s The chance of k I 9 Le r EEII How do we coun r The number of sequences of leng rh 4 wi rh 3 and 1 9 A gt This is The Pascal39s Triangle which gives as you may recall The binomial coefficien rs 3 8 nmcw wquot u Ln 3 3gtgtlt me 8 8 2 n ms 3 9 8 mg n 3 3 lt e 2 quot 33 Newton s Binomial Theorem n k 40 J Binomial Distribution For39 n independen r rr39iols each wi rh probability p of success and 1p of failure we have Psuccessesk pk1p 39k This defines rhe binomianp dis rr39ibu rion over39 The se r of n1 in reger39s O1n Binomial Distribution Pklt1Pgt quot ltp 110quot This r39epr39esen rs The chance Tha r in n draws There was some number of successes be rween zero and n A pair of coins will be Tossed 5 Times Find The probabili ry of ge r ring I on k of The Tosses k 0 To 5 I binomial514 A pair of coins will be Tossed 5 Times Find The probabiliTy of geTTing on k of The Tosses k 0 To 5 I 5 k W To fill ouT The disTr39ibuTion Table we could compuTe 6 quanTiTies for39 k 015 separaTely or39 use 1 Trick Consecu rive odds ratio r39ela res Pk and Pk l We can use This Table To find The following condiTional probabiliTy GT leasT 1 PaT leasT 3 l in firsT 2 Tosses P3 or more I amp10r 2 in firsT 2 Tosses P1 or39 in firsT 2 bi n514 CD 237 How useful is The binomial formula Try using your calculaTors To compu re P500 H in 1000 coin Tosses diPZClei 1000 1 P500 m 1000 1000 500 500 2 Your cacuafor may re Turn error when compufing 1000 This number is jusf Too big To be sfored The following is called the Stirling s approximation nlssZ 1m 2 I f is not very use fu if appfed directy nquot is a very big number if n is 1000 Question For39 a fair coin wi rh p wha r do we expec r in 100 Tosses So we expect abou r 50 H ExpecTed value or39 Mean 11 of a binomialnp dis rr39ibu rion u Trials Psuccess Z n p Ques rion Wha r is The mos r liker number39 of successes Recall Tha r P50in1oo 1 O797884561 d50n To see whe rher39 This is The mos r likely number of successes we need To compare This To Pk in 100 for every other k The mos r likely number39 of successes is called The mode of a binomial distribution If we can Show Tha r for39 some m Pl lt sPm1 s Pm gt Pm1 gt Pn Then m would be The mode In rr39oduc rion To probobili ry S ra r 134 Spring 2009 JIM PITMAN Follows Jim PiTmcm39s book ProbabiliTy SecTions 1415 LecTur39es Prepared by Emilia Huer ToSanchez Many Slides ar39e borrowed from Elchanan Mossel and Yelena ShveTs Named Distributions Uniform ab Dl39srriburion for a lt x lt y lt b Ppoim in xy y xba Hw 1 113 118 136 138 1315 1 46 148 1 55 Three draws from a magic ha r Note Draws are without replacemen39r Space of possible 393 draws39 from The hot IFJill In IE1 Willa WhaT is The chance ThaT on The 2quotd draw if we goT a E on The 1Sf draw we geT an Space of possible 393 draws39 from the hat El l I IE1 IE1 Ell The event I is we get an ipod on 2nd draw and R is the first draw is a rabbit Hal PIIR12 WhaT is The chance ThaT we geT an on The 15 draw if we goT a E on The 2quotd draw Space of possible 393 draws39 from The hot Eil l I lay I y liquot The event I is we get an ipod in 1st draw and R is the 2nd draw is a rabbit an PIR12 Coum ing formula for P A B 39 For39 a fini re se r S2 of equally likely ou rcomes under39 The uniform dis rr39ibu rion and A and B r39epr39esen red as subse rs of S2 The condifiona probabiify of A given B is PA I B ABB The pr39opor39 on of ou rcomes in B Tha r are also in A Her39e AB is The in rer39sec rion of A and B The Frequency In Terpre Taron If PA approximaTes The relaTive frequency of A in a long series of Trials Then PAIB approximaTes The relaTive frequency of Trials producing A among Those Trials which happen To resuIT in B OR In a long sequence of Trials among Those which belong To B The proporTion of Those ThaT also belong To A should be abouT PAIB Le r39s make 3 draws from The magic ha r many Times I Ell LEJI I I ABB 47 z 12 For39 a uniform measure we have A B B A B PAB 9 PAB B m 3 Example Two sided cards A haT ConTains 3 cards One card is black on boTh sides One card is whiTe on boTh sides One card is black on one side and whiTe on The oTher39 The cards are mixed up in The haT Then a single card is drawn and placed on The Table If The visible side 0d The card is black whaT is The chance ThaT The oTher39 side is whiTe Conditional probability in general The condi rional probabili ry of A given B is deno red by PA I B If is given by PA I B PAB PB Example Rich amp Famous In a cer39fain Town 10 of The inhabifanfs are rich 5 are famous and 3 are rich and famous If a Town39s person is chosen af random and she is rich whaf is The pr39obabilify she is famous PR 01 Example RICh 6 Famous PR amp F 003 PF I R 003O1 In a cer39fain Town 10 of g3 The i nhabifanfs are rich 39 39 5 are famous and 3 are rich and famous Thar If a Town39s person is chosen af random and she is rich whaf is The probabilify she is famous 39 39 o 39 39 I n u u I I I I I I I I u u n n n n n n n I n n n n n n n n n n n I I I I n I n n n I I I I 39 39 39 l l Example Rela rive areas A poin r is picked uniformly at random from The big r39ec rangle whose area is 1 Suppose rha r we are Told rha r rhe poin r is in B wha r is The chance Tha r if is in A Example Rela rive areas In o rher39 wor39ds Given Tha r The poin r is in B wha r is The condi rional probabili ry Tha r if is in A Consider The following exper39imenT we firsT pick one of The Two boxes and nexT we pick a ball from The boxed ThoT we picked WhoT39s The chance of geTTing o 4 Tr39ee Diagrams Podd box 1 P AV n box 12 0 O C C O l P2Iodd box Y I even box 12 12 0 C C C 0 P4 P4 and even box Peven box P4Ieven box 12 14 Consider The par39Ti rion BIBZBnQ AB1 AB2 ABn A PABI PABZPABn PA Rule of Average Conditional Probabilities If 81Bn is a disjoint Partition of Q then PA PABI PABZ PABn PA31PB1 PM BzPBzPA BnPBn The overall probability PA is the Weighted average of the conditional probabilities PAIBi with weights PB Consider The following exper39imenT we fir39sT pick one of The Two boxes and nexT we pick a ball from The boxed ThoT we picked WhoT39s The chance of geTTing a number39 less Thon 41 Independence When The probabili ry for39 A does no r depend on Whe rher39 or39 no r B occur39s we say Tha r A and B are independen r In o rher39 words A and B are independen r if PABPAIBC p No re rha r if A and B are i ndependen r Then PM PABPB PAIBCPBC PPpN90 P The above implies Tha r PABPA Independence If A is independen r of B Then A is also independen r of BC Ques rion If A is independen r of B is B independen r of A Multiplication Rule for39 Independen r evenTs PABPABPB PA PB Independence example Consider a ball picked uniformly Then Color39 RedGreen and Number39 1 or39 2 are INDEPENDENT PRedl112PRedl2 PGr39eenl112PGreenl2 Also Consider a ball picked uniformly Then Color39 RedGreen and Number39 1 or39 2 are INDEPENDENT P2 IGr39een 13P2 I Red P1IGreen23P1IRed Independence Consider a ball picked uniformly Then Color39 RedGreen and Number39 1 or39 2 are NOT INDEPENDENT P2IGr39een 13 a P2Red23 Independence Poin rs from a figure have coordino res X and Y If a poin r is picked of uniformly from a rec rongle Then The even rs X gt a and Y gt b are 9 V Independence PXgta amp Ygtb PXgta PYgtb la1b yl NIH Mu V O NIH lw H Pick a Box Then a ball O O O 0 CC 0 O Box 2 Box 3 Box 4 If a ball is drawn from a randomly picked box comes ou r To be green which box would you guess if come from and who r is The chance Tho r you are r39igh r Pick a Box Then a ball PBox 213 PBox 313 PBox 4 13 12 12 0 C 23 13 34 14 By The Rule of Average Conditional Probabilities PG Boll PG Ball I Box 2 PBox 2 PG Bolll Box 3 PBox 3 PG Ball Box 4 PBox 4 13 1323 13 2336 Given That The ball picked is green what is The Probability That it came from box i where i123 1312 2336 1323 2336 1334 2336 Pbox 2 I G Ball Pbox 2 G BaPG Ball Pbox 3 I G Ball Pbox 3 G BaPG Ball Pbox 4 I G Ball Pbox 4 G BaPG Ball BAYES39 RULE For39 a par ri rion Bl l3n of all possible ou rcomes PBi IA PABiPBi PAB1PB1 PM BzPBzPA BnPBn In rr39oduc rion To probabili ry S ra r 134 Spring 2009 Follows Jim PiTman39s book ProbabiliTy SecTions 21 LecTur39es Prepared by Emilia Huer39TaScmchez Many Slides ar39e borrowed from Elchanan Mossel and Yelena ShveTs JIM PITMAN T055 1 coin 100 Times wha r39s The chance of 60 Heads Assump rions n independence probabilities are fixed Each Time we pull an iTem ouT of The haT iT magically reappears sampling wiTh replacemenT WhaT39s The chance of drawing 3 I pods in 10 Trials l lL j i l 7 c l WI M l 3 l A 39l H ea x m 7 gr m H m l H h 2 I 7 39 39 law I39mk f A y v f 7 4n w l in 39 f in w39 quotH 39 w39 quotH 39 X v Am i f l r QM I A 339 X QM iv z x 1 W V Aquot 1 l d 1327 P3 in 10 deS of sequences wi rh 3 A E g Suppose we roll a die 4 Times Wha r39s The chance of k I 9 Le r l mll kO Suppose we roll a die 4 Times Wha r39s The chance of k I 9 Le r l mll k1 Hill at nil Elli IIII Suppose we roll a die 4 Times Wha r39s The chance of k I 9 Le r EEII k2 un nn all hill In In 11 Ill I Suppose we roll a die 4 Times Wha r39s The chance of k I 9 Le r EEII k3 Inna all II I quot IIII Suppose we roll a die 4 Times Wha r39s The chance of k I 9 Le r l mll k4 Suppose we roll a die 4 Times Wha r39s The chance of k I 9 Le r EEII How do we coun r The number of sequences of leng rh 4 wi rh 3 and 1 9 A gt This is The Pascal39s Triangle which gives as you may recall The binomial coefficien rs 3 8 nmcw wquot u Ln 3 3gtgtlt me 8 8 2 n ms 3 9 8 mg n 3 3 lt e 2 quot 33 Newton s Binomial Theorem n k 40 J Binomial Distribution For39 n independen r rr39iols each wi rh probability p of success and 1p of failure we have Psuccessesk pk1p 39k This defines rhe binomianp dis rr39ibu rion over39 The se r of n1 in reger39s O1n Binomial Distribution Pklt1Pgt quot ltp 110quot This r39epr39esen rs The chance Tha r in n draws There was some number of successes be rween zero and n A pair of coins will be Tossed 5 Times Find The probabili ry of ge r ring I on k of The Tosses k 0 To 5 I binomial514 A pair of coins will be Tossed 5 Times Find The probabiliTy of geTTing on k of The Tosses k 0 To 5 I 5 k W To fill ouT The disTr39ibuTion Table we could compuTe 6 quanTiTies for39 k 015 separaTely or39 use 1 Trick Consecu rive odds ratio r39ela res Pk and Pk l We can use This Table To find The following condiTional probabiliTy GT leasT 1 PaT leasT 3 l in firsT 2 Tosses P3 or more I amp10r 2 in firsT 2 Tosses P1 or39 in firsT 2

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