Concept of Random Variables

Transcription

1 Concep of andom Variables DOMAIN S is he enire ais ANE S is he posiive ais. Hence iven wo ses of numbers S and S S hen o every we assign a number () belonging o S S The generalizaion ( ) iven wo ses of objecs S ζ and S η consising of he elemens ζ S η S ζ η Then η is a funcion of ζ ; η η( ζ ) S. If every elemen of ζ corresponds o an elemen of S η ( S ζ - DOMAIN S - ANE η )

2 { X } he even - se of alloucomes ζ of our eperimen ha makes X lesshan or equal o { X } is he se of all oucomes ζ ha makes our random variable X inally { X } - is he se of all oucomes such ha X Noe { X } { X } { X } Eample; olling a die wih f f f ζ he evens if we define X ( f ) ; X ( f ) ; X ( f6 ) 6 Our random variable is X ( f i ) X i { X 35} { f f f 3} { X 5} { X 35} { f f 3} { X 4} { f 4} { X 35} Ses or evens can be defined as follows:

3 "A real random variable is a real funcion whose domain is S i.e. a resul of he process of assigning a real number o a random variable for every oucome of he eperimen and such ha Discree and Coninuous Probabiliy Disribuion uncion (PD) The disribuion funcion of he random variable. X is he funcion X ( ) P{ X } defined for any number from o +.

4 Eample Tossing of a coin S { h } P{ h} p P{ } q Define ζ h X ( ζ ) ζ ind he disribuion funcion. If hen { X } - cerain even { h } since X ( h ) X( ) X () () P{X } P{h } If < hen{ X } { } sincex ( h) > X ( ) < ( ) P{ X } P{ } q If < X hen { X} sincexh ( ) X ( ) ( )

5 Hence q ( ) X ( ) < < for for for q Eample { } T P Define ( ) oherwise { } { } ( ) { } T X P X no in he inerval < ( ) χ T

6 Define ( ) elsewhere beween for all -T X { } ( ) { } { } { } ( ) in for any S T X T X T { } ( ) { } { } ( ) T X X T < in for any ( ) < > T T T ( ) χ T

7 Properies of Disribuion uncions of andom Variables a) ( ) ; ( + ) b) if < { X } c) + ( ) d) Also if e) If < { X } { X } P{ X } { } P X ( ) ( ) coninuous i.e. ε from he righ + ( ) lim ( + ε ) ( ) ε { X χ} { X } + { < X } P{ X } P{ X } + P{ < X } P{ < X } P{ X } P{ X } ( ) ( ) P{ ε < X < χ} ( ) ( ε ) Leing ε P X { } ( ) ( ) if { } is disconinuous for { X } is jump a he poin P { X }.

8 f) Since { } { } { } { } ( ) ( ) ( ) ( ) { } ( ) ( ) + + < X P X P X X X ( ) ( ) T ( ) ( ) T T

9 The Probabiliy Densiy uncion (pdf) f X ( ) d X d ( ) ( + χ) ( ) lim Coninuous Type Variables a) b) c) Discree and Coninuous Probabiliy uncions d χ d χ f P ( ) ( ) d ( ) ( ) ( ) f ( ) ( ) f ( ) ( ) ( ) f ( ) { X } f ( ) d) for sufficien ly small P X f f ( ) χ χ d probabili y funcion d { X + } f ( ) P{ X } since χ χ d d

10 Probabiliy uncions of Discree and Coninuous andom Variables ) P ) 3) all s in S P ( s) ( s) P ( A + B) P( A) + P( B) or P muually eclusive ( A) + P( B) P( AB) no muually eclusive or Eample we have had several eamples of discree cases. Consider he following: for any S s : s 34 we define a probabiliy funcion as: oucome in a sample space defined by { } S P( s) k. 3 We can find k from aiom # above: s k k k 3 3 s { 4/ 3} k 3/ 4 3

11 Coninuous probabiliy funcion f S ( ) f ( ) d for all S f().875 Consider he funcion defined by he following Hence f ( ) k ( ) d k k ( ) ( + ) k.3 3. d k And if we wan he probabily of demise over age 6 P( > 6) z 6 P( > 6) k ( ) d k[ + ]

12 Eample; oll a pair of dice (red and green). Le A{odd on red} B{odd on green} C{sum is odd} Show ha any pair of hese evens is independen bu AB and C are no muually independen. Our sample space looks like he following elemens) ( S { } ( ) { } ( ) { } ( ) / / / C P C B P B A P A { } ( ) { } ( ) { } ( ) { } { } eclusive. muually NOT are evens e ha hes see we Hence 8 / 4 / 9 odd sum odd 4 / 9 odd sum odd / / 4 / odd odd ABC P ABC BC P BC AC P AC AB P AB

13 epeaed Independen Trials h Le { ose his safely in he i game } i 44 A i Then P { ose 44 } P { A A A44 } P( A ) P( A ) P( A3 ) 44 { P( A )} where all P( )' s are equal i A i To ge a value of P(A i ) consider he complemen of even A. P ( Ai ) P{ does no hi in ame i} P{ four ous } 4 { } P{ ose his in 44 consequiv e games } (.76 ). 57

14 Alernae soluion: Consider he possible oucomes of he eperimen. P ( A ) ( 3) i 4 4 OOOH OOHO OHOH HOHH OHOO HOOO HHOO HHHO HOHO 3 3 (.3)(.7) + 6(.3) (.7) + 4(.3) (.7) + (.3) (.9) + 6(.44) + 4(.89) OOHH HOOH OHHO And (.7599) OHHH HHOH HHHH 4.8 4

15 Discree Type andom Variables ( ) ( i ) i P { X } P ( ) ( ) and i i i called he probabiliy funcion i P i Probabiliy Disribuion uncion is i i n ( ) P[ X ] P{ X } i such ha. i i i i i

16 Dela Impulse uncions δ ( χ ) is a generalized funcion such ha i.e. Φ ( ) δ ( ) d Φ( ) Φ() Φ( ) where Φ ( χ ) any funcion coninuous a he origin. Shifing he origin Φ ( ) δ ( ) d Φ( ) Φ() Φ( ) o

17 δ ( ) or δ oherwise ( ) If ( ) sep funcion U ( ) oherwise δ du d ( ) where U ( ) if > U ( χ ) ( ) d( ) d δ ( ) and

18 If we had ( χ ) ( χ ) f() ( ) ( ) Hence because ( ) [ ( ) ( )] ( ) d f ( i ) i i i d δ We can hen say densiy funcion consiss of impulses f ( ) P δ i ( i ) a he poins i Noe i The difference P P { < X } f ( ) d { X } f ( )d χ + χ χ + + χ + i

19 In he coin ossing problem ( χ ) f ( χ ) q q p ( χ ) qu ( ) + pu ( ) > In he coninuous eample - > f ( χ ) qδ ( χ ) + pδ ( χ ) χ

20 In he coninuous eample ( χ) f ( χ ) T p T T χ ( ) χ ( ) f χ for < T elsewhere for < < T T elsewhere < T

21

22 Normal Densiy f ( ) Disribuions and Densiies ( χ η ) σ e σ π Assume η and σ 5 { X } P P χ χ η σ ( χ ) f ( y) dy + Φ 5 9 Φ Φ 5 5 Φ () +Φ( ) { 9 X 5}. 89

23 Poisson P k a a k k! k a a a { k} e ; f ( ) e δ ( k ); ( ) e such ha k! k k k k a k! f ( ) ( ) Eample: Toal No. of poins in a given inerval () P { k} e λ ( λ ) k λ - Poisson - disribued k wih papameer a k!

24 Binomial { } ( ) ( ) ( ) ( ) k k n k k n k k n k k f k q p k n f q p q p k n k P + where δ ( ) χ f n9 pq/

25 Uniform ( ) elsewhere f ( ) f ( )

26 amma f b cχ ( χ ) Aχ e U ( χ ) where for χ U ( χ ) is sep funcion for χ < b > c > - arbirary parameers A is such ha A b e c χ dχ

27 Negaive Eponenial c f ( ) Ae U( ) Bea f ( χ ) Aχ b c ( χ ) χ Γ( b + c + ) where A Γ( b + ) Γ( c + ) elsewhere

28 LaPlace f α α χ ( ) e Cauchy f / π α + ( )

29 ayleigh f α α ( ) e U( ) Mawell f ( ) e U( ) α 3 χ α π

30 Eample: Design of a all ower for wind loads. Maimum wind velociy near he sie is modeled by a coninuous probabiliy disribuion of he negaive eponenial form. ( ) ke f λ χ ( ) e f k k e k d ke λ λ λ χ λ λ λ The PD is found by inegraion ( ) ( ) -e e du e d f u u λ λ λ λ

31 Eample: Considering a square bay a by a in size. a a ( ) P[ X ] ( a ) shaded area oal area a a The densiy funcion f ( ) ( ) ( a ) d d a a a Noe a f() () a a a

32 Eample Hence: f P [ 7] λ.33 ( ).33e.33 ( ) e.33.9 e λ7 7λ ln. 7λ.3 ().5 () () f() ( ) ( ) e.33 probabili y of finding he ma. wind in any year greaer h an velociy.3. P[ 35<X<7]

33 Applicaions of Disribuion Laws Consider he occurrence of an even in a ime period a-b. /b-a a b a b elsewhere b a ) ( a b f < < a b a a ) ( a b a ) ( ) ( ) ( ] [ a b X Var a ab b d f X E b a + +

34 Eample: Le X delivery ime uniformly disribued over he inerval o 5 days. Cos C of he failure is C Co + C X. Prob[delivery akes o 5 days] ¼(5-) ¾ / 4 p() The Epeced coss are; 5 elsewhere E[ C] C o + C 5 + /(5 ) + ( ) 3 E[Coss] Co + C( ) 3 E[ X ] Var( X ) + m 3/ 3

35 Eample Consider he life-lengh X of a baery... f ( ) e /

36 amma funcion. z n Γ( n) e d Properies of he amma uncion:. Γ( n) ( n ) Γ( n ). Γ( n) ( n )! for n an ineger 3 Γ (/ ) π Hence 4. n e / θ E[ X ] d Γ( n) θ θ Γ() θ θ Γ() θ θ z / θ E[ X ] ) e d θ 3 Γ( 3) θ θ θ Var[ X] θ n f ( ) d e / θ d Inegraion by subsiuion y /θ ( θ ) e / θ d

37 Summarizing; e ( ) /θ < Eample A refinery has 3 processing plans. The amoun of sugar any one plan can process can be modeled as eponenial wih a mean value of 4 ons. find he probabiliy ha eacly of he 3 plans process han 4 ons. Prob[eacly use more han 4 ons] H I K J (. ) (. ). Considering a paricular plan how much sugar should be socked for ha plan so ha he probabiliy of running ou of sugar is only.5? z / 4 P[X>a] Prob[X is greaer han he amoun o be socked ] e d a 4 e a/4 Hence a.98 ons

38 elaionship beween Poisson and Eponenial- Assume a Poisson law wih a mean rae of occurrence λ. Le ime ill he firs even And Pr ob [ X > ] P[ Y in () ( λ) e λ λ e! Prob[ ] ( ) e X λ f ( ) e e X This is an Eponenial disribuion wih θ λ λ θ θ > λ. ailure ae uncion r(). f ( ) r( ) > () < ( )

39 or eponenial case / θ e r( ) + e θ / θ > θ

40 amma Probabiliy Disribuion: α f ( ) e / β Γ( α) β α where a and b are parameers z α / β As defined Γ( α) e d. Hence z z and Transformaion y β α / β α / β α α y α e d ( βy) e dy β y e dy β Γ( α) z α f ( ) β Γ( α) d α β Γ( α) z

41 Typical amma Densiies.

42 Eample Daa of 6 week summer rainfall oals.

43 z α / β E[ X ] f ( ) d e d α Γ( α) β Γ( α ) β z α α e / β d α + Γ( α + ) β αβ α Γ( α ) β Similarly E[ X ] α( α +) β Also if Var[ X ] αβ Y n i X E[ X ] nαβ Var[ X ] nαβ i z

44 Normal or aussian or Bell Disribuion f ( ) ep{ ( µ ) πα σ

45 Eample A machine auomaically fills 6 oz boles. There is some variaion in he amoun of liquid dispensed ino each bole. The average amoun disapensed was 6 oz. wih a sandard deviaion sof oz. ind: Prob[machine will dispense more han 7 oz in any one bole] z ( 6 ) P[ X > 7] e d 7 π P[ µ 7 µ 7 6 > ] P[ Z > ] P[ Z > ] 587. σ σ Anoher machine operaes has a dial seing of "amoun of liquid" wih a sandard deviaion s equal o. oz ind he proper seing so ha 7 oz will overflow only 5% of he ime. P[ X > 7]. 5 P µ 7 µ [ > ] σ σ 7 µ P[ Z > ]. 5 σ P[ Z > z ]. 5 zo µ. µ 56. seing o

46 Bea Probabiliy Disribuion uncion Γ( α + β) α β f ( ) ( ) Γ( α) Γ( β) Wih; z α β ( ) d elsewhere Γ( α) Γ( β) Γ( α + β) urher Γ( α + β ) E[ X ] f ( ) d Γ( α) Γ( β) Var[ X ] ( α αβ + β + ))( α + β ) α ( ) β d α α + β

47

48 Eample A gasoline bulk sorage anks hold a fied sup[ply. The anks are filled every Monday. Of ineres o he wholesaler is he proporion of his supply sold during he week Bea disribuion wih α 4 and β. ind he epeced value of his proporion. Is i highly likely ha he wholesaler will sell a leas 9% of he sock in a given week? Soluion: E[ X ] α α + β or he second par z z. 9 Γ( 4 + ) 3 P( X >. 9) ( ) d. 9 Γ( 4) Γ( ) 3 4 ( ) d (. 4). 8 Is no very likely ha 9% of he sock will be sold in a given week.

49 Weibull Probabiliy Disribuion γ γ f ( ) e θ γ θ > o elsewhere z γ γ θ ( ) e d θ e γ θ γ γ θ - e >

50 Transformaion YX γ. Hence f Y ( y) e θ y / θ y > E[X] for an X having he Weibull disribuion. Then γ / γ ( ) P( Y y) P( X y) P( X y ) γ ( y ) e X / γ γ / θ / ( y ) e E[ X[ E[ Y θ θ / γ y y θ / γ Γ( + ] e γ y θ ) θ y dy (+ / γ γ ) y > θ e θ y θ / γ dy Γ( + γ )

51 Daa Concerning Life Lenghs of Baeries.

52 Weibull Disribuion Commonly used as a model for life lenghs. / / ) ( ) ( ) ( γ θ θ γ θ γ θ γ γ λ λ e e f Noe for γ > his is a monoonically increasing funcion wih no upper bound..

53 Eample The lengh of service ime during which a cerain ype of hermiser produces resisance s wihin is specificaions has been observes o follow a Weibull disribuion wih θ 5 and γ (measuremens in hiusands of hours). (a) ind he probabiliy ha one of hese hermisers o be insalled in a sysem oday will funcion properly for over hours. (b)ind he epeced life lengh for hermisers of his ype. Soluion: (a) The Weibull disribuion has a closed form epression for (). Thus represens he life lengh of he hermiser in quesion. ( ) / 5 ( ) / 5 P( X > ) ( ) [ e ] e e. 4 Because θ 5 and γ E[ X ] θ γ (5) Γ( + ) Γ( ) (5) b) We know ha Thus he average service ime for hese hermisers is 67 hours. / ( γ ) (5) / Γ( 3 / ) (5) ( ) π / Γ( 6.7 )

54 Eamples of Disribuions for Discree andom Variables Bernoulli Disribuion. Le X denoe he condiion of he iem and Prob[defec] p Then: X wih probabiliy (-p) X wih probabily p The probabiliy funcion is p( ) p ( p) and p() denoies he probabily X This X is said o have a Bernoulli disribuion or o represen he oucom of a single Bernoulli rial E[ X ] p( ) p( ) + p( ) p E[ X ] ( ) p( ) ( ) p( ) p Var( X ) E[ X ] E [ X ] p p p( p) pq

55 Binomial Disribuion If we now consider inspecing n iems hen if; a) The eperimen consiss of n independen rials. b) Each rial can resul in one of only possible oucomes (success or failure). c) Probabiliy of success p is consan for each rial d) Trials are independen e) Y is defined as he number of successes in n rials Then P( Y y) p( y) y p ( p ) H n I K J Is he binomial disribuion y n y y 3... n

56 Eample Suppose a lo of fuses conains % defecives. If four fuses are randomly sampled from he lo find he probabiliy ha eacly one fuse is defecive. ind he probabiliy ha a leas fuse is defecive. a) p( ) (. ) (. ). H 4 I K J b) or P[Y ]p()+p()+p(3)+p(4)-p()-(.9).3496

57 eomeric Probabiliy Disribuion If we are ineresed in he occurrence of he firs success his is differen from he Binomial disribuion which is concerned wih he number of successes. Le Y denoe he number of rials o he firs success. Then Eample: p( s success) ( - p) y- p ecruiing firm finds 3% of applicans for a cerain job have advanced raining. If applicans are inerviewed a random find he probabioy ha he firs applican o have advanced raining is found on he 5 h inerview. Soluion: If he applican pool is large he probabily offinding a suiable applican will be relaively consan. Y is defined as he number of roials on which firs applican having advanced raining is found hen 4 p( Y 5) (.3) (.3).7

58 Negaive Binomial Disribuion Our ineress now is he number of rials for he nd success or 3 rd or 4 h ec. In hese cases we ge a Negaive Binomial disribuion as follow Le Y number of rials o he rh success P[Yy]P [( s (y-) rials conain (r-) successes and he y h rial a success] P[( s (y-) rials conain (r-)successes]p[y h rial a success] H y KJ r p y r ( p) p r Eample: If he problem consider previously is coninued o be considered suppose hree jobs ha require advanced raining are opened. ind he probabioy ha he hird qualified applican is found on he 5 h inerview. In his case r 3 and y p ( y 5) (.3) (.7) 3 I Simply a Binomial Probabiliy.79

59 Poisson Probabiliy Disribuion The Poisson disribuion was derived as one of he limiing cases of he binomial disribuion. Now le us spli up he inerval ino n sub-inervals wih P[one acciden in a sub-inerval] p P[no acciden in a sub-inerval] -p Assumming. ) Tha p is he same for all inervals ) The probabily of more han one acciden per inerval is zero 3) Occurrences of acciden per inerval are independen Then he oal number of accidens (successes) in inerval (n sub-inervals)is binomially disribued!! We assume ha as n increases p should decrease. (n -> and p -> ). We furher add resricion ha he mean np in he binomial case remains consan a λ (np λ) H I K J H I K J I H K J I + H K J n y I H K J n y y n n λ λ λ λ n( n )( n )...( n y ) λ lim lim n n! y y n n y n n n y λ y! lim λi λ y n H n K J I H n K J I n H n K J n H H I K J y I K J

60 Since n lim ( and all oher erms approach zero y λ λ p( y) y! E]Y] Var[Y] λ λ ) λ e n e

61 Eample A cerain manufacuring indusry has 3 accidens per week. ind he probabiliies ha here are no accidens in a given week four accidens less han four more han four. P[Y] P[ no accidens in a given week] 3 3 e. 5! 4 y P[Y4] y! e. y 3 y 3 P[Y 4] P[ Y 4 ] y y P[y4]

62 Hyper-geomeric Probabiliy Disribuion Assume a siuaion where he rials are dependen on wha ook place previously. or eample we have a relaively small lo consising of N iems of which k are defecive. Then if iems are drawn sequenially he oucome of he nd draw is influenced by wha happened on he s draw. Leing Y oal number of successes among n iems sampled of which k aare defecive. Then k N k H I yk J H n ykj P[ Y y] ; N H n I KJ I y k N y n N Eample: Personnel direcor selecs wo employees for a cerain job from 6 employees of which one is female and 5 are males. Wha is he probabiliy ha a woman is seleced? P[woman is seleced] P[Y] H I 5 K J H I K J 6 H I K J 3