UCLA STAT 110 A Applied Probability & Statistics for Engineers

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1 UCLA STAT 0 A Alied Probabilit & Statistics for Egieers Istructor: Ivo Diov, Asst. Prof. I Statistics ad eurolog Teachig Assistat: Maria Chag, UCLA Statistics Uiversit of Califoria, Los Ageles, Srig 003 htt:// Chaters 3 Discrete Variables, Probabilities, CLT Radom Variables (RV s Probabilit Desit Fuctios (PDF s) for discrete RV s Biomial, egativebiomial, Geometric, Hergeometric, Poisso distributios Slide Slide Frequec Distributios- damaged boxes Frequec Distributios- damaged boxes Te Total Relative Percetage Frequec Frequec A - Fla out B - Fla tor C - Ed smashed D Pucture E - Glue roblem F - Corer gouge G Comr. wrile H - Ti crushed I - Tot. destructio Slide 3 Total * 00 (* the relative frequecies do ot add to.0000 due to roudig) Slide 4 Frequec Distributios- damaged boxes Relative frequec for te A is: Percetage for te A is: ercet = = 0.96 Frequec Distributios- damaged boxes The frequec distributio of a variable is ofte reseted grahicall as a bar-chart/bar-lot. For examle, the data i the frequec table above ca be show as: The usefuless of relative frequecies ad ercetages is clear: for examle, it is easil see that corer gouge accouts for 59% of the total umber of damages. Slide 5 The vertical axis ca be frequecies or relative frequecies or ercetages. O the horizotal axis all boxes should have the same width leave gas betwee the boxes (because there is o coectio betwee them) the boxes ca be i a order. Slide 6

2 Exerimets, Models, RV s A exerimet is a aturall occurrig heomeo, a scietific stud, a samlig trial or a test., i which a object (uit/subject) is selected at radom (ad/or treated at radom) to observe/measure differet outcome characteristics of the rocess the exerimet studies. Model geeralized hothetical descritio used to aalze or describe a heomeo. A radom variable is a te of measuremet tae o the outcome of a radom exerimet. Defiitios The robabilit fuctio for a discrete radom variable X gives the chace that the observed value for the rocess equals a secific outcome, x. T P(X = x) [deoted r(x) or P(x)] for ever value x that the R.V. X ca tae E.g., umber of heads whe a coi is tossed twice x 0 r(x ) 4 4 Slide 7 Slide Stoig at oe of each or 3 childre Tossig a biased coi twice Samle Sace comlete/uique descritio of the ossible outcomes from this exerimet. Outcome GGG GGB GB BG BBG BBB Probabilit 4 4 For R.V. X = umber of girls, we have X 0 3 r(x ) 5 Slide 9 For each toss, P(Head) = P(Tail) = P(com(H))=- Outcomes: HH, HT, TH, TT Probabilities:., (-), (-), (-)(-) Cout X, the umber of heads i tosses X 0 r(x ) ( ) ( ) Slide Calculatig Iterval robabilities from cumulative robabilities x-values : To get 4 to, r(3 P(3< < X <9) - ) [= start with everthig u to P(X <9) = r(x - ) ad remove from 3 dow P(X<=3) r(x - 3) How to fid the uer-tail? Beroulli Trials A Beroulli trial is a exerimet where ol two ossible outcomes are ossible (0 / ). Examles: Coi tosses Comuter chi (0 / ) sigal. Poll suorters/ooets; es/o; for/agaist. Slide Slide 3

3 The two-color ur model The biased-coi tossig model balls i a ur, of which there are M blac balls M white balls Samle balls ad cout X = # blac balls i samle We will comute the robabilit distributio of the R.V. X Slide 4 toss toss toss r(h) = r(h) = r(h) = Perform tosses ad cout X = # heads We also wat to comute the robabilit distributio of this R.V. X! Are the two-color ur ad the biased-coi models related? How do we reset the models i mathematical terms? Slide 5 The aswer is: Biomial distributio The distributio of the umber of heads i tosses of a biased coi is called the Biomial distributio. Biomial(, ) the robabilit distributio of the umber of Heads i a -toss coi exerimet, where the robabilit for Head occurrig i each trial is. E.g., Biomial(6, 0.7) x Idividual r(x = x) Cumulative r(x - x) For examle P(X=0) = P(all 6 tosses are Tails) = ( 0.7) 6 = = 0.00 Slide 6 Slide 7 Biar radom rocess The biased-coi tossig model is a hsical model for situatios which ca be characterized as a series of trials where: Teach trial has ol two outcomes: success or failure; T = P(success) is the same for ever trial; ad Ttrials are ideedet. The distributio of X = umber of successes (heads) i such trials is Biomial(, ) Samlig from a fiite oulatio Biomial Aroximatio If we tae a samle of size from a much larger oulatio (of size ) i which a roortio have a characteristic of iterest, the the distributio of X, the umber i the samle with that characteristic, is aroximatel Biomial(, ). Y (Oeratig Rule: Aroximatio is adequate if / < 0..) Examle, ollig the US oulatio to see what roortio is/has-bee married. Slide Slide 9 3

4 Biomial Probabilities the momet we all have bee waitig for! Suose X ~ Biomial(, ), the the robabilit x ( P( X = x) = ( ) x Where the biomial coefficiets are defied b! =,! = 3... ( ) x ( x)! x! -factorial Slide 0 x ), 0 x Biomial Formula with examles Does the Biomial robabilit satisf the requiremets? ( ) ( = ) = x x ( ) = ( + (- ) ) x P X x x = x Exlicit examles for =, do the case =3 at home! x ( x) ( ) = Three terms i the sum x 0 0 ( ) + ( ) + ( ) = 0 ( ) + ( ) + = ( + ( -) ) = Slide Usual quadraticexasio formula Examles Birthda Paradox The Birthda Paradox: I a radom grou of eole, what is the chage that at least two eole have the same birthda? E.x., if =3, P>0.5. Mai cofusio arises from the fact that i real life we rarel meet eole havig the same birthda as us, ad we meet more tha 3 eole. The reaso for such high robabilit is that a of the 3 eole ca comare their birthda with a other oe, ot just ou comarig our birthda to abod else s. There are -Choose- = 0*9/ was to select a air or eole. Assume there are 365 das i a ear, P(oe-articular-air-same- B-da)=/365, ad P(oe-articular-air-failure)=-/365 ~ For =0, 0-Choose- = 90. E={o eole have the same birthda is the evet all 90 airs fail (have differet birthdas)}, the P(E) = P(failure) 90 = = Hece, P(at-least-oe-success)=-0.59=0.4, quite high. ote: for =4 P>0.9 Slide Prize ($) x 3 Probabilit r(x) What we would "exect" from 00 games add across row umber of games wo $ wo Sum Total rize moe = Sum; Exected values The game of chace: cost to la:$.50; Prices {$, $, $3}, robabilities of wiig each rice are {0.6, 0.3, 0.}, resectivel. Should we la the game? What are our chaces of wiig/loosig? Average rize moe = Sum/00 = =.5 Theoreticall Fair Game: rice to la EQ the exected retur! Slide 3 TABLE 5.4. Average Wiigs from a Game coducted times umber Prize wo i dollars(x ) of games 3 Average wiigs laed frequecies er game ( ) (Relative frequecies) (.64) (.5) (.).7, (.573) (.36) (.).53 0, (.5995) (.305) (.099) , (.5959) (.3040) (.00) , (.59) (.306) (.00).500 (.6) (.3) (.).5 (x) So far we looed at the theoretical exectatio of the game. ow we simulate the game o a comuter to obtai radom samles from our distributio, accordig to the robabilities {0.6, 0.3, 0.}. Defiitio of the exected value, i geeral. The exected value: E(X) = x all x P( x) = x P( x) dx all X = Sum of (value times robabilit of value) Slide 4 Slide 5 4

5 Examle I the at least oe of each or at most 3 childre examle, where X ={umber of Girls} we have: X 0 3 r(x ) 5 Slide 6 E( X ) = x P( x) x 5 = =.5 The exected value ad oulatio mea µ X = E(X) is called the mea of the distributio of X. µ X = E(X) is usuall called the oulatio mea. µx is the oit where the bar grah of P(X = x) balaces. Slide 7 Poulatio stadard deviatio For the Biomial distributio... mea E(X) =, sd( X) = (- ) The oulatio stadard deviatio is sd( X) = E[(X - µ) ] ote that if X is a RV, the (X-µ) is also a RV, ad so is (X-µ). Hece, the exectatio, E[(X-µ) ], maes sese. X~Biomial(, ) X=Y +Y +Y Y, where Y ~Beroulli(), E(Y )= E(X) = E(Y +Y +Y Y )= Slide 9 Slide 30 Biomial ad Multiomial Distributios Multiomial Distributio T ossible outcomes (, E ) TEach outcome has robabilit i ( + +, = ) TI ideedet trials, X + X + + X = where X i = # of times that E i occurs. x x x f ( x,, x ;,,, ) = x, x,, x with i= x i =, i = i= Biomial ad Multiomial Distributios Ex. Suose we have 9 eole arrivig at a meetig. P(b Air) = 0.4, P(b Bus) = 0. P(b Automobile) = 0.3, P(b Trai) = 0. P(3 b Air, 3 b Bus, b Auto, b Trai) =? P( b air) =? Margial distributio of X i : Bi(, i ) Slide 35 Slide 36 5

6 Liear Scalig (affie trasformatios) ax + b Liear Scalig (affie trasformatios) ax + b For a costats a ad b, the exectatio of the RV ax + b is equal to the sum of the roduct of a ad the exectatio of the RV X ad the costat b. E(aX + b) = a E(X) +b Ad similarl for the stadard deviatio (b, a additive factor, does ot affect the SD). SD(aX +b) = a SD(X) Slide 37 Wh is that so? E(aX + b) = a E(X) +b SD(aX +b) = a SD(X) E(aX + b) = (a x + b) P(X = x) = a x P(X = x) + b P(X = x) = a x P(X = x) + b P(X = x) = ae(x) + b = ae(x) + b. Slide 3 Liear Scalig (affie trasformatios) ax + b Liear Scalig (affie trasformatios) ax + b Examle: E(aX + b) = a E(X) +b SD(aX +b) = a SD(X). X={-,, 0, 3, 4, 0, -, }; P(X=x)=/, for each x. Y = X-5 = {-7, -, -5,, 3, -5, -9, -3} 3. E(X)= 4. E(Y)= 5. Does E(X) = E(X) 5? 6. Comute SD(X), SD(Y). Does SD(Y) = SD(X)? Slide 39 Ad wh do we care? E(aX + b) = a E(X) +b SD(aX +b) = a SD(X) -comletel geeral strateg for comutig the distributios of RV s which are obtaied from other RV s with ow distributio. E.g., X~(0,), ad Y=aX+b, the we eed ot calculate the mea ad the SD of Y. We ow from the above formulas that E(Y) = b ad SD(Y) = a. -These formulas hold for all distributios, ot ol for Biomial ad ormal. Slide 40 Liear Scalig (affie trasformatios) ax + b Meas ad Variaces for (i)deedet Variables! Ad wh do we care? E(aX + b) = a E(X) +b SD(aX +b) = a SD(X) -E.g., sa the rules for the game of chace we saw before chage ad the ew a-off is as follows: {$0, $.50, $3}, with robabilities of {0.6, 0.3, 0.}, as before. What is the ewl exected retur of the game? Remember the old exectatio was equal to the etrace fee of $.50, ad the game was fair! Y = 3(X-)/ {$, $, $3} {$0, $.50, $3}, E(Y) = 3/ E(X) 3/ = 3 / 4 = $0.75 Ad the game became clearl biased. ote how eas it is to comute E(Y). Slide 4 Meas: T Ideedet/Deedet Variables {X, X, X3,, X0} Y E(X + X + X3 + + X0) = E(X)+ E(X)+ E(X3)+ + E(X0) Variaces: T Ideedet Variables {X, X, X3,, X0}, variaces add-u Var(X +X + X3 + + X0) = Var(X)+Var(X)+Var(X3)+ +Var(X) T Deedet Variables {X, X} Variace cotiget o the variable deedeces, Y E.g., If X = X + 5, Var(X +X) =Var (X + X +5) = Var(3X +5) =Var(3X) = 9Var(X) Slide 4 6

7 For the Biomial distributio... SD E(X) = Slide 43 SD( X) = (- ) X~Biomial(, ) X=Y +Y +Y 3 + +Y, where Y ~ Beroulli(), Var(Y ) = (-) x + (0-) x(-) Var(Y ) = (-)(- + ) = (-) Var(X) = Var(Y ) + + Var(Y ) = (-) SD(X)=Sqrt[Var(X)] = Sqrt[(-)] Samle vs. theoretical mea & varaice The Exected value: (oulatio mea) E( X ) = x P( x) = x P( x) dx all x all x Samle mea X = x = (Theoretical) Variace Var( X ) = ( x µ ) P( x) = ( ) x µ P( x) dx X X all x all x (Samle) variace Var( X ) = = = ( x X ) = ( x X ) P( x) Slide 44 Poisso Distributio Defiitio Fuctioal Brai Imagig Positro Emissio Tomograh (PET) Used to model couts umber of arrivals () o a give iterval The Poisso distributio is also sometimes referred to as the distributio of rare evets. Examles of Poisso distributed variables are umber of accidets er erso, umber of sweestaes wo er erso, or the umber of catastrohic defects foud i a roductio rocess. Slide 45 Slide 46 Fuctioal Brai Imagig - Positro Emissio Tomograh (PET) Fuctioal Brai Imagig Positro Emissio Tomograh (PET) Isotoe Eerg (MeV) Rage(mm) /-life Al. C mi recetors 5O.7.5 mi stroe/activatio F mi eurolog 4I ~ das ocolog Slide 47 htt:// Slide 4 7

8 Fuctioal Brai Imagig Positro Emissio Tomograh (PET) Poisso Distributio Mea Used to model couts umber of arrivals () o a give iterval e Y~Poisso( ), the P(Y=) =, = 0,,,! Mea of Y, µ Y =, sice e E( Y ) = = e = e =!! ( )! = e = 0 = = e ( )! = 0 = 0 = e! = e = Slide 49 Slide 5 Poisso Distributio - Variace Y~Poisso( ), the P(Y=) = e Variace of Y, σ Y = ½!, sice σ = = Y Var( Y ) = 0 ( ) e!, = 0,,, For examle, suose that Y deotes the umber of bloced shots (arrivals) i a radoml samled game for the UCLA Bruis me's basetball team. The a Poisso distributio with mea=4 ma be used to model Y. =... = Poisso Distributio - Examle For examle, suose that Y deotes the umber of bloced shots i a radoml samled game for the UCLA Bruis me's basetball team. Poisso distributio with mea=4 ma be used to model Y Slide 5 Slide 53 Poisso as a aroximatio to Biomial Suose we have a sequece of Biomial(, ) models, with lim( ), as ifiit. For each 0<=<=, if Y ~ Biomial(, ), the T P(Y =)= ( ) T But this coverges to: e ( ) WHY?! Thus, Biomial(, ) Poisso() Poisso as a aroximatio to Biomial Rule of thumb is that aroximatio is good if: T >=00 T <=0.0 T = <=0 The, Biomial(, ) Poisso() Slide 54 Slide 55

9 Examle usig Poisso arox to Biomial Suose P(defective chi) = 0.000=0-4. Fid the robabilit that a lot of 5,000 chis has > defective! Y~ Biomial(5,000, 0.000), fid P(Y>). ote that Z~Poisso( = =5,000 x 0.000=.5) P( Z > ) = P( Z ) = 0.5 e 0! e!.5 Slide 56 z = e! z.5 e z!.5 = = ormal aroximatio to Biomial Suose Y~Biomial(, ) The Y=Y + Y + Y Y, where T Y ~Beroulli(), E(Y )= & Var(Y )=(-) T E(Y)= & Var(Y)=(-), SD(Y)= ((-)) / T Stadardize Y: Y Z=(Y-) / ((-)) / Y B CLT Z ~ (0, ). So, Y ~ [, ((-)) / ] ormal Arox to Biomial is reasoable whe >=0 & (-)>0 ( & (-) are OT too small relative to ). Slide 57 ormal aroximatio to Biomial Examle Roulette wheel ivestigatio: Comute P(Y>=5), where Y~Biomial(00, 0.47) T The roortio of the Biomial(00, 0.47) oulatio havig more tha 5 reds (successes) out of 00 roulette sis (trials). T Sice =47>=0 & (-)=53>0 ormal arox is justified. Z=(Y-)/Sqrt((-)) = Roulette has 3 slots red blac eutral 5 00*0.47)/Sqrt(00*0.47*0.53)=. P(Y>=5) P(Z>=.) = True P(Y>=5) = 0.77, usig SOCR (demo!) Biomial arox useful whe o access to SOCR avail. ormal aroximatio to Poisso Let X ~Poisso() & X ~Poisso(µ) X + X ~Poisso(+µ +µ) Let X, X, X 3,, X ~ Poisso(), ad ideedet, Y = X + X + + X ~ Poisso(), E(Y )=Var(Y )=. The radom variables i the sum o the right are ideedet ad each has the Poisso distributio with arameter. B CLT the distributio of the stadardized variable (Y ) / () / (0, ), as icreases to ifiit. So, for >= 00, Z = {(Y ) / () / } ~ (0,). Y ~ (, () / ). Slide 5 Slide 59 ormal aroximatio to Poisso examle Let X ~Poisso() & X ~Poisso(µ) X + X ~Poisso(+µ +µ) Let X, X, X 3,, X 00 ~ Poisso(), ad ideedet, Y = X + X + + X ~ Poisso(400), E(Y )=Var(Y )=400. B CLT the distributio of the stadardized variable (Y 400) / (400) / (0, ), as icreases to ifiit. Z = (Y 400) / 0 ~ (0,) Y ~ (400, 400). P( < Y < 400) = (std z & 400) = P( ( 400)/0 < Z < ( )/0 ) = P( -0< Z <0) = 0.5 Poisso or ormal aroximatio to Biomial? Poisso Aroximatio (Biomial(, ) Poisso() ): ( ) e WHY?! T>=00 & <=0.0 & = <=0 ormal Aroximatio (Biomial(, ) (, ((-)) / ) ) T >=0 & (-)>0 Slide 60 Slide 6 9

10 Geometric, Hergeometric, egative Biomial X ~ Geometric(), the the robabilit mass fuctio is Probabilit of first failure at x th trial. x P( X = x) = ( ) ; E( X ) = ; Var( X ) = Ex: Stat det urchases 40 light bulbs; 5 are defective. Select 5 comoets at radom. Fid: P(3 rd bulb used is the first that does ot wor) =? Slide 6 Geometric, Hergeometric, egative Biomial Hergeometric X~HerGeom(, M,) Total objects:. Successes: M. Samle-size: (without relacemet). X = umber of Successes i samle M M M E( X ) = x P ( X = x) = M M Var( X ) = Ex: 40 comoets i a lot; 3 comoets are defectives. Select 5 comoets at radom. P(obtai oe defective) = P(X=) =? Slide 63 Hergeometric Distributio & Biomial Biomial aroximatio = to Herheometric T is small (usuall < 0.), the aroaches HerGeom( x;,, ) Bi( x;, ) Ex: 4,000 out of 0,000 residets are agaist a ew tax. 5 residets are selected at radom. P(at most 7 favor the ew tax) =? Geometric, Hergeometric, egative Biomial egative biomial mf [X ~ egbi(r, ), if r= Geometric ()] x P( X = x) = ( ) umber of failures util the r th success (egative, sice umber of successes (r) is fixed & umber of trials (X) is radom) x + r r x P( X = x) = ( ) r r( ) r( ) E( X ) = ; Var( X ) = Slide 64 Slide 65 Logormal (Y) µ,σ Relatio amog Distributios ormal (X) µ,σ X = ly X Y = e Uiform(X) α, β α U = X β α Beta α, β α = β = µ Z = X σ χ Uiform(U) 0, = = X = ( β α) U + α i Z i ormal (Z) 0, Chi-square ( χ ) α = /, β = Gamma α, β = α = X = β lu Weibull γ, β γ = Exoetial(X) β Slide 66 0