Chapter 3 - continued Chapter 3 sections

1 Chapter 3 - continuedChapter 3 Random Variables and Discrete Continuous The Cumulative Distribution Bivariate Marginal Conditional DistributionsJust skim: multivariate Distributions (generalization of bivariate- random vectors) Functions of a Random Functions of Two or More Random VariablesSKIP: Markov ChainsSTA 611 Random Variables and Distributions1 / 16 Chapter 3 - Bivariate DistributionsBivariate discrete distributionsDef: Discrete joint distribution / joint pfLetXandYbe random variables. If there are at most countablepossible outcomes(x,y)for the pair(X,Y), we say thatXandYhaveadiscrete joint probability function (joint pf)isf(x,y) =P(X=xandY=y) =:P(X=x,Y=y) (x,y) R2As for univariate case we havef(x,y) 0 and All(x,y) R2f(x,y) =1andP((X,Y) C) = (x,y) Cf(x,y)STA 611 Random Variables and Distributions2 / 16 Chapter 3 - Bivariate DistributionsExample - Three coin tossesA fair coin is tossed three times. LetX=number of heads on the first tossY=total number of headsThe pff(x,y)can be given in a table:yx01230182818010182818 Can easily see that (x,y)f(x,y) =1 STA 611 Random Variables and Distributions3 / 16 Chapter 3 - Bivariate DistributionsBivariate continuous distributionsDef: Continuous joint distribution / joint pdfTwo random variablesXandYhave acontinuous joint distributionifthere exists a non-negative functionfsuch that for everyC R2P((X,Y) C) = Cf(x,y)dxdyThe functionfis called thejoint probability density function (joint pdf).

2 A joint pdf must satisfy:f(x,y) 0 <x< , <y< and f(x,y)dxdy=1 Mixed discrete and continuous variables: Use integrals forcontinuous dimension, and sums for discrete 611 Random Variables and Distributions4 / 16 Chapter 3 - Bivariate DistributionsExampleVerify thatf(x,y) ={8xyif 0<y<x<10otherwiseis a joint pdfSTA 611 Random Variables and Distributions5 / 16 Chapter 3 - Bivariate DistributionsBivariate cumulative distribution functionDef: Joint cumulative distribution functionThejoint cumulative distribution function (joint cdf)of two randomvariablesXandYisF(x,y) =P(X x,Y y) (x,y) R2 Relationship between joint cdf s and joint pdf s:Continuous:F(x,y) = y x f(r,s)drdsandf(x,y) = 2F(x,y) x y= 2F(x,y) y xDiscrete:F(x,y) = r x s yf(r,s)STA 611 Random Variables and Distributions6 / 16 Chapter 3 - Bivariate DistributionsExampleFind the joint cdf for the following joint pdff(x,y) ={8xyif 0<y<x<10otherwiseSTA 611 Random Variables and Distributions7 / 16 Chapter 3 - Marginal DistributionsMarginal distributions - discrete random variablesTheoremLet(X,Y)be a discrete random vector with joint pffX,Y(x,y), then themarginal pfsofXandYare given byfX(x) =P(X=x) = y Rf(x,y)andfY(y) =P(Y=y) = x Rf(x,y)Example: Find the marginal distributions for the coin toss exampleSTA 611 Random Variables and Distributions8 / 16 Chapter 3 - Marginal DistributionsMarginal distributions - continuous random variablesTheoremLet(X,Y)be a continuous random vector with joint pdffX,Y(x,y), thenthemarginal pdfsofXandYare given byfX(x) = f(x,y)dyfor <x< andfY(y) = f(x,y)dxfor <y< Example.}}

3 Find the marginal distributions forf(x,y) =8xyfor 0<y<x<1 STA 611 Random Variables and Distributions9 / 16 Chapter 3 - Marginal DistributionsIndependenceIndependence for random variables is defined in the same way as foreventsDef: Independent random variablesTwo random variables areindependentif for every two setsAandBinRthe events{s:X(s) A}and{s:Y(s) B}are independenteventsTheoremRandom variablesXandYare independent if and only ifFX,Y(x,y) =FX(x)FY(y)STA 611 Random Variables and Distributions10 / 16 Chapter 3 - Marginal DistributionsIndependenceThe following holds for both discrete and continuous random variables:TheoremTwo random variablesXandYwith joint pf/pdff(x,y)and marginalpf s/pdf sfX(x)andfY(y)are independent if and only iff(x,y) =fX(x)fY(y)for ALL(x,y) R2 Examples: Are the following random variables independent?1 XandYin the tossing coin example2 XandYwith joint pdff(x,y) =6xy2for 0<y<1 and 0<x<13 XandYwith joint pdff(x,y) =8xyfor 0<y<x<1 STA 611 Random Variables and Distributions11 / 16 Chapter 3 - Marginal DistributionsIndependenceA helpful theoremTheoremLetXandYbe random variables with joint pf/pdff(x,y)and supportthat is a rectangleRinR2(possibly unbounded).

4 ThenXandYare independent if and only iffcan be written asf(x,y) =h1(x)h2(y)for all(x,y) RSTA 611 Random Variables and Distributions12 / 16 Chapter 3 - Conditional DistributionsConditional distributionsDef: Conditional distributionLetXandYbe random variables with joint pf/pdff(x,y). LetfY(y)bethe marginal pf/pdf ofYand letybe a value such thatfY(y)>0. Thentheconditional pf/pdf of X given that Y=yis defined asf(x|y) =f(x,y)fY(y)Note that in the continuous case we are conditioning onsomething that has probability 0. We need to show that thecontinuous case off(x|y)is indeed a pdfExamples: Find the conditional pf/pdf:X|Y=2 from the tossing coin exampleX|Y=ywhere the joint pdf isf(x,y) =8xyfor 0<y<x<1 STA 611 Random Variables and Distributions13 / 16 Chapter 3 - Conditional DistributionsIndependence and conditional distributionsTheoremRandom variablesXandYare independent if and only iff(x|y) =fX(x)We have the law of total probability for random variables ( in the book)We also have Bayes theorem for random variables ( in the book)STA 611 Random Variables and Distributions14 / 16 Chapter 3 - multivariate DistributionsMultivariate Distributions - extension of bivariateRandom vector:X= (X1,X2.)

5 ,Xn)Joint cdf:F(x) =F(x1,x2,..,xn) =P(X1 x1,X2 x2,..,Xn xn)Discrete joint pf:f(x) =f(x1,x2,..,xn)=P(X1=x1,X2=x2,..,Xn=xn) =P(X=x)Continuous joint pf:f(x) =f(x1,x2,..,xn) = nF(x1,x2,..,xn) x1 xnP(X C) = C f(x1,x2,..,xn)dx1 dxnSTA 611 Random Variables and Distributions15 / 16 Chapter 3 - multivariate DistributionsMultivariate Distributions - extension of bivariateMarginal pdf - integrate out all the others, :f1(x1) = f(x1,x2,..,xn)dx2 dxnX1,..,Xnareindependentif for every setA1,..,AninRP(X1 A1,..,Xn An) =P(X1 A1) P(Xn An)X1,..,Xnare independent if and only ifF(x1,x2,..,xn) =F1(x1) F2(x2) Fn(xn)X1,..,Xnare independent if and only iff(x1,x2,..,xn) =f1(x1) f2(x2) fn(xn)Conditional pdfsf(x|y) =f(x1,x2,..,xn|y1,y2,..,yk)=f(x1,x2,..,x n,y1,y2,..,yk)fY(y1,y2,..,yk)=f(x,y)fY(y )STA 611 Random Variables and Distributions16 / 16