Chapter 4: Multiple Random Variables - NTPU

1 Chapter 4: Multiple Random Variables1 Yunghsiang S. HanGraduate Institute of Communication Engineering,National Taipei UniversityTaiwanE-mail: from the lecture notes by Prof. Mao-Ching ChiuY. S. HanMultiple Random Vector Random VariablesConsider the two dimensional Random variableX= (X, Y). Find the regions of the planes correspondingto the eventsA={X+Y 10},B={min(X, Y) 5}andC={X2+Y2 100}.Graduate Institute of Communication Engineering, National Taipei UniversityY. S. HanMultiple Random Variables2 Graduate Institute of Communication Engineering, National Taipei UniversityY.

2 S. HanMultiple Random Variables3 Let then-dimensional Random variableXbeX= (X1, X2, .. , Xn) andAkbe a one dimensionalevent that involvesXk. Events withproduct formis defined asA={X1 A1} {X2 A2} {Xn An}.P[A] =P[{X1 A1} {X2 A2} {Xn An}] =P[X1 A1, .. , Xn An]. Some events may not be of product Institute of Communication Engineering, National Taipei UniversityY. S. HanMultiple Random Variables4 Some two-dimensional product form eventsGraduate Institute of Communication Engineering, National Taipei UniversityY. S. HanMultiple Random Variables5 probability of non-product-form event Bis partitioned into disjoint product-form events suchasB1, B2.

3 , Bn,andP[B] P"[kBk#=XkP[Bk]. Approximation becomes exact asBk s become Institute of Communication Engineering, National Taipei UniversityY. S. HanMultiple Random Variables6 Non-product-form eventsGraduate Institute of Communication Engineering, National Taipei UniversityY. S. HanMultiple Random Variables7 Independence Two Random variablesXandYare independent ifP[X A1, Y A2] =P[X A1]P[Y A2]. Random variablesX1, X2, .. , Xnare independent ifP[X1 A1, .. , Xn An] =P[X1 A1] P[Xn An].Graduate Institute of Communication Engineering, National Taipei UniversityY.]

4 S. HanMultiple Random Pairs of Random VariablesPairs of Discrete Random Variables Random variableX= (X, Y) Sample spaceS={(xj, yk) :j= 1,2, .. , k= 1,2, ..}is countable. Joint probability mass function (pmf)ofXispX,Y(xj, yk)=P[{X=xj} {Y=yk}] =P[X=xj, Y=yk]j= 1,2, ..k= 1,2, ..Graduate Institute of Communication Engineering, National Taipei UniversityY. S. HanMultiple Random Variables9 probability of eventAisP[X A] =X(xj,yk) ApX,Y(xj, yk). Marginal probability mass functionispX(xj) =P[X=xj]=P[X=xj, Y= anything]=P[{X=xjandY=y1} {X=xjandY=y2} ]= Xk=1pX,Y(xj, yk).

5 Graduate Institute of Communication Engineering, National Taipei UniversityY. S. HanMultiple Random Variables10 SimilarlypY(yk) = Xj=1pX,Y(xj, yk).Graduate Institute of Communication Engineering, National Taipei UniversityY. S. HanMultiple Random Variables11 Joint cdf ofXandY Joint cumulative distribution function ofXandYisgiven asFX,Y(x1, y1) =P[X x1, Y y1]Graduate Institute of Communication Engineering, National Taipei UniversityY. S. HanMultiple Random Variables12 Graduate Institute of Communication Engineering, National Taipei UniversityY.

6 S. HanMultiple Random ,Y(x1, y1) FX,Y(x2, y2), ifx1 x2andy1 ,Y( , y1) =FX,Y(x1, ) = ,Y( , ) = (x) =FX,Y(x, ) =P[X x, Y < ] =P[X x];FY(y) =FX,Y( , y) =P[Y y].5. continuous from the rightlimx a+FX,Y(x, y) =FX,Y(a, y)limy b+FX,Y(x, y) =FX,Y(x, b)Graduate Institute of Communication Engineering, National Taipei UniversityY. S. HanMultiple Random Variables14 Graduate Institute of Communication Engineering, National Taipei UniversityY. S. HanMultiple Random Variables15 Example:Joint cdf ofX= (X, Y) is given asFX,Y(x, y) =((1 e x)(1 e y)x 0, y the marginal cdf :FX(x) = limy FX,Y(x, y) = 1 e xx (y) = limx FX,Y(x, y) = 1 e yy Institute of Communication Engineering, National Taipei UniversityY.)

7 S. HanMultiple Random Variables16 probability of regionB={x1< X < x2, Y y1}FX,Y(x2, y1) =FX,Y(x1, y1) +P[x1< X < x2, Y y1] P[x1< X < x2, Y y1] =FX,Y(x2, y1) FX,Y(x1, y1) probability of regionA={x1< X x2, y1< Y y2}FX,Y(x2, y2) =P[x1< X x2, y1< Y y2]+FX,Y(x2, y1) +FX,Y(x1, y2) FX,Y(x1, y1)P[x1< X x2, y1< Y y2]=FX,Y(x2, y2) FX,Y(x2, y1) FX,Y(x1, y2) +FX,Y(x1, y1)Graduate Institute of Communication Engineering, National Taipei UniversityY. S. HanMultiple Random Variables17 Graduate Institute of Communication Engineering, National Taipei UniversityY.

8 S. HanMultiple Random Variables18 Joint pdf of Two Jointly continuous RandomVariables Random variableX= (X, Y) Joint probability density functionfX,Y(x, y) is definedsuch that for every eventAP[X A] =Z ZAfX,Y(x , y )dx dy .Graduate Institute of Communication Engineering, National Taipei UniversityY. S. HanMultiple Random Variables19 Graduate Institute of Communication Engineering, National Taipei UniversityY. S. HanMultiple Random =Z+ Z+ fX,Y(x , y )dx dy .2. FX,Y(x, y) =Zy Zx fX,Y(x , y )dx dy .3. fX,Y(x, y) = 2FX,Y(x, y) x P[a1< X b1, a2< Y b2] =Zb2a2Zb1a1fX,Y(x , y )dx dy.

9 Pdf sfX(x) =ddxFX(x) =ddxFX,Y(x, )Graduate Institute of Communication Engineering, National Taipei UniversityY. S. HanMultiple Random Variables21=ddxZx Z+ fX,Y(x , y )dy dx =Z+ fX,Y(x, y )dy .6. fY(y) =Z+ fX,Y(x , y)dx .Graduate Institute of Communication Engineering, National Taipei UniversityY. S. HanMultiple Random Variables22 Graduate Institute of Communication Engineering, National Taipei UniversityY. S. HanMultiple Random Variables23 Example: Let the pdf ofX= (X, Y) befX,Y(x, y) =(1 0 x 1 and 0 y 10 the joint : Consider five <0 ory <0,FX,Y(x, y) = 0;2.)

10 (x, y) unit interval,FX,Y(x, y) =Ry0Rx01dx dy =xy;3. 0 x 1 andy >1,FX,Y(x, y) =R10Rx01dx dy =x; >1 and 0 y 1,FX,Y(x, y) =y; >1 andy >1,FX,Y(x, y) =R10R101dx dy = Institute of Communication Engineering, National Taipei UniversityY. S. HanMultiple Random Variables24 Graduate Institute of Communication Engineering, National Taipei UniversityY. S. HanMultiple Random Variables25 Example: Random variablesXandYare jointlyGaussianfX,Y(x, y) =12 p1 2e (x2 2 xy+y2)/2(1 2) < x, y < .Find the marginal pdf Institute of Communication Engineering, National Taipei UniversityY.