Chapter 2 Multivariate Distributions and Transformations

1 Chapter 2 Multivariate Distributionsand Joint, Marginal and Conditional Distri-butionsOften there arenrandom variablesY1,..,Ynthat are of interest. For exam-ple,age, blood pressure, weight, genderandcholesterol levelmight be someof the random variables of interest for patients suffering from heart <nbe then dimensional Euclidean space. Then thevectory= (y1,..,yn) <nifyiis an arbitrary real number fori= 1,.., ,..,Ynare discrete random variables, then thejointpmf(probability mass function) ofY1,..,Ynisf(y1,..,yn) =P(Y1=y1,..,Yn=yn)( )for any (y1,..,yn) < joint pmffsatisfiesf(y) f(y1,..,yn) 0 y <nand f(y1,..,yn) = :f(y)>0 For any eventA <n,P[(Y1.)]

2 ,Yn) A] = f(y1,..,yn).y:y Aandf(y)>029 Definition cdf(cumulative distribution function) ofY1,..,YnisF(y1,..,yn) =P(Y1 y1,..,Yn yn) for any (y1,..,yn) < ,..,Ynare continuous random variables, then thejoint pdf(probability density function) ofY1,..,Ynis a functionf(y1,..,yn)that satisfiesF(y1,..,yn) = yn y1 f(t1,..,tn)dt1 dtnwhere theyiare any real numbers. A joint pdffsatisfiesf(y) f(y1,..,yn) 0 y <nand f(t1,..,tn)dt1 dtn= any eventA <n,P[(Y1,..,Yn) A] = f(t1,..,tn)dt1 ,..,Ynhas a joint pdf or pmff, then thesupportofY1,..,YnisY={(y1,..,yn) <n:f(y1,..,yn)>0}.IfYcomes from a family of distributionsf(y| ) for ,then the supportY ={y:f(y| )>0}may depend on.

3 Theorem ,..,Ynhave joint cdfF(y1,..,yn) and joint pdff(y1,..,yn).Thenf(y1,..,yn) = n y1 ynF(y1,..,yn)wherever the partial derivative pmfof any subsetYi1,..,Yikof thecoordinates (Y1,..,Yn) is found by summing the joint pmf over all possiblevalues of the other coordinates where the valuesyi1,..,yikare held fixed. Forexample,fY1,..,Yk(y1,..,yk) = yk+1 ynf(y1,..,yn)wherey1,..,ykare held fixed. In particular, ifY1andY2are discrete RVswith joint pmff(y1,y2),then the marginal pmf forY1isfY1(y1) = y2f(y1,y2)( )wherey1is held fixed. The marginal pmf forY2isfY2(y2) = y1f(y1,y2)( )30wherey2is held 2,double integrals are used to find marginalpdfs (defined below) and to show that the joint pdf integratesto 1.

4 If theregion of integration is bounded on top by the functiony2= T(y1), onthe bottom by the functiony2= B(y1) and to the left and right by the linesy1=aandy2=bthen f(y1,y2)dy1dy2= f(y1,y2)dy2dy2= ba[ T(y1) B(y1)f(y1,y2)dy2] the inner integral, treaty2as the variable, anything else, includingy1, is treated as a the region of integration is bounded on the left by the functiony1= L(y2), on the right by the functiony1= R(y2) and to the top and bottom bythe linesy2=candy2=dthen f(y1,y2)dy1dy2= f(y1,y2)dy2dy2= dc[ R(y2) L(y2)f(y1,y2)dy1] the inner integral, treaty1as the variable, anything else, includingy2, is treated as a pdfof any subsetYi1.

5 ,Yikof the co-ordinates (Y1,..,Yn) is found by integrating the joint pdf over all possiblevalues of the other coordinates where the valuesyi1,..,yikare held fixed. Forexample,f(y1,..,yk) = f(t1,..,tn)dtk+1 dtnwherey1,..,ykare held fixed. In particular, ifY1andY2are continuous RVs with joint pdff(y1,y2),then the marginal pdf forY1isfY1(y1) = f(y1,y2)dy2= T(y1) B(y1)f(y1,y2)dy2( )wherey1is held fixed (to get the region of integration, draw a line parallel tothey2axis and use the functionsy2= B(y1) andy2= T(y1) as the lowerand upper limits of integration). The marginal pdf forY2isfY2(y2) = f(y1,y2)dy1= R(y2) L(y2)f(y1,y2)dy1( )31wherey2is held fixed (to get the region of integration, draw a line parallel tothey1axis and use the functionsy1= L(y2) andy1= R(y2) as the lowerand upper limits of integration).

6 Definition pmfof any subsetYi1,..,Yikof thecoordinates (Y1,..,Yn) is found by dividing the joint pmf by the marginalpmf of the remaining coordinates assuming that the values ofthe remainingcoordinates are fixed and that the denominator> example,f(y1,..,yk|yk+1,..,yn) =f(y1,..,yn)f(yk+1,..,yn)iff(yk+1,..,yn) > particular, the conditional pmf ofY1givenY2=y2is a function ofy1andfY1|Y2=y2(y1|y2) =f(y1,y2)fY2(y2)( )iffY2(y2)>0, and the conditional pmf ofY2givenY1=y1is a function ofy2andfY2|Y1=y1(y2|y1) =f(y1,y2)fY1(y1)( )iffY1(y1)> pdfof any subsetYi1,..,Yikof thecoordinates (Y1,..,Yn) is found by dividing the joint pdf by the marginalpdf of the remaining coordinates assuming that the values ofthe remainingcoordinates are fixed and that the denominator> example,f(y1.)

7 ,yk|yk+1,..,yn) =f(y1,..,yn)f(yk+1,..,yn)iff(yk+1,..,yn) > particular, the conditional pdf ofY1givenY2=y2isa function ofy1andfY1|Y2=y2(y1|y2) =f(y1,y2)fY2(y2)( )iffY2(y2)>0, and the conditional pdf ofY2givenY1=y1is a function ofy2andfY2|Y1=y1(y2|y1) =f(y1,y2)fY1(y1)( )32iffY1(y1)> : Common the joint pmff(y1,y2) =P(Y1=y1,Y2=y2) is given by a table, then the functionf(y1,y2) is a joint pmf iff(y1,y2) 0, y1,y2and if (y1,y2):f(y1,y2)>0f(y1,y2) = marginal pmfs are found from the row sums and column sums usingDefinition , and the conditional pmfs are found with the formulas givenin Definition : Common the joint pdff(y1,y2) =kg(y1,y2) on its support, findk, find the marginal pdfsfY1(y1) andfY2(y2)and find the conditional pdfsfY1|Y2=y2(y1|y2) andfY2|Y1=y1(y2|y1).

8 Also,P(a1< Y1< b1,a2< Y2< b2) = b2a2 b1a1f(y1,y2) : Often support of the marginal pdf does not depend on the 2nd the conditional pdf can depend on the 2nd variable. Forexample, the support offY1|Y2=y2(y1|y2) could have the form 0 y1 continuous random variablesY1andY2is the region wheref(y1,y2)> support is generally given by one to three inequalities suchas 0 y1 1,0 y2 1,and 0 y1 y2 each variable, set theinequalities to equalities to get boundary lines. For example 0 y1 y2 1yields 5 lines:y1= 0,y1= 1, y2= 0,y2= 1,andy2= onthe vertical axis andy1is on the horizontal axis for determine thelimits of integration, examine thedummy variableused in the inner integral, saydy1.

9 Then within the region of integration,draw a line parallel to the same (y1) axis as the dummy variable. The limitsof integration will be functions of the other variable (y2), never of the dummyvariable (dy1). Expectation, Covariance and IndependenceFor joint pmfs withn= 2 random variablesY1andY2, the marginal pmfsand conditional pmfs can provide important information about the joint pdfs the integrals are usually too difficult for the joint, conditional33and marginal pdfs to be of practical use unless the random variables areindependent. (An exception is the Multivariate normal distribution and theelliptically contoured Distributions .)

10 See Sections and )For independent random variables, the joint cdf is the product of themarginal cdfs, the joint pmf is the product of the marginal pmfs, and thejoint pdf is the product of the marginal pdfs. Recall that is read for all. Definition ) The random variablesY1,Y2,..,YnareindependentifF(y1 ,y2,..,yn) =FY1(y1)FY2(y2) FYn(yn) y1,y2,.., ) If the random variables have a joint pdf or pmffthen the random variablesY1,Y2,..,Ynare independent iff(y1,y2,..,yn) =fY1(y1)fY2(y2) fYn(yn) y1,y2,.., the random variables are not independent, then they particular random variablesY1andY2areindependent, writtenY1Y2,if either of the following conditions )F(y1,y2) =FY1(y1)FY2(y2) y1, )f(y1,y2) =fY1(y1)fY2(y2) y1, , that the supportYof (Y1,Y2.