136 Multivariate DistributionsYj=uj(X 1 ,...,Xn)forj=1,...,mand eachE(Yi) exists, thenE⎡
⎣∑mj=1kjYj⎤
⎦=∑mj=1kjE[Yj], (2.6.5)wherek 1 ,...,kmare constants.
We next discuss the notions of marginal and conditional probability density
functions from the point of view ofnrandom variables. All of the preceding defini-
tions can be directly generalized to the case ofnvariables in the following manner.
Let the random variablesX 1 ,X 2 ,...,Xnbe of the continuous type with the joint
pdff(x 1 ,x 2 ,...,xn). By an argument similar to the two-variable case, we have for
everyb,
FX 1 (b)=P(X 1 ≤b)=∫b−∞f 1 (x 1 )dx 1 ,wheref 1 (x 1 ) is defined by the (n−1)-fold integral
f 1 (x 1 )=∫∞−∞···∫∞−∞f(x 1 ,x 2 ,...,xn)dx 2 ···dxn.Therefore,f 1 (x 1 ) is the pdf of the random variableX 1 andf 1 (x 1 ) is called the
marginal pdf ofX 1. The marginal probability density functionsf 2 (x 2 ),...,fn(xn)
ofX 2 ,...,Xn, respectively, are similar (n−1)-fold integrals.
Up to this point, each marginal pdf has been a pdf of one random variable.
It is convenient to extend this terminology to joint probability density functions,
which we do now. Letf(x 1 ,x 2 ,...,xn)bethejointpdfofthenrandom variables
X 1 ,X 2 ,...,Xn, just as before. Now, however, take any group ofk<nof these
random variables and find the joint pdf of them. This joint pdf is called the marginal
pdf of this particular group ofkvariables. To fix the ideas, taken=6,k=3,and
let us select the groupX 2 ,X 4 ,X 5. Then the marginal pdf ofX 2 ,X 4 ,X 5 is the joint
pdf of this particular group of three variables, namely,
∫∞
−∞∫∞−∞∫∞−∞f(x 1 ,x 2 ,x 3 ,x 4 ,x 5 ,x 6 )dx 1 dx 3 dx 6 ,if the random variables are of the continuous type.
Next we extend the definition of a conditional pdf. Supposef 1 (x 1 )>0. Then
we define the symbolf 2 ,...,n| 1 (x 2 ,...,xn|x 1 )bytherelation
f 2 ,...,n| 1 (x 2 ,...,xn|x 1 )=f(x 1 ,x 2 ,...,xn)
f 1 (x 1 ),andf 2 ,...,n| 1 (x 2 ,...,xn|x 1 ) is called thejoint conditional pdf ofX 2 ,...,Xn,
givenX 1 =x 1. The joint conditional pdf of anyn−1 random variables, say
X 1 ,...,Xi− 1 ,Xi+1,...,Xn,givenXi=xi, is defined as the joint pdf ofX 1 ,...,Xn
divided by the marginal pdffi(xi), provided thatfi(xi)>0. More generally, the
joint conditional pdf ofn−kof the random variables, for given values of the remain-
ingkvariables, is defined as the joint pdf of thenvariables divided by the marginal