Mathematical Methods for Physics and Engineering - Matematica.NET

30.13 GENERATING FUNCTIONS FOR JOINT DISTRIBUTIONSAs would be expected, X is uncorrelated with either W or Y , colour and face-value beingtwo independent characteristics. Positive correlations are to be expected between W andY and between X and Z; both correlations are fairly strong. Moderate anticorrelationsexist between Z and both W and Y , reflecting the fact that it is impossible for W and Yto be positive if Z is positive. ◭Finally, let us suppose that the random variables X i , i =1, 2,...,n, are relatedto a second set of random variables Y k = Y k (X 1 ,X 2 ,...,X n ), k =1, 2,...,m.Byexpanding each Y k as a Taylor series as in (30.137) and inserting the resultingexpressions into the definition of the covariance (30.133), we find that the elementsof the covariance matrix for the Y k variables are given byCov[Y k ,Y l ] ≈ ∑ ∑( )( )∂Yk ∂YlCov[X i ,X j ].∂Xi j i ∂X j(30.140)It is straightforward to show that this relation is exact if the Y k are linearcombinations of the X i . Equation (30.140) can then be written in matrix form asV Y = SV X S T , (30.141)where V Y and V X are the covariance matrices of the Y k and X i variables respectivelyand S is the rectangular m × n matrix with elements S ki = ∂Y k /∂X i .30.13 Generating functions for joint distributionsIt is straightforward to generalise the discussion of generating function in section30.7 to joint distributions. For a multivariate distribution f(X 1 ,X 2 ,...,X n ) ofnon-negative integer random variables X i , i =1, 2,...,n, we define the probabilitygenerating function to beΦ(t 1 ,t 2 ,...,t n )=E[t X11 tX2 2 ···tXn n ].As in the single-variable case, we may also define the closely related momentgenerating function, which has wider applicability since it is not restricted tonon-negative integer random variables but can be used with any set of discreteor continuous random variables X i (i =1, 2,...,n). The MGF of the multivariatedistribution f(X 1 ,X 2 ,...,X n ) is defined asM(t 1 ,t 2 ,...,t n )=E[e t1X1 e t2X2 ···e tnXn ]=E[e t1X1+t2X2+···+tnXn ](30.142)and may be used to evaluate (joint) moments of f(X 1 ,X 2 ,...,X n ). By performinga derivation analogous to that presented for the single-variable case in subsection30.7.2, it can be shown that∂m1+m2+···+mn M(0, 0,...,0)E[X m11 Xm2 2···Xn mn]= . (30.143)∂t m11 ∂tm2 2 ···∂tmn n1205