Mathematical Methods for Physics and Engineering : A Comprehensive Guide

30.13 GENERATING FUNCTIONS FOR JOINT DISTRIBUTIONS

As would be expected,Xis uncorrelated with eitherWorY, colour and face-value being
two independent characteristics. Positive correlations are to be expected betweenWand
Yand betweenXandZ; both correlations are fairly strong. Moderate anticorrelations
exist betweenZand bothWandY, reflecting the fact that it is impossible forWandY
to be positive ifZis positive.

Finally, let us suppose that the random variablesXi,i=1, 2 ,...,n, are related

to a second set of random variablesYk=Yk(X 1 ,X 2 ,...,Xn),k=1, 2 ,...,m.By

expanding eachYkas a Taylor series as in (30.137) and inserting the resulting

expressions into the definition of the covariance (30.133), we find that the elements

of the covariance matrix for theYkvariables are given by

Cov[Yk,Yl]≈

∑

i

∑

j

(

∂Yk

∂Xi

)(

∂Yl

∂Xj

)

Cov[Xi,Xj].

(30.140)

It is straightforward to show that this relation is exact if theYkare linear

combinations of theXi. Equation (30.140) can then be written in matrix form as

VY=SVXST, (30.141)

whereVYandVXare the covariance matrices of theYkandXivariables respec-

tively andSis the rectangularm×nmatrix with elementsSki=∂Yk/∂Xi.

30.13 Generating functions for joint distributions

It is straightforward to generalise the discussion of generating function in section

30.7 to joint distributions. For a multivariate distributionf(X 1 ,X 2 ,...,Xn)of

non-negative integer random variablesXi,i=1, 2 ,...,n, we define the probability

generating function to be

Φ(t 1 ,t 2 ,...,tn)=E[tX 11 t 2 X^2 ···tXnn].

As in the single-variable case, we may also define the closely related moment

generating function, which has wider applicability since it is not restricted to

non-negative integer random variables but can be used with any set of discrete

or continuous random variablesXi(i=1, 2 ,...,n). The MGF of the multivariate

distributionf(X 1 ,X 2 ,...,Xn) is defined as

M(t 1 ,t 2 ,...,tn)=E[et^1 X^1 et^2 X^2 ···etnXn]=E[et^1 X^1 +t^2 X^2 +···+tnXn]

(30.142)

and may be used to evaluate (joint) moments off(X 1 ,X 2 ,...,Xn). By performing

a derivation analogous to that presented for the single-variable case in subsection

30.7.2, it can be shown that

E[X 1 m^1 Xm 22 ···Xnmn]=

∂m^1 +m^2 +···+mnM(0, 0 ,...,0)

∂tm 11 ∂tm 22 ···∂tmnn

. (30.143)