2
$\begingroup$

Suppose I have two independent integer random variables $X_1$, $X_2$ (with constraint that $X_1+X_2\le N,0\le X_1\le N,0\le X_2\le N$), with probability generating functions $g_1(z)$, $g_2(z)$. Now I have a joint-distribution $P(X_1-X_2,2X_1+X_2-N)$, whose probability generating function is $G(z_1,z_2)$. $N$ is a constant.

What is the relation between $G(z_1,z_2)$ and $g_1(z)$, $g_2(z)$?

The following question is related but not the same: multivariate probability generating function

$\endgroup$
1
  • $\begingroup$ The constraints tend not to marry well with independence; $\endgroup$ Apr 10, 2014 at 5:58

1 Answer 1

1
$\begingroup$

Note that $$ G(z_1,z_2)=E(z_1^{X_1-X_2}z_2^{2X_1+X_2-N})=z_2^{-N}E((z_1z_2^2)^{X_1}(z_1^{-1}z_2)^{X_2}). $$ Since $X_1$ and $X_2$ are independent, $$ G(z_1,z_2)=z_2^{-N}g_1(z_1z_2^2)g_2(z_1^{-1}z_2). $$ This does not use the constraints.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.