probability generating function for multivariate distribution

Question

I would like to ask how can we derive PGF of any multivariate distribution? and can anyone give an example of deriving the PGF of a multivariate distribution? That will be great. Thanks advance.

Answer 1

You have in general $$G(\mathbf{z}) = G(z_1,\ldots,z_d)=\operatorname{E}\bigl (z_1^{X_1}\cdots z_d^{X_d}\bigr) = \sum_{x_1,\ldots,x_d=0}^{\infty}p(x_1,\ldots,x_d)z_1^{x_1}\cdots z_d^{x_d}$$

So suppose for example $$P(\mathbf{X}=(0,0))=\frac45, \,P(\mathbf{X}=(1,0))=\frac35, \,P(\mathbf{X}=(0,1))=\frac25, \, P(\mathbf{X}=(1,1))=\frac15,$$ then rather simply you have $$G(\mathbf{z})=\frac45 + \frac35 z_1 + \frac25 z_2 + \frac15 z_1z_2.$$

Sometimes a particular multivariate distribution will allow some simplification. In particular, if the components $X_1,X_2,\ldots, X_d$ of $\mathbf{X}$ are independently distributed then you have $$G(\mathbf{z}) = G_1(z_1)G_2(z_2)\cdots G_d(z_d).$$