Generating function expectation

[jimmy1]

A probability distribution,$$f(x)$$ ,can be represented as a generating function,$$G(n)$$, as $$\sum_{x} f(x) n^x$$. The expectation of $$f(x)$$ can be got from $$G'(1)$$.

A bivariate generating function, $$G(m,n)$$ of the joint distribution $$f(x,y)$$ can be represented as $$\sum_{x} \sum_{y} f(x,y) n^x m^y$$.

Now my question is how can I get the expectation of $$f(x,y)$$ from the above generating function?

 

[Hurkyl]

Er, are you sure you're asking the right question? What meaning did you have in mind for "the expectation of f(x, y)"? Do you mean to think of f as the probability distribution for an R²-valued random variable, or something like that? Anyways, I would start by writing down the definition of expected value, and work from there.

 

[tacman]

To get the expectation of f(x,y) from the bivariate generating function G(m,n), we can use the partial derivatives of G(m,n) with respect to m and n. The expectation can be calculated as E[f(x,y)] = G_m(1,1) + G_n(1,1) - G(1,1), where G_m(1,1) and G_n(1,1) represent the partial derivatives of G(m,n) with respect to m and n evaluated at 1, and G(1,1) is the value of G(m,n) at m=1 and n=1. This formula is derived from the definition of expectation, which is the sum of the product of the possible values of a random variable and their corresponding probabilities. In this case, the possible values are x and y, and their probabilities are given by the coefficients of the terms in the generating function G(m,n). By taking the partial derivatives and evaluating at 1, we are essentially finding the coefficients of the terms in the generating function that represent the probabilities of x and y, and then multiplying them by their corresponding values and summing them up to get the expectation.