# Probability generating function.

## Main Question or Discussion Point

For any integer valued RV X Summation n=0 to infinity of s^n P(X=<n) = (1-s)^-1 * Summation k=0 to infinity of P(x=k)s^k

Why does Sum k=0 to infinity P(x=k)s^k = sum n=0 to infinity of P(X=< n)

$$\sum_{n=0}^{\infty} s^n P(X\leq n) = \sum_0^{\infty} s^n \sum_{k=0}^n P(X=k)$$