Probability generating function.

[nikki92]

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)

 

[mfb]

$$\sum_{n=0}^{\infty} s^n P(X\leq n) = \sum_0^{\infty} s^n \sum_{k=0}^n P(X=k)$$
You can re-write this double sum and use k as primary index. The inner sum (over n) has an explicit formula then, and you get the correct result.