Probability Generating Function of Discrete Uniform Distribution

Theorem
Let $X$ be a discrete random variable with the discrete uniform distribution with parameter $n$.

Then the p.g.f. of $X$ is:
 * $\ds \map {\Pi_X} s = \frac {s \paren {1 - s^n} } {n \paren {1 - s} }$

Proof
From the definition of p.g.f:


 * $\ds \map {\Pi_X} s = \sum_{x \mathop \ge 0} \map {p_X} x s^x$

From the definition of the discrete uniform distribution:
 * $\forall k \in \N, 1 \le k \le n: \map {p_X} k = \dfrac 1 n$

So:

Hence the result.