# 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:

$\map {\Pi_X} s = \dfrac {s \paren {1 - s^n} } {n \paren {1 - s} }$

## 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:

 $\ds \map {\Pi_X} s$ $=$ $\ds \sum_{k \mathop = 1}^n \frac 1 n s^k$ $\ds$ $=$ $\ds \frac s n \sum_{k \mathop = 0}^{n - 1} s^k$ $\ds$ $=$ $\ds \frac s n \paren {\frac {1 - s^n} {1 - s} }$ Sum of Geometric Sequence

Hence the result.

$\blacksquare$