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:

$\displaystyle \Pi_X \left({s}\right) = \frac {s \left({1 - s^n}\right)} {n \left({1 - s}\right)}$

Proof

From the definition of p.g.f:

$\displaystyle \Pi_X \left({s}\right) = \sum_{x \ge 0} p_X \left({x}\right) s^x$

From the definition of the discrete uniform distribution:

$\forall k \in \N, 1 \le k \le n: p_X \left({k}\right) = \dfrac 1 n$

So:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \Pi_X \left({s}\right)\)	\(=\)	\(\displaystyle \sum_{k = 1}^n \frac 1 n s^k\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \frac s n \sum_{k = 0}^{n-1} s^k\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \frac s n \left({\frac {1 - s^n} {1 - s} }\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	by Sum of Geometric Progression

Hence the result.

$\blacksquare$