Probability generating function of discrete uniform random variable
Andrew Henderson 
Let X be uniform over {$\ x_1,x_2,\ldots,x_n$}
I have to prove that $\ G_X(t)=(t^{x_1}+\ldots +t^{x_n})/n$, where$\ G$ is the probability generating function... so far:
$$\ G_X(t)=\sum_{k=0}^\infty t^k P(X=k)=\sum_{k=0}^∞ t^k \frac{1}{n}=\frac{1}{n}\sum_{k=0}^∞ t^k $$
Help would be appreciated
$\endgroup$1 Answer
$\begingroup$Note that $P(X=k )=\begin{cases} 0 & ,k \not \in \{ x_1, \ldots, x_n\} \\ \frac1n & k \in \{ x_1, \ldots, x_n\} \end{cases}$.
Hence
$$\ G_X(t)=\sum_{k=0}^\infty t^k P(X=k)=\sum_{i=1}^n t^{x_i} \frac{1}{n}$$
Remark:
I am assuming $x_i$ are distinct and take non-negative integer value.
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