Paper 2, Section I, F

Define the probability generating function $G(s)$ of a random variable $X$ taking values in the non-negative integers.

A coin shows heads with probability $p \in(0,1)$ on each toss. Let $N$ be the number of tosses up to and including the first appearance of heads, and let $k \geqslant 1$. Find the probability generating function of $X=\min \{N, k\}$.

Show that $E(X)=p^{-1}\left(1-q^{k}\right)$ where $q=1-p$.

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