[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

Let [math]X[/math] be the first time that a failure occurs in

an infinite sequence of Bernoulli trials with probability [math]p[/math] for success. Let [math]p_k = P(X = k)[/math] for [math]k = 1[/math], 2, \dots. Show that [math]p_k = p^{k - 1}q[/math] where [math]q = 1 - p[/math]. Show that [math]\sum_k p_k = 1[/math]. Show that [math]E(X) = 1/q[/math]. What is the expected number of tosses of a coin required to obtain the first tail?