[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]

For a sequence of Bernoulli trials, let [math]X_1[/math] be the number

of trials until the first success. For [math]j \geq 2[/math], let [math]X_j[/math] be the number of trials after the [math](j - 1)[/math]st success until the [math]j[/math]th success. It can be shown that [math]X_1[/math], [math]X_2[/math], \dots is an independent trials process.

• What is the common distribution, expected value, and variance for [math]X_j[/math]?
• Let [math]T_n = X_1 + X_2 +\cdots+ X_n[/math]. Then [math]T_n[/math] is the time until the [math]n[/math]th success. Find [math]E(T_n)[/math] and [math]V(T_n)[/math].
• Use the results of (b) to find the expected value and variance for the number of tosses of a coin until the [math]n[/math]th occurrence of a head.