[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]

You are offered the following game to play: a fair coin is

tossed until heads turns up for the first time (see Example). If this occurs on the first toss you receive 2 dollars, if it occurs on the second toss you receive [math]2^2 = 4[/math] dollars and, in general, if heads turns up for the first time on the [math]n[/math]th toss you receive [math]2^n[/math] dollars.

• Show that the expected value of your winnings does not exist (i.e., is given by a divergent sum) for this game. Does this mean that this game is favorable no matter how much you pay to play it?
• Assume that you only receive [math]2^{10}[/math] dollars if any number greater than or equal to ten tosses are required to obtain the first head. Show that your expected value for this modified game is finite and find its value.
• Assume that you pay 10 dollars for each play of the original game. Write a program to simulate 100 plays of the game and see how you do.
• Now assume that the utility of [math]n[/math] dollars is [math]\sqrt n[/math]. Write an expression for the expected utility of the payment, and show that this expression has a finite value. Estimate this value. Repeat this exercise for the case that the utility function is [math]\log(n)[/math].