## Friday, 31 January 2014

In Lecture 7 I mentioned that $EX$ may not exist, and that for a random variable $X$ such that $X\geq 0$ it is possible for $EX=\infty$. Here's a famous example arising from the case $EX=\infty$. See also, Examples sheet 3, #19.
How much would you be willing to pay to play the following game? We will toss a fair coin repeatedly. We start you with £1 and then double this every time the coin shows a head. We stop when the first tail occurs. You keep your winnings, denoted by the random variable $X$. Now
$EX = (1/2)1+(1/4)2+(1/8)4+(1/16)8 +\cdots = \infty$.
This is the St. Petersburg paradox  posed by Nicolas Bernoulli 1713.The paradox concerns what the mathematics predicts versus what a rational person might really do in practice. One resolution is that a person's utility for money is not linear. We should really be looking at $Eu(X)$, where perhaps $u(X)=\log X$. In that case $Eu(X) < \infty$.