All of Statistics, 3.8.1
Suppose we play a game where we start with $c$ dollars. On each play of the game you either double or halve your money, with equal probability. What is your expected fortune after n trials?
Let $f(n)$ be the amount of money left after $n$ trials. We are interested in the expected value of this, $E[f(n)]$, which we can write as:
$E[f(n)] = \frac{1}{2} \cdot \frac{E[f(n-1)]}{2} + \frac{1}{2} \cdot 2 \cdot E[f(n-1)]$
$ = \frac{5}{4} E[f(n-1)]$
$ = \frac{5}{4}^{(n-1)} E[f(0)]$
$E[f(n)] = c \cdot \frac{5}{4}^{(n-1)}$
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