Suppose I toss a fair coin independently until I get a tail. Let X be the number of time the coin is tossed, if the coin is tossed x times, a reward of $2^x$ will be given, the question is what is the expected money I get?
A systematic approach would be to think of X is a geometric random variables and let $g(x)=2^x$ then use the formula
$E[g(x)]=\sum_xg(x)P_X(x)=\sum_x1$
which can not even be evaluated. Can anyone tell me where I was wrong?
Alternative approach leading to the same result.
$2^X$ is nonnegative so at forehand we know that $0\leq\mathbb E2^X\leq+\infty$.
Let $T$ denote the event that the first toss gives a tail and $H$ the event that the first toss gives a head.
$\mathbb{E}2^{X}=\mathbb{E}\left(2^{X}\mid T\right)P\left(T\right)+\mathbb{E}\left(2^{X}\mid H\right)P\left(H\right)=2^{1}\cdot\frac{1}{2}+\mathbb{E}2^{X+1}.\frac{1}{2}=1+\mathbb{E}2^{X}$
This can only be true if $\mathbb{E}2^{X}=+\infty$.
