This is a two part problem that expands on the St Petersburg Paradox. I have attempted this problem and ca not seem to arrive at the answers given in my book. (a) c/2-c (b) 2 log2
Below is the problem
A fair coin is flipped until the first tail appears; we win $2 if it appears on the first toss, $4 on the second and in general $2^k if it first appears on the kth toss. Let the random variable X denote our winnings. How much should we pay in order for this to be a fair game.++++++++++
I have already figured out the paradoxical nature of this part of the problem. That the series is divergent. Now where I'm having the trouble is
(a)what if the amounts won are c^k instead of 2^k where 0<c<2
(b) the amounts won are log 2^k
p= probability of winning = 1/2
hence of losing = q = 1 -1/2
expectations of winning:
E(X) = (1/2)*2 + (1- 1/2)*(1/2)*4 + (1- 1/2)*(1- 1/2)*(1/2)*2^3 + ....
=> E(X) = 1+1+1+1+1+.... infinity = infinity
The solution answers the question(s) below.