I want to compute the expected value of the following game. I have a symmetric 1-D random walk, I can choose to stop at any time and my winnings will be the value of the random walk. What is the value of this game for large number of steps N?
I have run some simulations and have heuristic arguments that both show that it should be proportional to sqrt(N)
, but I'm particularly interested in the coefficient.
X_i
is equal to at stepi
, whereX_0 = 0
and eachX_i - X_(i-1)
is plus or minus one with equal probability. $\endgroup$