Let X1, X2,... be independent and identically distributed Bernoulli random variables with values ±1 that have equal probability of 1/2. Let K1 and K2 be positive integers, and define N as follows:

N = min{n:Sn = K1 or −K2}

where Sn =n k=1 Xk n = 1, 2,...

is called a symmetric random walk.

a. Show that E[N] < ∞.

b. Show that P[Sn = K1] = K2 K1+K2

.