For continuous time birth-death processes on {0,1,2,...}, the first passage time T+n from n to n + 1 is always a mixture of (n + 1) independent exponential random variables. Furthermore, the first passage time T0,n+1 from 0 to (n + 1) is always a sum of (n + 1) independent exponential random...

It is known that the first passage time of a birth death process from n to n+1 has a completely monotone density, however, the discrete analogue for a simple random walk does not hold. In this paper, first passage times of simple random walks from n to n+1 and from 0 to n are characterized. This...