Interarrival Times of a Poisson Process

Use the memoryless property implied by stationary, independent increments.

Compute $P(\text{no arrival in }[0,t])=e^{-\lambda t}$ and interpret it as a survival function.

1. 1. Compute the survival probability

   The waiting time $T$ for the first arrival exceeds $t$ exactly when no arrivals occur in $[0,t]$, which has probability $P(N(t)=0)=e^{-\lambda t}$.

2. 2. Recognize the exponential distribution

   A random variable with survival function $e^{-\lambda t}$ is Exponential$(\lambda )$; by stationary independent increments, every interarrival time has this same distribution.

   $$T\sim \text{Exponential}(\lambda )$$

Answer: Exponential(lambda)