Poisson Process Counts

The number of arrivals in an interval of length $t$ depends only on the length of that interval.

By definition, counts over an interval of length $t$ are Poisson with mean equal to rate times length.

1. 1. Recall the defining property

   A Poisson process with rate $\lambda$ has stationary, independent increments, and the count over any interval of length $t$ is Poisson with mean $\lambda t$.

2. 2. Apply to [0,t]

   Since $N(t)=N(t)-N(0)$ is the count over an interval of length $t$, it follows the stated distribution.

   $$N(t)\sim \text{Poisson}(\lambda t)$$

Answer: Poisson(lambda*t)