Expected Time to the nth Poisson Arrival

The $n$th arrival time is the sum of $n$ independent interarrival times.

Each interarrival time is Exponential$(\lambda )$ with mean $1/\lambda$.

1. 1. Write the arrival time as a sum

   The time of the $n$th arrival is $T_n={\sum }_{k=1}^n{\tau }_k$, where each ${\tau }_k$ is an independent Exponential$(\lambda )$ interarrival time.

2. 2. Sum the expectations

   By linearity of expectation, $E[T_n]=n\cdot E[{\tau }_1]=n/\lambda$.

   $$E[T_n]=\frac{n}{\lambda }$$

Answer: n/lambda