Question on Poisson Processes

Question

Good evening, I am sort of stuck in one problem of Poisson Processes and I hope I could get some help (no it is not a homework). Suppose that the customers arrive at the ticket booth independently. Let $T_{j}$ be the arrival time at the j-th customer. Find the density function of $T_{1}$.If you can find this density, then find the general formula for the density function of $T_{j}$ for any j. Find the mean arrival time $E[T_{j}]$ for linear arrival rate $\lambda(t)=t$. For some reason, this question seems way too hard, although there are some leads to do it. Thank you

Answer 1

Hint: $T_j$ is the sum of $j$ iid random variables, each with a _ distribution.