Poisson Process - Finding conditional expectation and conditional variance

Question

Consider a Poisson process $X={X(t);t\ge0}$ of rate $λ=5$. Here $X(t)$ is the number of customers arrived up to the time$=t$. Suppose that $X(1)=5$ (so 5 customers arrived by the end of the first hour). Find the conditional expectation and conditional variance for (total waiting time) W=W₁+W₂+...+W₅, given that $X(1)=5.$

I'm lost on how to solve this problem. Any help is appreciated!

Answer 1

Hint: conditioned on $X(1)=5$, the interarrival times $W_1,\ldots,W_5$ are distributed as the order statistics of 5 independent and uniformly distributed random variables over $[0,1]$.