Customers arrive at a service facility according to a Poisson process of rate $\lambda = 5$ customers/hour. Let N(t) be the number of customers that have arrived up to time t hours. Let $W_1,W_2,W_3,...$ be the successive arrival times of the customers.
a) Find the expected arrival time of the 7th customer $E[W_7]$
b) Given N(1) = 4, determine the expected arrival time of the 7th customer, $E[W_7|N(1)=4]$
so far I have calculated that each one customer arrives at a rate of $\frac{1}{5}$ and used that to calculate $E[W_7] = \sum_{1}^7\frac{1}{5} = \frac{7}{5}$
I am currently having trouble figuring out how to approach b). Any help would be appreciated
Hint: $$\require{cancel} \xcancel{\operatorname{E}[W_7 \mid N(1) = 4] = 4 + \operatorname{E}[W_6].}$$
Edit. I apologize--I misread the question. The notation $N(1) = 4$ means that we observed exactly four customers arriving within $1$ hour. Then this means we are waiting for another $3$ customers to arrive, so the correct relationship is $$\operatorname{E}[W_7 \mid N(1) = 4] = \color{red}{1} + \operatorname{E}[W_{\color{red}{3}}].$$ This is because the expected additional waiting time after one hour corresponds to having to wait to see $3$ more customers, which by the memorylessness property, is just $\operatorname{E}[W_3]$. Then we have to add back the $1$ hour that has already elapsed.