Drug incidents occur in a bathroom as a Poisson process at the rate of $2$/hour and Harry goes in to detect them as a Poisson process at the rate of $1$/hour. Assume that
- the 2 processes are independent
- if a drug incident has occured before, Harry can detect it whenever he goes in
What is the expected time until Harry detects an incident of drug use?
My take from this question is that I need to find $$E(Y_n|Y_{n-1}<X_1\le Y_n)=\sum_yyP(Y_n-1<X_1,X_1\le y)\frac{P(Y_n=y)}{P(Y_{n-1}<X_1\le Y_n)}$$ Can't figure how to solve this, hints would be great.
Let $D_1$ be the first time when the drug incident happened and $T_1$ be the first time after $D_1$ when Harry goes in to detect them. Clearly, $\mathbb{E}(D_1)=\frac{1}{2}$ (this is the expected time for the first drug incident to happen). Now conditional on $D_1$, the first arrival of Harry will happen after an exponential wait time with mean $1$ (since the processes are independent). Hence the expected amount of time for this to happen is $1$. Therefore expected amount of time before Harry detects a drug incident is after $3/2$ hours.