# Poisson Process. Expected time of three fishermen catching at least three fish.

Three fishermen are fishing, we model the fishing as a Poisson Process of rate $2.5$ fish/hour. The fishermen leave only when each of them them has caught at least 3 fish, we call this leaving time $T$. Calculate $\mathbb{E}[T]$.

My attempt:

I started out calculating the distribution $F_T$:

$F_T(t)=\mathbb{P}(X_1(t)\ge 3, X_2(t)\ge 3, X_3(t)\ge 3)=(\mathbb{P}(X_1(t)\ge 3))^3,$ where I have assumed independence of the Poisson processes corresponding to each fisherman $X_i$. Then I integrate $(1-F_T)$ to get the expectation.

Question: Is my reasoning correct? The numerical calculation yields an expected time of 3.56 hours. Also, is there any easier way to do this using arrival waiting or interarrival times?

I think you started off OK in finding $F_T(t)$. To find $E(T)$, you need to differentiate $F_T(t)$ to get $f_T(t)$ and use that to obtain $E(T)$.
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