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I am currently learning about Poisson processes and I was thinking about the following question.

The waiting time for your lunch to be collected by delivery drivers A and B are distributed as $\text{Expo}(\lambda_1)$ and $\text{Expo}(\lambda_2)$. The probability that A delivers the food to you is $p_1$, while the probability that B delivers the food is $p_2$. What is the mean waiting time for your lunch?

I wish that the mean waiting time was naively $p_1/\lambda_1+p_2/\lambda_2$, but I doubt this is the case. I’ve been reading myself silly about bus waiting times, birth-death rates, survival times etc. and am still not able to reconcile the facts in my head. Could someone help me out and lead me in the right direction? Thanks!

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Ah love a bit of actuarial science. We can use Tower Property of conditional expectation. Let $X$ be r.v. "delivery driver $A$ or $B$"

$T|(X=A) = Exp(\lambda_1)$,

$T|(X=B) = Exp(\lambda_2)$.

$E[T] = E[E[T|X]] = E[\frac{1}{\lambda_1}\mathbb{1}_{(X=A)}+\frac{1}{\lambda_2}\mathbb{1}_{(X=B)}] = \frac{p_1}{\lambda_1} + \frac{p_2}{\lambda_2}$

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