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!


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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