# Combination of Exponential distributions question (with different probabilities)?

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!

## 1 Answer

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

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

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

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