Expectation of the maximum of two exponential random variables

Question

Let $Z:=\text{max}(X,Y)$ where $X,Y$ are independent random variables having exponential distribution with parameters $\lambda$ and $\mu$ respectively.

My question is:

What is the expectation of $Z$, i.e. what is $\mathbb{E}(Z)$?

Answer 1

Hint:

Make use of: $$\mathbb EZ=\int_0^{\infty}P(Z>z)dz$$

and of course:$$P(Z>z)=P(X>z)+P(Y>z)-P(X>z\wedge Y>z)$$

By independence of $X,Y$ this results in:$$P(Z>z)=P(X>z)+P(Y>z)-P(X>z)P(Y>z)$$