# Expected Value of the maximum of two exponentially distributed random variables

I want to find the expected value of $\text{max}\{X,Y\}$ where $X$ ist $\text{exp}(\lambda)$-distributed and $Y$ ist $\text{exp}(\eta)$-distributed. X and Y are independent. I figured out how to do this for the minimum of $n$ variables, but i struggle with doing it for 2 with the maximum.

(The context in which this was given is waiting for the later of two trains, with their arrival times being exp-distributed).

Thanks!

The minimum of two independent exponential random variables with parameters $\lambda$ and $\eta$ is also exponential with parameter $\lambda+\eta$.
Also $\mathbb E\big[\min(X_1,X_2)+\max(X_1,X_2)\big]=\mathbb E\big[X_1+X_2\big]=\frac{1}{\lambda}+\frac{1}{\eta}$. Because $\mathbb E\big[\min(X_1,X_2)\big]=\frac{1}{\lambda+\eta}$, we get $\mathbb E\big[\max(X_1,X_2)\big]=\frac{1}{\lambda}+\frac{1}{\eta}-\frac{1}{\lambda+\eta}.$
Let $$V=\max\{X,Y\}$$. Then $$\mathbb{P}(V\leq t)=\mathbb{P}(X\leq t,Y\leq t)=\mathbb{P}(X\leq t)\mathbb{P}(Y\leq t).$$ Now find $$f_V(t)$$ and then $$\int_{0}^{+\infty}tf_V(t)dt$$, which should be $$\frac{1}{\lambda}+\frac{1}{\eta}-\frac{1}{\lambda+\eta}$$.
The sample $(X,Y)$ have a density given by $f_X(x)f_Y(y)$ since $X$ and $Y$ are independent. You have to compute $$\iint_{\Bbb R^2}\max\{x,y\}f_X(x)f_Y(y)dxdy.$$ Cut this integral in two parts.