Conditional Expected value for exponential distribution

Question

I meet a problem when solving a exponential distribution problem.

The problem is to calculate a conditional expectation value for two independent exponential distribution with rate ${\mu _1},{\mu _2}$. I am going to calculate the expectation value that $E[{X _2}|{X _2}>{X _1}]$

My thought is to first calculate the probability that $Pr[{X _2}>{X _1}]$ and use total probability theorem to compute $Pr[{X_2}|{X_2}>{X_1}]$ and then calculate the expected value via conditional expectation but I cannot figure out how to start it. Could anybody give me a direction on how to solve this problem? Thanks!

Answer 1

Start with the joint distribution of $X_1$ and $X_2$, which will be a distribution over the (cartesian) product of the ranges of $X_1$ and $X_2$, i.e. over $(0,\infty)^2$. The area that you're interested in are those points $(x_1,x_2) \in (0,\infty)^2$ with $x_2 > x_1$.

$E[X_2|X_2 > X_1]$ is thus $$\begin{eqnarray} \int_0^\infty \int_{x_1}^\infty x_2f_1(x_1)f_2(x_2) dx_2x_1 \end{eqnarray} $$ where $f_1$ and $f_2$ are the densities of your two exponential distributions.