Conditional expectation of an exponential random variable given it is smaller than another independent exponential r.v.

Question

We have two exponentially distributed random variables, $X\sim \exp(\theta)$ and $Y\sim \exp(\mu)$, $X$ and $Y$ are independent from each other. What is $E[X|X<Y]$ and $E[X|X>Y]$?

Any help would be very much appreciated!

Many thanks.

Answer 1

Consider $E(X\mid X<Y)$. What you need are $P(X<Y)$ and $E(X1_{X<Y})$. As an example, we consider $E(X1_{X<Y})$, for which we note that \begin{align*} E(X1_{X<Y}) &= E\big(E(X1_{X<Y} \mid X) \big)\\ &=E\big(X E(1_{X<Y} \mid X) \big)\\ &=E\left(X\, e^{-\mu X} \right). \end{align*} The remaining is now straightforward.