$X,Y$ are independent exponential random variables with respective rates $\lambda,\mu$. Let $M=\text{min}(X,Y)$.
How to calculate $E[MX\;|\;M=X]$.
In general, how to calculate $E[X\;|\;B]$ where $X$ is a continuous RV and $B$ is an event?
$X,Y$ are independent exponential random variables with respective rates $\lambda,\mu$. Let $M=\text{min}(X,Y)$.
How to calculate $E[MX\;|\;M=X]$.
In general, how to calculate $E[X\;|\;B]$ where $X$ is a continuous RV and $B$ is an event?
$E[M X \mid M = X]$ is the same thing as $E[X^2 \mid X < Y]$, which is:
$ E[X^2 \mid X < Y] = \int_0^\infty x^2 p(x \mid X < Y) dx. $
Now, via Bayes' Rule, $p(x \mid X < Y)$ is $p(x \mid X < Y) = \frac{p(X < Y \mid x) p(x)}{p(X < Y)}.$
$P(X < Y \mid X = x)$ is just $P(Y > x) = 1 - P(Y \le x) = e^{-\mu x}$, and $p(x) = \lambda e^{-\lambda x}$.
$p(X < Y)$ is a little trickier, but we can do it as $ p(X < Y) = \int_0^\infty \int_0^\infty I(x
Putting that together, we have $ p(x \mid X < Y) = e^{-\mu x} \lambda e^{-\lambda x} \frac{\lambda + \mu}{\lambda} = (\lambda + \mu) e^{-(\lambda + \mu) x}, $ so that $E[X^2 \mid X < Y]$ is just the second moment of an exponential with rate $\lambda + \mu$ (call it $Z$), $E[Z^2] = Var[Z] + E[Z]^2 = \frac{2}{(\lambda + \mu)^2}.$
Moderately surprisingly, this means that $E[X^2 \mid X < Y] = E[Y^2 \mid Y < X]$.
For the second question, if by $B$ is an event means it is a discrete distribution, let $p(B_i)$ be the probabilities of each value of each value of $B$. Then $E[X|B]=\sum_i p(B_i)E[X(B_i)]$ where $E[X(B_i)]$ is the expectation of $X$ given that $B=B_i$