Question about finding expected value and variance of x given the mean.

Question

Suppose Y is distributed as an exponential random variable with mean 0.5 and given Y = y, X is distributed as an exponential random variable with mean y. What is E(X) and Var(X)?

Answer 1

As d.k.o. says, use the Law of Iterated Expectations: $$\newcommand{\E}{\operatorname{\sf E}}\begin{align}\E(X) = &\; \E(\E(X\mid Y))\\ = &\; \int_0^\infty \E(X\mid Y) f_Y(y)\operatorname d x\end{align}$$

Likewise the Law of Iterated Variance:

$$\newcommand{\Var}{\operatorname{\sf Var}}\begin{align}\Var(X) = & \; \E(\Var (X\mid Y))+\Var(\E(X\mid Y)) \\ = & \;\int_0^\infty \Var(X\mid Y) f_Y(y)\operatorname d y+\int_0^\infty \Big(\E(X\mid Y)-\underbrace{\E(\E(X\mid Y))}_{:=\E(X)}\Big)^2 f_Y(y)\operatorname d y\end{align}$$

Now, for exponential distributions of given means these terms are :

$$\begin{align}\E(X\mid Y) = & y \\ \Var(X\mid Y) = & y^2 \\ f_Y(y) = & 2 e^{-2y} \quad\Bbb I_{(0<y)}\end{align}$$