Correlated Normal distributions

Question

Consider a number $X$ being selected according to normal distribution $\frac{1}{\sqrt{2 \pi \sigma^2}}exp\left[-\frac{X^2}{2 \sigma^2}\right]$. Suppose the outcome is $X_1$. Then, a new number $Y$ is picked according to probability distribution $\frac{1}{\sqrt{2 \pi \sigma^2}}exp\left[-\frac{(Y-X_1)^2}{2 \sigma^2}\right]$. Lets say it is $Y_1$.

What can be said about the average of the quantity $(Y_1-X_1)^2$?

Answer 1

So you have $X \sim \cal{N}(0, \sigma^2)$. Then $Y \mid X=X_1 \sim \cal{N}(X_1, \sigma^2)$. Remember the double expectation theorem says that $$ \DeclareMathOperator{\E}{\mathbb{E}} \E \E [Y \mid X] = \E Y $$ Using this we can compute $$ \E (Y-X_1)^2 = \E \E [(Y-X_1)^2 \mid X_1] = \E [\sigma^2 \mid X_1] = \sigma^2 $$