Answer by Robert Israel for Computing Distribution of Conditional Expectation...
Note that $Y \sim {\mathcal N}(0, \sigma_y^2)$ where $\sigma_y^2 = \sigma_x^2 + \sigma_z^2$ and$\text{Cov}(X,Y) = \sigma_x^2$. Thus $X = \dfrac{\sigma_x^2}{\sigma_y^2} Y + W$ where $W$ is independent...
View ArticleComputing Distribution of Conditional Expectation of Gaussian RV
I am trying to compute distribution of the following random variable\begin{align*}E[(X-E[X|Y])^2|Y]\end{align*}where $X \sim \mathcal{N}(0,\sigma^2_x)$ and $Z \sim \mathcal{N}(0,\sigma^2_Z)$ where $Z$...
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