partial derivative of conditional expectation

Question

Can someone help me where to find the necessary background material or explain me why $\frac{\partial E(X|Y)}{\partial Y} = \frac{\mathrm{Cov}(X,Y)}{\mathrm{Var}(X)}$ for the linear Gaussian case?

Answer 1

You've mixed up $X$ and $Y$ on one side of the equation or the other. I'll swap them on the LHS: $\frac{\partial E(Y|X)}{\partial Y} = \frac{\mathrm{Cov}(X,Y)}{\mathrm{Var}(X)}$

Suppose $Y = \beta_0 + \beta_1 X + \epsilon$, with $X$ and $\epsilon$ independent, and with $E(\epsilon) = 0$.

The left hand side is $\beta_1$, by taking $E(\cdot|X)$ of both sides.

Now compute the covariance.

$\mathrm{Cov}(X,Y) = \beta_1 \mathrm{Cov}(X,X) = \beta_1 \mathrm{Var}(X)$