Condition Expectation of normal variables Let X,Y be jointly normal. Then I know that $E(X|Y)=E(X)+\frac{Cov(X,Y)}{V(Y)}(Y-E(Y))$.
Do I need joint normality for this result? Does it also hold, if just X is normal and Y is normal?
 A: Counter example: Let $Y\sim\mathcal{N}(0,1)$. Define the random variable $X$ as follows:
\begin{equation}
X = \begin{cases}
Y, & \text{if } |Y| \geq 1\\
-Y, & \text{if } |Y| < 1.
\end{cases}.
\end{equation}
Note that by symmetry, $-Y$ is also $\mathcal{N}(0,1)$. Then, we compute the cdf of $X$ as follows:
\begin{align}
\mathbb{P}\{X\leq x\} &= \mathbb{P}\{X\leq x, |Y|\geq 1\} + \mathbb{P}\{X\leq x, |Y|< 1\} \\
&=  \mathbb{P}\{Y\leq x, |Y|\geq 1\} + \mathbb{P}\{-Y\leq x, |Y|< 1\} \\
&=  \mathbb{P}\{Y\leq x, |Y|\geq 1\} + \mathbb{P}\{Y\leq x, |Y|< 1\} \\
&=  \mathbb{P}\{Y\leq x\}.
\end{align}
Thus, $X\sim\mathcal{N}(0,1)$. Next, computing the conditional expectation, we get
\begin{align}
\mathbb{E}\{X|Y\} &= Y\delta_{ |Y|\geq 1} - Y\delta_{ |Y| < 1},
\end{align}
where $\delta$ denote the indicator function. Further, the RHS gives
\begin{equation}
 \mathbb{E}\left\{X\right\}+\frac{\text{Cov}\{X,Y\} }{\text{Var}\{Y\} } \left(Y-\mathbb{E}\{Y\}\right) = \text{Cov}\{X,Y\} Y.
\end{equation}
Since $\text{Cov}\{X,Y\} $ is a real number independent of $X$ and $Y$, we get that LHS$\neq$RHS.
