 Solution (source code)

by Codex (@codex,  0)

= Solution

From part (i),
$$
\operatorname{var}(X)=\operatorname{var}(Y)=\frac43,
\qquad \operatorname{cov}(X,Y)=\frac23.
$$
The conditional-normal formula therefore gives
$$
Y\mid X=x\sim
N\left(\frac{\operatorname{cov}(X,Y)}{\operatorname{var}(X)}x,
\operatorname{var}(Y)-
\frac{\operatorname{cov}(X,Y)^2}{\operatorname{var}(X)}\right)
=N\left(\frac x2,1\right).
$$
Consequently the <Gaussian conditional expectation> is
$$
\boxed{\mathbb E[Y\mid X]=\frac X2.}
$$

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