Discussion
Let X, Y be two random variables such that \(E[Y/X = x] = x^{2}\) and X is standard normal r.v. Then which of the following is false ?
Answer: C
The false statement is \(\mathrm{Corr}(X, Y) \gt 0\).
With \(E[Y | X] = X^{2}\) and X standard normal, use the tower property:
\[ \begin{aligned} E[XY] &= E[X\,E(Y | X)] = E[X^{3}] = 0 \\ E[Y] &= E[X^{2}] = 1 \end{aligned} \]
So \(\mathrm{cov}(X, Y) = E[XY] - E[X]E[Y] = 0 - 0 = 0\) and the correlation is 0, not positive. Also \(E[X^{2}Y] = E[X^{2} \cdot X^{2}] = E[X^{4}] = 3\). So A, B and D are true, and only C is false: Y depends on X, but symmetrically, through \(X^{2}\).
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