Covariance of X and Y^2

For a bivariate normal \( (X,Y) \) with standard normal marginals and correlation \( \rho \), the covariance is

\( \operatorname{Cov}(X,Y^2)=\mathbb{E}[XY^2]-\mathbb{E}[X]\mathbb{E}[Y^2] \).

Since the marginals are standard normal, \( \mathbb{E}[X]=0 \) and \( \mathbb{E}[Y^2]=1 \), so

\( \operatorname{Cov}(X,Y^2)=\mathbb{E}[XY^2] \).

Using conditional expectation

The conditional distribution of \(Y\) given \(X=x\) is

\( Y\mid X=x \sim \mathcal N(\rho x,\,1-\rho^{2}) \).

Hence

\( \mathbb{E}[Y^2\mid X=x]=(1-\rho^{2})+(\rho x)^{2}=1-\rho^{2}+\rho^{2}x^{2}. \)

Taking expectation over \(X\) gives

\( \mathbb{E}[XY^2]=\mathbb{E}[X(1-\rho^{2}+\rho^{2}X^{2})]=(1-\rho^{2})\mathbb{E}[X]+\rho^{2}\mathbb{E}[X^{3}]. \)

For a standard normal variable all odd moments are zero, so the right‑hand side is zero and

\( \mathbb{E}[XY^2]=0. \)

Thus

\( \boxed{\operatorname{Cov}(X,Y^2)=0}. \)

Alternative (Isserlis/Wick) argument

For zero‑mean jointly Gaussian variables any product with an odd total number of factors has expectation zero. Since \(XY^2\) contains three factors, \( \mathbb{E}[XY^2]=0 \) and the covariance is zero.

Interpretation

Even though \(X\) and \(Y\) may be correlated, \(Y^2\) is uncorrelated with \(X\). Uncorrelated does not imply independence unless \( \rho=0 \).

21. Covariance in Bivariate Normal

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