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Suppose that $X$ and $Y$ are scalar random variables that are jointly normally distributed:

$$ \begin{pmatrix} X \\ Y \end{pmatrix} \sim N \left( \begin{pmatrix} \mu_x \\ \mu_y \end{pmatrix}, \begin{pmatrix} \sigma_x^2 & \sigma_{xy} \\ \sigma_{xy} & \sigma_y^2 \end{pmatrix} \right) $$

I would like a formula for $\operatorname{Var}(XY)$ in terms of $\mu_x, \mu_y, \sigma_x^2, \sigma_{xy}, \sigma_y^2$. The answers at Variance of product of k correlated random variables are too general and don't help much with this particular problem. The second answer at Variance of product of dependent variables starts but doesn't finish the derivation.

I can reduce to the problem of finding $E[X^2Y^2]$, using the definition of the covariance $\sigma_{xy}$: \begin{align*} \operatorname{Var}(XY) &= E[ (XY)^2 ] - E[XY]^2 \\ &= E[(XY)^2] - (\sigma_{xy}+\mu_x\mu_y)^2 \end{align*}

I have an idea about how to do that, but I'm not sure if it's correct. Write $Y = E[Y \mid X] + \varepsilon$ where $\varepsilon \sim N(0, \sigma_\varepsilon^2)$ and is independent of $X$. In fact, $$ E[Y \mid X] = \mu_y + \frac{\sigma_{xy}}{\sigma_x^2} (X - \mu_x) $$ so $\sigma_\varepsilon^2 = \sigma_y^2 - (\sigma_{xy})^2/\sigma_x^2$. Then \begin{align*} Y^2 &= E[Y \mid X]^2 + 2 E[Y \mid X] \,\varepsilon + \varepsilon^2 \\ E[X^2Y^2] &= E[ X^2 \, E[Y \mid X]^2 ] + E[ 2 X^2 E[Y \mid X] \, \varepsilon] + E[X^2 \varepsilon^2 ]. \end{align*}

The third term is just $$ E[X^2] E[\varepsilon^2] = (\sigma_x^2 + \mu_x^2)\sigma_\varepsilon^2. $$ The second term is zero. For the first term, rewrite $E[Y \mid X] = \alpha + \beta X$, where $\beta = \sigma_{xy}/\sigma_x^2$ and $\alpha = \mu_y - \beta \mu_x$, to get $$ \alpha^2 \, E[X^2] + 2 \alpha \beta \, E[X^3] + \beta^2 \, E[X^4] $$ or $$ \alpha^2 (\mu_x^2 + \sigma_x^2) + 2 \alpha \beta (\mu_x^3 + 3 \mu_x \sigma_x^2) + \beta^2 (\mu_x^4 + 6 \mu_x^2 \sigma_x^2 + 3 \sigma_x^4). $$ Putting these all together and simplifying: $$ E[(XY)^2] = 2 (\sigma_{xy})^2 + 4 \sigma_{xy} \mu_x \mu_y + (\mu_x^2 + \sigma_x^2)(\mu_y^2 + \sigma_y^2). $$ Two questions:

  1. Is the derivation here correct?
  2. Is there a simpler way to derive this result?
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    $\begingroup$ In what sense could those references be "too general"? The second one directly answers your question; or just set $k=2$ in the first one and you have your answer. Also, they both come with shorter derivations and therefore answer your second question. $\endgroup$
    – whuber ♦
    Commented Aug 10, 2018 at 19:37
  • $\begingroup$ Thanks @whuber. I don't see how to derive this from the first reference. The second one gives a few steps and then cites "standard results that can be found in many texts and reference books" for the rest, which I was unable to find in the texts I have on hand. $\endgroup$
    – Frank
    Commented Aug 10, 2018 at 19:44
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    $\begingroup$ I don't understand, because there's nothing to be derived from the first reference: there are three answers and all you need do is set $k=2.$ $\endgroup$
    – whuber ♦
    Commented Aug 10, 2018 at 19:46
  • $\begingroup$ Am I missing something in the first reference? In the first answer, I see a formula that's in terms of objects like $E[(\delta_x)^i(\delta_y)^j]$, which need to be derived. The second requires independence. In the third, I see a formula for the case where $\mu_x = \mu_y = 0$. $\endgroup$
    – Frank
    Commented Aug 10, 2018 at 19:49
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    $\begingroup$ Are we reading the same posts? Please follow the duplicate link at the head of this thread. $\endgroup$
    – whuber ♦
    Commented Aug 10, 2018 at 19:58

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