Conditional expectation of product of independent RVs

Question

Let $A$, $B$, $\epsilon_a$ and $\epsilon_b$ be independent normal random variables:

$A \sim N(\mu_A, \sigma^{2}_A)$

$B \sim N(\mu_B, \sigma^{2}_B)$

$\epsilon_a \sim N(0, \sigma^{2}_{\epsilon_A})$

$\epsilon_b \sim N(0, \sigma^{2}_{\epsilon_B})$

Let Z be defined as $Z \equiv A + B + \epsilon_a + \epsilon_b$.

I am interested in computing the expectation: $E(AB|Z)$.

From questions like this I realize that I (most likely) cannot simply write $E(AB|Z) = E(A|Z)E(B|Z)$, which would be easy to solve using the bivariate normal distribution.

How can I go about computing $E(AB|Z)$ in terms of means and variances?

Answer 1

One can deduce the desired identity from the following pair of results.

Let $X$, $Y$ and $Z$ denote independent centered normal random variables with respective variances $x^2$, $y^2$, and $z^2$.

Linear case: Let $S=X+Y$. There exists some real numbers $(a,u)$ such that $$X=aS+uU$$ where $U$ is standard normal and independent of $S$. Thus,

$$E(X\mid S)=aS$$ The parameter $a$ solves $$x^2=a(x^2+y^2)$$

Quadratic case: Let $T=X+Y+Z$. There exists some real numbers $(a,b,u,v,w)$ such that $$X=aT+uU\qquad Y=bT+vU+wW$$ where $U$ and $W$ are standard normal independent random variables, independent of $T$. Thus, $$XY=abT^2+((av+b)uU+awW)T+uvU^2+uwUW$$ which implies that

$$E(XY\mid T)=abT^2+uv$$ The parameters $(a,b,uv)$ solve $$x^2=a(x^2+y^2+z^2)\qquad y^2=b(x^2+y^2+z^2)\qquad uv=-ab(x^2+y^2+z^2)$$

Note that $$E(XY\mid T)=ab(T^2-E(T^2))$$