Do there exist any systems for symbolically solving expectations?
This is sort of a follow-up to my question List of Tricks for Solving Messy Expectations? Basically, I'm looking for ways to solve a messy expectation after I've exhausted all obvious routes.
EDIT: BACKGROUND
I'm trying to solve the following for $\alpha$ (constrained to be greater than 0) as a function of $\sigma_X^2$, $\sigma_Y^2$, and $p$
$E\left[\ln(F) F^\alpha X^2 (X + Y)^2 + \ln(F) F^{2\alpha}(X + Y)^4\right] = 0$
where:
$Y \sim N(0,\sigma_Y^2)$
$ X = \left\{ \begin{array}{cc} N(0,\sigma_X^2) & p \\ 0 & (1-p) \end{array}\right.$ where it's assumed $\sigma_X^2 \gg \sigma_Y^2$ and that $p$ is very small. (i.e., $X$ is a jump process with most of its mass at 0)
$F = F_{|Z|}(|X+Y|)$ where $Z \sim N(0,\sigma_Y^2)$ (i.e., $F$ is the CDF for the absolute value of a normal)
EDIT: EVEN MORE BACKGROUND
The equation I'm trying to solve above is the first order condition for the min-MSE problem:
$\min_{\alpha > 0} \left(X^2 - \widehat{X^2}\right)^2 $
where $ \widehat{X^2} = F_{|Z|}\left(|R|\right)^{\alpha}R^2$ and $R = X + Y\,$ is the only observed variable.
Basically, I'm trying to estimate the square of the jump, $X^2$ (given that I can only observe the aggregated process $R$) by smoothing down $R^2$. If $|R|$ is large, the smoothing function $F_{|Z|}\left(|R|\right)^{\alpha}$ should be close to 1 and the estimate of $X^2$ would be close to $R^2$. If $|R|$ is small, the estimate of $X^2$ would be close to 0.
