Let $X$ and $Y$ be jointly normal random variables with mean zero, variances $\sigma_x^2$ and $\sigma_y^2$, correlation $\rho$.

Then, given a constant $\delta$:

$\mathrm{E}\left[x\ |\ y>\delta\right]=\mathrm{E}\left[\mathrm{E}_y[x\ |\ y]\ |\ y>\delta\right] = \frac{\rho\sigma_x}{\sigma_y}\mathrm{E}[y\ |\ y>\delta]$

It is not clear to me how we went from inner expectation to the expression with $\frac{\rho\sigma_x}{\sigma_y}$ in it and factored out $x$ from the expectation.


1 Answer 1


The conditional expectation of $X$ given $Y=y$ for mean zero bivariate normal $(X,Y)$ is

$$ \frac{\rho\sigma_x}{\sigma_y}y, $$ see e.g. here. We therefore do not need to factor out $x$, it no longer shows up in the conditional expectation.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.