Formula of $\text{Var}[X|Y,Z]$ for $X\sim \mathcal N(\mu_X,\sigma_X^2)$, $Y\sim \mathcal N(\mu_Y,\sigma_Y^2)$, $Z\sim \mathcal N(\mu_Z,\sigma_Z^2)$?

Asked Apr 05 '23 at 06:40

Active Apr 05 '23 at 07:56

Viewed 42 times

How do I condition the variance of a normally distributed random variable on two other normally distributed random variables?

How do I condition the expectation of a normally distributed random variable on two other normally distributed random variables?

NOTE: $Y$ and $Z$ are correlated ($\rho_{Y,Z}$)

edited Apr 05 '23 at 07:56

asked Apr 05 '23 at 06:40

anonymous

Welcome to CV. What have you tried? Please tell us your attempts and also kindly add the [tag:self-study] to your post. – User1865345 Apr 05 '23 at 07:27
Thank you. I added self-study. I was looking into the multivariate normal distribution and the multi-dimensional projection theorem, but I am not convinced I'm on the right track and, if I were, I do not know how to apply it here. – anonymous Apr 05 '23 at 07:42
If the three together have a multivariate normal distribution with covariance matrix $\Sigma$ then you should probably say so – Henry Apr 05 '23 at 08:06
@Henry What do you mean by "together"? – anonymous Apr 05 '23 at 08:22
By together, I am saying the joint random variable $(X,Y,Z)$ has a multivariate normal distribution on its support $\mathbb R^3$ – Henry Apr 05 '23 at 09:10
I'd like my question to be addressed as comprehensively as possible, that is, in the case where the joint random variable exists but also in the case it does not (what's the difference?). – anonymous Apr 05 '23 at 10:20

0 Answers0