Conditional expectation of a gaussian variable

Question

There are two gaussian variables,$ X$ and $Y$, where $X \sim N(\mu_x,\sigma_x^2)$ and $Y \sim N(\mu_y, \sigma_y^2)$. Given that $X+Y = z$, what is the conditional expectation of $X$. That is, $E(X|X+Y = z)$? Thank.

Answer 1

Assuming that initially $X$ and $Y$ are independent, then $$E(X|X+Y = z) = \mu_x + (z - \mu_x - \mu_y)\frac{ \sigma_x^2}{\sigma_x^2+\sigma_y^2}$$ and subject to the constraint $X+Y=z$, you will have $X$ conditionally normally distributed with that mean and with variance $\dfrac{\sigma_x^2 \sigma_y^2}{\sigma_x^2 + \sigma_y^2}$ with the conditional variance curiously not depending on $z$.

See my question on Cross Validated for an excellent answer from whuber explaining why this is the conditional expectation.