How to find conditional expectation?

Question

We have ξ and η independent random variables with normal distribution ~ N(0,1). How to find variance of E(2ξ+η|ξ+η)?

Answer 1

First, it would seem that $$ E[2\xi+\eta|\xi+\eta] = (\xi+\eta)+E[\xi|\xi+\eta] = (\xi+\eta) + \frac{\xi+\eta}2 = \frac{3(\xi+\eta)}2. $$ In first step, I use $E[\xi+\eta|\xi+\eta] = \xi+\eta$, and in the second step, I used that $\xi$ and $\eta$ are i.i.d., so if you know their sum, your best guess for either one is the average (it is also easy to prove this fact).

It should now be simple to compute the variance straight from definition.