Conditional expectation of independent identically distributed random variables

Question

Let $X, Y, Z$ -- independent identically distributed random variables.

I need to calculate the conditional expectation $\mathbb{E}(3X - 3Y + Z | X + Y + Z)$.

I use linearity property: $3\mathbb{E}(X| X + Y + Z) - 3\mathbb{E}(Y | X + Y + Z) + \mathbb{E}( Z | X + Y + Z)$. But what about the right side conditional expectation?

Answer 1

Use the formula $E(X|X+Y+Z)=E(Y|X+Y+Z)=E(Z|X+Y+Z)=\frac13(X+Y+Z)$, just like on the seminar. :)