Find expectation, $\mathbb{E}[(X_1Y+X_2)^2|X_1,X_2]$

Question

Given random variables $X_1, X_2, Y$ with $\mathbb{E}[Y|X_1,X_2] = 5X_1+X_1X_2$ and $\mathbb{E}[Y^2|X_1X_2] = 25X_1^2 X_2^2 + 15$. Find $$\mathbb{E}[(X_1Y+X_2)^2|X_1,X_2].$$

What I did was to expand $(X_1Y+X_2)^2$. Without knowing $X_1, X_2, Y$ are independent, what's the next step?

Answer 1

Hint:

So, if we expand $(X_1Y+X_2)^2$ we get:

$$ X_1^2Y^2+2X_1X_2Y + X_2^2$$

Now, we can treat $X_1,X_2$ as fixed because they are given in the conditional. Therefore, when we take expectations we get:

$$E[X_1^2Y^2+2X_1X_2Y + X_2^2]=X_1^2E[Y^2] + 2X_1X_2E[Y] + X_2^2$$

Can you take it from here? The key was to realize that you are given the values for $X_1,X_2$ they are no longer random.