Conditional expectation of maximum given the sum

Question

Let $X_1,X_2$ be i.i.d. random variables with uniform law on $\{ 1, \dots N \}$. I want to compute

$$ E \left[ \max \{ X_1, X_2 \} \vert X_1 + X_2 \right].$$

How do I approach this? Do I need to consider particular cases?

Answer 1

Guide:

For $k=2,3,\dots,2N-1,2N$ find the function $f$ prescribed by:

$$f\left(k\right):=\mathbb{E}\left[\max\left\{ X_{1},X_{2}\right\} \mid X_{1}+X_{2}=k\right]=$$$$\sum_{i=1}^{N}\sum_{j=1}^{N}\max\left\{ i,j\right\} P\left(X_{1}=i,X_{2}=j\mid X_{1}+X_{2}=k\right)$$

Then: $$\mathbb{E}\left[\max\left\{ X_{1},X_{2}\right\} \mid X_{1}+X_{2}\right]=f\left(X_{1}+X_{2}\right)$$