Suppose that X1, X2 are i.i.d. exponential random variables with mean 1, i.e., their densities are given by $e^{-x}$ for x > 0. Compute
$E[X_1|X_1 + X_2]$.
$P(X-1 < 3|X-1 + X-2)$.
$E[X-1| \min(X-1, t)]$.
$E[X-1| \max(X1, t)]$.
My work:For the first one, I think since $X_1$ is measurable on $\sigma(X_1+X_2)$ then $E[X_1|X_1+X_2]=E[X_1]$. And this imply that $X_1$ is independent to $X_1 + X_2$, which is not make sense. I am wondering where I am wrong here.
My solution for the first one
$$\mathbb{E}[X_1|X_1+X_2]+\mathbb{E}[X_2|X_1+X_2]=\mathbb{E}[X_1+X_2|X_1+X_2]=x_1+x_2$$
As per independence the rv are also exchangeable, thus
$$\mathbb{E}[X_1|X_1+X_2]=\frac{x_1+x_2}{2}$$