I have $(X_i)$ an i.i.d sequence of random variables in $L^1$.
Now I want to compute $\Bbb{E}\left(\sum_{k=1}^n X_k|X_1\right)$. I know that this is equal to $\sum_{k=1}^n\Bbb{E}\left( X_k|X_1\right)$ i.e. the problem reduces to compute $\Bbb{E}(X_k|X_1)$. But somehow I don't see how I can compute this conditional expectation. Could maybe someone help me?
We have the following definition:
If $X\in L^1$ and $B\subset A$ a sub sigma algebra, then $\Bbb{E}(X|B)$ is the unique r.v. $\xi\in L^1$ such that for all $Q\in L^\infty$ $$\Bbb{E}(XQ)=\Bbb{E}(\xi Q)$$
Best Answer
Since the random variables are independent, the conditional expectation equals the expectation for $k\ge 2$, i.e. $\mathbb E[X_k|X_1]=\mathbb E[X_k]$ almost surely. Since the random variables have the same law, we have $\mathbb E[X_k]=\mathbb E[X_1]$, so we have $\mathbb E[\sum_{k=1}^nX_k|X_1]=X_1+(n-1)\mathbb E[X_1]$ almost surely, where we used $\mathbb E[X_1|X_1]=X_1$ (since $X_1$ is $X_1$-measurable).