Find the expected value for these random variables.

Question

Let $X_{1}, X_{2}, ..., X_{n}$ be independent and identically distributed random variables with finite expected value. Find $\mathbb{E}[X_{1}|(X_{1}+X_{2}+ ... + X_{n})=x]$. Assuming that $f_{(X_{1}+X_{2}+ ... + X_{n})}(x)\neq 0$, I must have the expected value is $\frac{x}{n}$. ¿Could you give me a hint to do it, please?

Answer 1

By symmetry, $$\mathbb E[X_1\mid X_1+\dots+X_n]=\mathbb E[X_2\mid X_1+\dots+X_n]=\dots=\mathbb E[X_n\mid X_1+\dots+X_n].$$ But by linearity $$\sum_{i=1}^n\mathbb E\left[X_i\,\middle|\,\sum_{j=1}^n X_j=x\right]=\mathbb E\left[\sum_{i=1}^nX_i\,\middle|\,\sum_{j=1}^nX_j=x\right]=x,$$ so we have $\mathbb E[X_i\mid X_1+\dots+X_n=x]=\frac{x}{n}$.