Let X_{1,...,}X_{n }be independent and identically distributed random variables. Find E[X_{1}|X_{1}+...+X_{n}=x]
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I believe it will be $\dfrac{x}{n}$
$\displaystyle \sum_{k=1}^n~E[X_k | X_1 + \dots +X_n = x] = x$ $E[X_1 | X_k + \dots +X_n = x] = E[X_k | X_1 + \dots +X_n = x],~\forall k \in [1,n]$ Thus $E[X_1 | X_k + \dots +X_n = x] = \dfrac x n$