Conditional expected value, joint distribution, independent variables

Question

For $x_1,...,x_n\in N(a,1)$, i.i.d., is it correct that $$E\left(x_1 \,\Big\lvert\, \frac{x_1+\cdots+x_n}{n}=M\right)=\int_ℝ \frac{xf_{x_1}(x)f_{x_2+\cdots+x_n}(Mn-x)}{f_{\frac {x_1+\cdots+x_n}{n}}(M)}dx\,?$$

I'm doing an exercise in statistics and feeling a bit lost. After that I substitute Normal Distributions' pdf-s and try to solve the integral, but with no success.

Answer 1

Suppose that $x_1,\dots,x_n$ are i.i.d. (not necessarily normal), then by symmetry \begin{align*} \mathbb E[x_1|x_1+\dots+x_n = nM] &= \frac{1}{n} \sum_{i=1}^n \mathbb E[x_i|x_1+\dots+x_n = nM]\\ &=\frac{1}{n} E\left[\sum_{i=1}^nx_i\middle|x_1+\dots+x_n = nM\right]\\ &= M \end{align*}