Expectation of a Multivariate Random Variable of Standard Gaussian Distribution

Question

Let $(X_1,...,X_n)^T$ be a multivariate r.v. with standard gaussian distribution on $\mathbb{R}^n$

What is $\mathbb{E}(X_1^2 | X_1 + ... + X_n)?$

Answer 1

Consider the variable $(X_1,Y)$, where $Y=X_1+\dots + X_n$. Notice that $(X_1,Y)\sim N_2(0,\begin{pmatrix}1 & 1 \\ 1 & n\end{pmatrix})$ and using the conditional distribution for a bivariate normal distribution we get $$X_1|Y=y \sim N(\frac{y}{n} , 1-\frac{1}{n}).$$ If we for fixed $y$ consider a random variable $Z_y \sim N(\frac{y}{n} , 1-\frac{1}{n})$, then the distribution of $X_1^2|Y=y$ is the same as the distribution of $Z_y^2$, and therefore $$\mathbb{E}[X_1^2 \: | \: Y=y] = \mathbb{E}[Z_y^2] = (1-\frac{1}{n}) + \frac{y^2}{n^2}.$$ Finally we conclude that $$\mathbb{E}[X_1^2 \: | \: X_1 + \dots + X_n] = (1-\frac1n)+\frac{(X_1+\dots+X_n)^2}{n^2}.$$