Let $D:=\{(x,\dots,x)\mid x\in\mathbb R\}\subset\mathbb R^d$ and $X$ be a random variable and normally distributed $X\sim N(\mu,\sigma^2 E_d) $ with $\sigma\neq 0$, $\mu\in\mathbb R^d$ and $E_d$ the $d\times d$ identity matrix.
How can you calculate the distribution of the ortogonal projection $P_DX $ from X onto D?
$P_D$ is a symmetric idempotent matrix, i.e. $P_D^2 = P_D^T=P_D$. So if $X\sim N(\mu,\sigma^2 E_d)$, then $$ P_D X\sim N(P_D\mu,\, P_D(\sigma^2 E_D)P_X^T) = N(P_D\mu,\,\sigma^2 P_D). $$ (Generally if $X\sim N(\mu,\,V)$ then we would have $AX\sim N(A\mu,\,AVA^T)$.)
(Since it's the projection onto $\{(x,\ldots,x)\mid x\in\mathbb R\}$, I suppose I should mention that the projection of any point $(x_1,\ldots,x_n)$ onto that particular subspace is $(\bar x,\ldots,\bar x)$ where $\bar x = (x_1+\cdots+x_n)/n$.)