Mean of the partial sum of the normalized Gaussian random vector

Question

Let $X_1, X_2, \dots, X_d$ be $d$ independent $N(0, 1)$ random variables, and let $Y=\frac{1}{\|X\|}X$, where $X=(X_1, X_2, \dots, X_d)$. Let the vector $Z \in \mathbf{R}^{k}$ be the projection of $Y$ onto its first $k$ coordinates, and let $L=\|Z\|^2$. Show that $\mathbf{E}(L)=k/d$.

Answer 1

Note that $1=\|Y\|^2=\sum_{i=1}^d Y_i^2$. By symmetry (there's a lot to go around) $EY_i^2$ is independent of $i$ and so $EY_i^2 = 1/d$. But $L=\|Z\|^2=\sum_{i=1}^kY_i^2$, so $EL=k/d$.