For the proof of Johnson-Lindenstrauss algorithm by Dasgupta and Gupta, they comment in their paper that for a vector $Z \in R^k $, the expected squared length, $E[L]$ (where $L = \|Z\|^2$) of the projected vector of $Y$ on first $k$ coordinates is $k/d$.

Given -

$X_1, X_2, ... , X_d$ be $d$ independent Gaussian $N(0,1)$ random variables and let $Y$ be $\frac{1}{\|x\|} (X_1, X_2, ... , X_d)$

Can someone give an elementary proof for this statement?