Why Expected squared length of a projected vector on reduced dimensionality coordinates is k/d?

Question

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?

Answer 1

This not really appropriate for MO, but the proof follows immediately from additivity of expectation. The expected length squared of the vector is $L,$ that means that the expected square of a coordinate is $L/d,$ and the sum of $k$ of them is $Lk/d.$