# Mean of the partial sum of the normalized Gaussian random vector

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 $$\mathbf{E}(L)=k/d$$.

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 $$EL=k/d$$.