Expectation of scaled sum of squares of iid random variables

Question

Let $X_1, \dots, X_n$ be iid standard normal random variables. Consider the vector $X = (X_1, \dots, X_n)$ and the vector

$Y = \frac{1}{\|X\|}(X_1, \dots, X_k)$

for $k < n$. What is $\mathbb{E}[\|Y\|^2] = \mathbb{E}\left[\frac{\sum_{i=1}^k X_i^2}{\sum_{i=1}^n X_i^2} \right]$?

Answer 1

I don't know the closed form of this but concentration of measure phenomenon ensures that the expectation will converge to $\frac{k}{n}$ exponentially fast in $k$. So using $k/n$ might be a reasonable approximation.