Assume that $V$ is a vector $V=[v_1, v_2, .., v_n]$ where each element is drawn independently forom a uniform normal distribution $v_i \sim \mathcal{N}(\mu,\,\sigma^{2})$. Suppose that $\mu = 0$ and $\sigma = 1$. Let X be a normalized version of $V$ so $X=\frac{V}{|V|}$. What is the probability that there is only 1 possible selection ok $k$ elemnets out of $n$ s.t their sum is less than a constant $r$. This means that any other $k$ selection will need to have its value larger then r.
This problem is really breaking my head. I assume that after the normalisation $x$s will not be independent random variables any more but I have no idea how to even start this.