Linear combination of coordinates of random unit vector

Question

Let $v\in \mathbb{R}^n$ be uniformly distributed on the unit sphere. Let $\lambda_1,...,\lambda_n$ be given real numbers. What is the distribution of $$X=\sum_{i=1}^n\lambda_iv_i^2\;?$$ Does it happen to belong to any known family of distributions? I think this is a very flexible way to model the distribution with compact support. When $n=2$, $X$ is just the celebrated arcsine distribution supported on $(\lambda_{\min},\lambda_{\max})$. What about for general $n$? I also think $X$ can capture the ''spreadness'' of the sequence $\lambda_1,..,\lambda_n$.

Answer 1

The distribution of $X$ is the distribution of the ratio $$\frac{\sum_{i=1}^n\lambda_iZ_i^2}{\sum_{i=1}^nZ_i^2} $$ of two quadratic forms in iid standard normal random variables $Z_1,\dots,Z_n$ (because the distribution of $(v_1,\dots,v_n)$ is the same as that of $(Z_1,\dots,Z_n)\big/\sqrt{\sum_{i=1}^nZ_i^2}$). The distribution of such ratios was studied by Gurland; also see e.g. Watson.