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 $\lambda\_1,..,\lambda\_n$. 



Linear combination of coordinates of random unit vector
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](https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=3&ved=2ahUKEwiDoeDbrP3hAhUOEawKHTKNBtsQFjACegQIAxAC&url=https%3A%2F%2Fcowles.yale.edu%2Fsites%2Fdefault%2Ffiles%2Ffiles%2Fpub%2Fcdp%2Fs-0354.pdf&usg=AOvVaw1xMd5R-PTGs0rvjHtme9W8); also see e.g. [Watson](http://adsabs.harvard.edu/full/1955AuJPh...8..402W). 


