TITLE: Generalized $\chi^2$ distribution?
QUESTION [5 upvotes]: Consider each $X_i \sim N(0,1)$. Then the random variable $Y=\sum_{i=1}^n X_i^2$ is a $\chi^2$ distribution with $n$ degree of freedom. Is there any probability distribution about a weighted sum of the square of standard normal random variables $Z=\sum_{i=1}^n w_i X_i^2$?

REPLY [2 votes]: Distributions of norm squared of multi-normal distribution had been extensively studied in radio-communication theory. See paper of S.O. Rice, "Distribution of Quadratic Forms in Normal Random Variables".
It is quite easy to find the characteristic function of $Z$, since
$$
  \mathbb{E} \left( \mathrm{e}^{i t X_i^2} \right) = \frac{1}{\sqrt{1-2 i t}} 
$$
Therefore
$$
  \mathbb{E} \left( \mathrm{e}^{i t Z} \right) = \left( \prod_{i=1}^n 1-2 i t w_i \right)^{-1/2}
$$
Inversion in closed form is possible only in special cases.