Given $X\_i \sim \mathcal{N}(\mu\_i,\sigma\_i^2)$, for $i = 1,\dots,n$. How does one find the distribution of $D = \sum\_{i=1}^n X\_i^2$? In the case that all the standard deviations are the same (i.e. $\sigma\_i = \sigma\_1$ for all $i$), the random variable $D/\sigma\_1^2$ has a [noncentral chi-squared distribution](http://en.wikipedia.org/wiki/Noncentral_chi-squared_distribution). But when I don't have this simplifying assumption I don't know what to do. Is there a well known distribution for this?



Sum of Squares of Normal distributions
All you could conceivably want to know about the subject (and many things you might not) are in Mathai + Provost, Quadratic Forms in Random Variables.


