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This question already has an answer here:

Is there a PDF for a distribution defined as a sum of squares of random variables pulled from a family of normal distributions with different standard deviation?

Is there a way of analytically writing the PDF of such a distribution as a hypergeometric function without prior normalization to make all the underlying normal distributions have unit variance?

A possible case of application would be an N-dimensional length of displacement from center under the effect of a random, normally distributed forces along each dimension, each with it's own characteristic variance and mean, different between dimensions.

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marked as duplicate by gung - Reinstate Monica, whuber Aug 20 '14 at 21:09

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    $\begingroup$ See stats.stackexchange.com/q/67533/9964, which is basically the same question. The short answer is I don't think there's a closed form but there are approximations or computational ways to compute the PDF. $\endgroup$ – Dougal Aug 20 '14 at 20:25
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The regular noncentral chi-square, where all the SDs are equal, is messy enough to write analytically. It is a Poisson-weighted sum of central chi-square densities. That comes about as a result of applying integration by parts to the joint density of the terms. In turn, that relies on the fact that when the SDs are equal, the exponential part of that simplifies considerably from what it is when the SDs are unequal.

Closely related to all this is the Satterthwaite method, whereby linear functions of chi-squares with unequal scales are approximated by a chi-square with fractional df. This method exists because the distribution of the linear combination is analytically intractable. And that's a central chi-square case.

I think one can pretty confidently say that there is no closed-form pdf for a case where the SDs are unequal. There does exist sophisticated software such as MathStatica that could probably find it if it can be done.

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  • $\begingroup$ Do you have a source where the computation for a non-central chi-square in the simple case is carried out explicitly? $\endgroup$ – Andrei Kucharavy Aug 20 '14 at 22:14
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    $\begingroup$ The actual computation? Or the derivation? For the derivation, you can find it in many linear models texts, e.g. Monahan, A Primer on Linear Models. For computation, refer to Algorithm AS170 published in Applied Statistics. You can find it online by searching for it. $\endgroup$ – rvl Aug 21 '14 at 0:12

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