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I am struggling trying to find the (equation of the) pdf of the sum of (what I believe to be) two non-central chi-squared random variables.

The formula given on wikipedia (http://en.wikipedia.org/wiki/Noncentral_chi-squared_distribution) shows that the random variable associated with a sum of squared normal variables normalized by their variances is "non-central chi squared". The problem I am having is that I do not want to normalize my normal random variables by their variances; I just want to add together the squared values. For example, if X1 and X2 are normally distributed random variables with non identical means and non identical variances, what is the pdf of X1^2+X2^2?

So far my solution has been to calculate the individual pdfs of X1^2 and X2^2 and perform a numerical convolution to find the pdf X1^2+X2^2 - it just seems silly not to have an analytical solution if one has already been created for this exact problem.

Thanks,

Jeff.

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# Difficulty with summation of non-central chi-squared random variables

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