Although this question has been extensively studied since 100 years ago, I am getting tired of not finding an answer for this in the literature:

If $X_i$s are independent normal random variables with mean $\mu_i$ and variance $\sigma_i^2$, then what is the probability density function (PDF) of $z=\sum_{i=1}^{n}X_i^2$?

Can you please give me a closed form of the PDF and the CDF?

  • 1
    $\begingroup$ I think here you have a solution to your question. $\endgroup$ – Masacroso Apr 1 '17 at 17:50
  • $\begingroup$ @Masacroso Thanks. But they are talking about standard normal random variables. In my question, I don't have normal random variables. $\endgroup$ – Susan_Math123 Apr 1 '17 at 17:57
  • $\begingroup$ If $\sigma_i$ are identical, then you have (scaled) non-central chi square. $\endgroup$ – BGM Apr 1 '17 at 20:26

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