1

I want to analyze the distribution of $$X = \sum_i X_i^2$$ where independent $X_i \sim \mathcal{N}(\mu_i, \sigma_i^2)$.

If $\mu_i=0$, I can derive the distribution by passing Gamma distribution like this answer.

If $\sigma_i^2=1$, $X$ is non central chi-squared.

But for general means and variances, is there any method to derive the distribution of $X$ ?

kjetil b halvorsen
  • 63,378
  • 26
  • 142
  • 467
Sabrina Cantu
  • 11
  • 1
  • 1
    When all the $\sigma_i^2$ are equal, the answer is at https://stats.stackexchange.com/questions/116334/why-are-these-two-random-variables-identically-distributed/116336#116336. Otherwise the answer is complicated and requires approximation, as indicated by the situation where all the $\mu_i=0$: https://stats.stackexchange.com/questions/72479/generic-sum-of-gamma-random-variables/72486#72486 – whuber Jan 27 '18 at 16:28

0 Answers0