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 $X$ ?


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