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Let's assume $Y=\sum_{i=1}^{N}\alpha_iX_i^2$.

where $X_i$ has Gaussian distribution with mean $0$ and variance $1$, i.e., $\mathcal{N}(0,1)$ and $\alpha_i$s are constants.

When $\alpha_i$s are all $1$, the distribution of $Y$ is Chi-Squared. My question is what is the distribution of $Y$ when $\alpha_i\neq 1$?

kjetil b halvorsen
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Jacob
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  • Re your title: when $\alpha\neq 1$ its (plainly) not chi-squared even for a single $\alpha X^2$ (consider the ratio of variance to mean for example). – Glen_b Jul 03 '18 at 23:29
  • @Glen_b So what is the distribution if it's not Chi-Squared? I couldn't find anything similar in literature. – Jacob Jul 03 '18 at 23:30
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    discussed many times on site already -- e.g. https://stats.stackexchange.com/search?q=weighted+sum+chi+squared ... for example see https://stats.stackexchange.com/questions/72479/generic-sum-of-gamma-random-variables – Glen_b Jul 03 '18 at 23:33
  • Also see https://stats.stackexchange.com/questions/147466/distribution-of-sum-of-mean-squared-errors-weighted-sum-of-chi-squared-distrib ... there's a number of others. – Glen_b Jul 03 '18 at 23:34

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