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I came up with a question: What is the distribution of $(X−Y)^2+(Z−Y)^2$, where $X$,$Y$ and $Z$ are independent normal distributions with their own means and variance? The common part is $Y$ in both of the summands that introduces correlations, otherwise it is χ2. Thanks for the help!

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  • $\begingroup$ How does the histogram look like if you sample, say $10^5$, observations from $X, Y, Z$, compute and plot that expression? Maybe you can guess from that empirical data. $\endgroup$
    – TrungDung
    Dec 18 '20 at 8:31
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    $\begingroup$ @Xi'an HI, i think the link you provided works only for independent variables, but in my case they are not. $\endgroup$
    – Jing Tang
    Dec 18 '20 at 11:48
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    $\begingroup$ The highly upvoted answer by djs there points out that this is a generalized chi-squared distribution and offers links to literature. $\endgroup$
    – whuber
    Dec 18 '20 at 13:17
  • $\begingroup$ Dear whuber, I don't find the answer by djs. Could you please help? Thanks. $\endgroup$
    – Jing Tang
    Dec 18 '20 at 15:46
  • $\begingroup$ The answer by djs is here. $\endgroup$ Dec 18 '20 at 16:10