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This question already has an answer here:

$X$ and $Y$ are independent normal variable with zero mean and common variance $k$, Find the Probability Density of Function of $Z = X^2 + Y^2$? We can although find the PDF of a random variable if it is function of one random variable with know PDF, but how to calculate for $2$ variables. I was not able to proceed. How to use the given data of common variance?

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marked as duplicate by Math-fun, Graham Kemp probability Sep 21 '16 at 13:06

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https://en.wikipedia.org/wiki/Chi-squared_distribution

It's known as the Chi-squared distribution. The sum of squares of Gaussians is quite common so it is well studied. That's for zero mean and unit variance but to get a different variance it's just a scaling factor.

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