Question

Prove sum of squares of gaussian to be chi distribution

Answer 1

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Theorem.  Let Xi denote n independent random variables that follow these chi-square distributions:  

X1∼χ2(r1)X2∼χ2(r2)⋮Xn∼χ2(rn)

Then, the sum of the random variables:

Y=X1+X2+⋯+Xn

follows a chi-square distribution with r1 + r2 + ... + rn degrees of freedom. That is:

Y∼χ2(r1+r2+⋯+rn)

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