Question: Deduce that, if X1, X2, . . . , Xn are independent random variables having the normal distribution with mean 0 and variance 1, then

Deduce that, if X1, X2, . . . , Xn are independent random variables having the normal distribution with mean 0 and variance 1, then Z = X2 1 + X2 2 + · · · + X2 n

has the χ2 distribution with n degrees of freedom. Hence or otherwise show that the sum of two independent random variables, having the χ2 distribution with m and n degrees of freedom, respectively, has the χ2 distribution with m + n degrees of freedom.

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