If has p.d.f. , then

To find the probability of a function (assuming single-valued, for now) of being in a specific range, we find the limits within which would lie, i.e.

(where .) Hence,

upon substituting . If , then

(since the Normal distribution is symmetrical). Similarly,

From the above,

We recognise that this is the p.d.f. of the chi-squared distribution with . Hence

Alexander Frolkin 2001-02-01