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Abstract If a series u1, u2,...u2n + 1 constitute a random sample from a normally distributed population, they any linear function A = S 12n + 1 (arur) will also be normally distributed; moreover its mean will be zero if S(ar) = 0, and its variance will be equal to that of the original population if S (ar2) = 1. Any other liner function B = S 12n + 1 (brur) will be distributed independently of the first if S(arbr) = 0, and in this case the sum of the squares, x = A2 + B2, will be distributed so that the chance of exceeding any particular value of x is e -x/e, where c is the mean value of x, equal to twice the variance of the population sampled.
Ronald Aylmer Fisher (Thu,) studied this question.