The exact distribution of a statistic based on the $r$ smallest of $n$ independent observations from a unit uniform distribution is derived. In life-testing terminology, this statistic includes as special cases (i) the sum of the $r$ earliest failure times, (ii) the total observed life up to the $r$th failure, and (iii) the sum of all $n$ failure times. The density, cumulative distribution function (c.d.f.) and first four moments of the general statistic are summarized in Sec. 2. Section 3 gives the derivation of the density and c.d.f. The moments are obtained from the moment generating function in Sec. 4. Asymptotic normality under certain conditions is proved in Sec. 5 and illustrations of the rapidity of approach to normality are given in Sec. 6.