A remarkable method for measuring half-lives of radioactive nuclei was proposed several years ago that entailed statistical sampling of the source activity. A histogram of half-life estimates, calculated from pairs of activity measurements separated in time, took the unexpected form of a nearly perfect Cauchy distribution, the midpoint of which corresponded very closely to the true value of the half-life. No theoretical justification of the method was given. In this article I derive the exact probability density function (pdf) of the two-point half-life estimates, show how (and under what conditions) a Cauchy distribution emerges from the exact pdf—which, mathematically, shows no resemblance to a Cauchy function—and discuss the utility of the statistical sampling method. The analysis shows that the exact pdf, under the conditions leading to an empirical Cauchy lineshape, is an unbiased estimator of the true half-life.