On solutions to min (X,{text{ Y}})mathop = limits^d aX and min (X,{text{ Y}})mathop = limits^d aXmathop = limits^d bY

Suppose that the minimum of a pair of independent non-negative random variables X and y has the same distribution, up to a scale factor, as the first of the two random variables. The restricted class of possible distributions for X and Y is identified. If in addition it is required that X and Y have distributions only differing by a scale factor, it is shown under mild regularity conditions that X and Y have Weibull distributions.