Inverting Khintchine’s relationship and generating length biased data

Abstract

If X>0 follows a distribution with decreasing density, then Khintchine’s theorem states that it has the same distribution as U×S where U and S>0 are independent, U following the uniform distribution on (0,1). In this letter, an explicit function of X and independent V∼U(0,1) is discovered which has the same distribution as S. This result is then used to find an explicit function of two independent uniform random variables which follows the length biased form of a general distribution on R+ with finite mean.