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.