On An Intriguing Distributional Identity

For a continuous random variable X with support equal to (a, b), with c.d.f. F, and g: .1 . .2 a continuous, strictly increasing function, such that .1..2.(a, b), but otherwise arbitrary, we establish that the random variables F(X) . F(g(X)) and F(g. 1(X)) . F(X) have the same distribution. Further developments, accompanied by illustrations and observations, address as well the equidistribution identity U . .(U) = d.. 1(U) . U for U . U(0, 1), where . is a continuous, strictly increasing and onto function, but otherwise arbitrary. Finally, we expand on applications with connections to variance reduction techniques, the discrepancy between distributions, and a risk identity in predictive density estimation.