Here is an interesting probability problem: Suppose is a random variable with absolutely continuous distribution function . What is ?
Since is absolutely continuous, is differentiable a.e. and has a density, i.e. there is a nonnegative function such that
for each , where a.e. By the law of the unconscious statistician,
Replacing with the integral of its density and applying Tonelli’s theorem, we have
Adding the two equations, we get