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

and hence

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