Sunday, July 14, 2019

probability - Expectation of a random variable in terms of its distribution function

Here is a theorem on expectation of a random variable in terms of its distribution function





Theroem: Let X be a (continuous or discrete) non-negative random variable with distribution function F. Then, E(|X|)< if and only if 01F(x)dx<, and in that case,
E(X)=01F(x)dx.




Then, a corollary of the Theorem is given as:




Corollary: For any random variable X, E(|X|)< if and only if the integrals 01F(x)dx and 0F(x)dx both converge, and in that case
E(X)=01F(x)dx0F(x)dx





I understand the Theorem, but I do not see how the Corollary follows from the Theorem. I understand the first claim of the Corollary, but I do not see why
E(X)=01F(x)dx0F(x)dx
holds in that case.



I have that:



E(|X|)=0P{|X|>x}dx=0P{X>x}+0P{X<x}dx=0P{X>x}0P{X<x}dx,
but then I could not conclude (1) since the integrand in the second integral of the last line in (2) is P{X<x}, which is equal to F(x) if X is a continuous random variable, but not equal to F(x) if X is discrete variable.



What am I missing?

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