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 ∫∞01−F(x)dx<∞, and in that case,
E(X)=∫∞01−F(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 ∫∞01−F(x)dx and ∫0−∞F(x)dx both converge, and in that case
E(X)=∫∞01−F(x)dx−∫0−∞F(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)=∫∞01−F(x)dx−∫0−∞F(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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