Theorem: Let F(x) be the distribution function of X.
If X is any r.v. (discrete, continuous, or mixed) defined on the interval [a,∞) (or some subset of it), then
∫ [1 - F(x)]dx + a
1) Is this formula true for any real number a? In particular, is it true for a<0?
2) When is this formula ever useful (computationally)? Why don't just get the density function then integrate to find E(X)?
Thanks for clarifying!
[also under discussion in talk stats forum]