In Pitman's Probability, the tail sum formula for expectation is introduced for a nonnegative (0,1,...) discrete random variable $X$:
$E(X) = \sum_{i=0}^\infty P(X > i).$
I wonder if there is a similar formula for nonnegative continuous random variable $X$:
$E(X) = \int_0^\infty P(X > x) dx?$
If no, are there some conditions for it to hold? And how can it be proved?
Here is my thought:
If the cdf $F$ of $X$ is bijective, then $X=F^{-1}(U)$ for some random variable $U$ uniformly distributed over $[0,1)$. So $E(X) = \int_0^1 F^{-1}(u) du.$
To prove the tail sum formula, it suffices to prove $\int_0^1 F^{-1}(u) du = \int_0^\infty P(X > x) dx.$ But I am stuck here.
What's more, is the condition that the cdf $F$ of $X$ is bijective really necessary for tail sum formula to hold?
- Can tail sum formula be generalized to a random variable that is not necessarily nonnegative?
Thanks!