Expectation of a non negative random variable

For any non negative continuous random variable having distribution function $F(X)$ we have the followings:

1. $E[X]={\int }_0^{\mathrm{\infty }}Pr[X>t]dt$

2. $E[X^r]=r{\int }_0^{\mathrm{\infty }}{t}^{r-1}Pr[X>t]dt$

3. $E[\min (X,T)]=T-{\int }_0^{\mathrm{\infty }}F(T)𝑑t$

4. where $T$ is a constant.

Title	Expectation of a non negative random variable
Canonical name	ExpectationOfANonNegativeRandomVariable
Date of creation	2013-03-22 19:10:52
Last modified on	2013-03-22 19:10:52
Owner	georgiosl (7242)
Last modified by	georgiosl (7242)
Numerical id	7
Author	georgiosl (7242)
Entry type	Theorem
Classification	msc 60C05
Classification	msc 05A10
Classification	msc 60-00