Given that $X\ge 0, q>0 $ and $P$ is the probability measure, I need to prove the following in a probability theory perspective:
- $E X = \int_{0}^\infty P(X>x) \,dx$
- $EX = \int_{0}^\infty xf(x) \,dx$
- $EX^q = \int_{0}^\infty qx^{q-1}P(X>x) \,dx$
In proving 1, I want to start with the definition of expected value which is: $EX = \int_{\Omega} X \,dP$, but not sure how to proceed. I checked with the other examples on this site describing this, but they use the existence of density function to prove 1. Can we prove 1 without using it? Also I think if I can prove 3 first, I can get 1 by setting $q=1$. Please help