Proof of expected value expression

[grimster]

The usual expression of the expected value of X is:
E[X] = (sum) x*p(x)

i'm supposed to show that, for X a random non-negative discrete random(stochastic) variable, we have that:
E[X]=(sum: i from 1 to infinity) P(X>=i)

i have absolutely no idea how to do this. does anyone want to push me in the right direction?

 

[mathman]

P(X>=i)=sum(j=i,inf) P(X=j). Plug this into your sum over i, interchange i and j and you will get what you want, since the sum over i will simply be j.