is larger than or equal to 0 as r.v. is non-negative and is larger than or equal to because the conditional expectation only takes into account of values larger than or equal to which r.v. can take.
Hence intuitively , which directly leads to .
Probability-theoretic proof
Method 1:
From the definition of expectation:
:
However, X is a non-negative random variable thus,
:
From this we can derive,
:
From here, dividing through by allows us to see that
:
Method 2:
For any event , let be the indicator random variable of , that is, if occurs and otherwise.
Using this notation, we have if the event occurs, and if