is larger than or equal to 0 as the random variable 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.
Property 1:
Given a non-negative random variable , the conditional expectation because . Also, probabilities are always non-negative, i.e., . Thus, the product:
.
This is intuitive since conditioning on still results in non-negative values, ensuring the product remains non-negative.
Property 2:
For , the expected value given is at least . Multiplying both sides by , we get:
.
This is intuitive since all values considered are at least , making their average also greater than or equal to .
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