Expectation Preserves Inequality

Theorem
Let $X$, $Y$ be random variables.

Let $\map \Pr {X \ge Y} = 1$.

Then:


 * $\expect X \ge \expect Y$

Proof
Note that we have:


 * $\map \Pr {X - Y \ge 0} = 1$

From Expectation of Non-Negative Random Variable is Non-Negative, we then have:


 * $\expect {X - Y} \ge 0$

From Sum of Expectations of Independent Trials, we have:


 * $\expect X + \expect {-Y} \ge 0$

From Expectation of Linear Transformation of Random Variable, we have:


 * $\expect X - \expect Y \ge 0$

That is:


 * $\expect X \ge \expect Y$