Expectation Preserves Inequality
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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:
- $\expect X \ge \expect Y$
$\blacksquare$