Expectation of Bounded Random Variable
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Theorem
Let $X$ be a random variable.
Let $a$ and $b$ be real numbers with $b \ge a$.
Let:
- $\map \Pr {a \le X \le b} = 1$
Then:
- $a \le \expect X \le b$
where $\expect X$ denotes the expectation of $X$.
Proof
From:
- $\map \Pr {a \le X \le b} = 1$
it follows that:
- $\map \Pr {X \ge a} = 1$
That is:
- $\map \Pr {X - a \ge 0} = 1$
From Expectation of Non-Negative Random Variable is Non-Negative, we therefore have that:
- $\expect {X - a} \ge 0$
From Expectation of Linear Transformation of Random Variable, we have:
- $\expect X - a \ge 0$
That is:
- $\expect X \ge a$
Note that we also have:
- $\map \Pr {X \le b} = 1$
That is:
- $\map \Pr {b - X \ge 0} = 1$
Again applying Expectation of Non-Negative Random Variable is Non-Negative, we have:
- $\expect {b - X} \ge 0$
From Expectation of Linear Transformation of Random Variable, we have:
- $b - \expect X \ge 0$
giving:
- $\expect X \le b$
Putting these inequalities together, we have:
- $a \le \expect X \le b$
$\blacksquare$