Expectation of Bounded Random Variable

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:


 * $a \le \expect X \le b$