Expectation of Almost Surely Constant Random Variable

Theorem
Let $X$ be an almost surely constant random variable.

That is, there exists some $c \in \R$ such that:


 * $\map \Pr {X = c} = 1$

Then:


 * $\expect X = c$

Proof
Note that since $\map \Pr {X = c} = 1$, we have $\map \Pr {X \ne c} = 0$ from Probability of Event not Occurring.

Therefore:


 * $\map {\mathrm {supp} } X = \set c$

We therefore have: