# 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:

 $\ds \expect X$ $=$ $\ds \sum_{x \mathop \in \map {\mathrm {supp} } X} x \map \Pr {X = x}$ Definition of Expectation of Discrete Random Variable $\ds$ $=$ $\ds c \map \Pr {X = c}$ $\ds$ $=$ $\ds c$

$\blacksquare$