# Expectation of Almost Surely Constant Random Variable

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## 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 | |||||||||||

\(\ds \) | \(=\) | \(\ds c \map \Pr {X = c}\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds c\) |

$\blacksquare$