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

$\blacksquare$