Expectation of Function of Discrete Random Variable

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Theorem

Let $X$ be a discrete random variable.

Let $\expect X$ be the expectation of $X$.

Let $g: \R \to \R$ be a real function.


Then:

$\displaystyle \expect {g \sqbrk X} = \sum_{x \mathop \in \Omega_X} \map g x \, \map \Pr {X = x}$

whenever the sum is absolutely convergent.


Proof

Let $\Omega_X = \Img X = I$.

Let $Y = g \sqbrk X$.

Thus:

$\Omega_Y = \Img Y = g \sqbrk I$

So:

\(\ds \expect Y\) \(=\) \(\ds \sum_{y \mathop \in g \sqbrk I} y \, \map \Pr {Y = y}\)
\(\ds \) \(=\) \(\ds \sum_{y \mathop \in g \sqbrk I} y \sum_{ {x \mathop \in I} \atop {\map g x \mathop = y} } \map \Pr {X = x}\) Probability Mass Function of Function of Discrete Random Variable
\(\ds \) \(=\) \(\ds \sum_{x \mathop \in I} \map g x \, \map \Pr {X = x}\)

From the definition of expectation, this last sum applies only when the last sum is absolutely convergent.

$\blacksquare$


Sources