Expectation of Function of Discrete Random Variable

Theorem

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

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

Then:

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

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 $

$=$

$\ds \sum_{y \mathop \in g \sqbrk I} y \sum_{ {x \mathop \in I} \atop {\map g x \mathop = y} } \map \Pr {X = x}$

$\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.

$\blacksquare$