\begin{definition}[Definition:Expectation/Discrete]
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space. 
Let $X$ be a real-valued discrete random variable on $\struct {\Omega, \Sigma, \Pr}$. 
The '''expectation of $X$''', written $\expect X$, is defined as:
:$\expect X := \ds \sum_{x \mathop \in \image X} x \map \Pr {X = x}$
whenever the sum is absolutely convergent, that is, when:
:$\ds \sum_{x \mathop \in \image X} \size {x \map \Pr {X = x} } < \infty$
where $\map \Pr {X = x}$ is the probability mass function of $X$.
Note that the index of summation does not actually need to be limited to the image of $X$, as:
:$\forall x \in \R: x \notin \image X \implies \map \Pr {X = x} = 0$
Hence we can express the expectation as:
:$\expect X := \ds \sum_{x \mathop \in \R} x \map \Pr {X = x}$
Also, from the definition of probability mass function, we see it can also be written:
:$\expect X:= \ds \sum_{x \mathop \in \R} x \map {p_X} x$
\end{definition}