Definition:Expectation/Discrete

Definition
Let $X$ be a discrete random variable.

The expectation of $X$ is written $\expect X$, and is defined as:
 * $\expect X := \displaystyle \sum_{x \mathop \in \image X} x \, \map \Pr {X = x}$

whenever the sum is absolutely convergent, that is, when:
 * $\displaystyle \sum_{x \mathop \in \image X} \size {x \, \map \Pr {X = x} } < \infty$

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:
 * $\expect X := \displaystyle \sum_{x \mathop \in \R} x \, \map \Pr {X = x}$

Also, from the definition of probability mass function, we see it can also be written:
 * $\expect X:= \displaystyle \sum_{x \mathop \in \R} x \, \map {p_X} x$

Also see
It can also be seen that the expectation of a discrete random variable is its first moment.