Definition:Expectation/Discrete

Definition

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:

$\expect X:= \ds \sum_{x \mathop \in \R} x \map {p_X} x$

Also known as

The expectation of a random variable $X$ is also called the expected value of $X$ or the mean of $X$, and (for a given random variable) is often denoted $\mu$.

The terminology is appropriate, as it can be seen that an expectation is an example of a normalized weighted mean.

This follows from the fact that a probability mass function is a normalized weight function.

Various forms of $E$ can be seen to denote expectation:

$\map E X$
$\map {\mathrm E} X$
$E \sqbrk X$
$\mathop {\mathbb E} \sqbrk X$

and so on.

Also see

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

Historical Note

The concept of expectation was first introduced by Christiaan Huygens in his De Ratiociniis in Ludo Aleae ($1657$).

The notation $\expect X$ was coined by William Allen Whitworth in his Choice and Chance: An Elementary Treatise on Permutations, Combinations, and Probability, 5th ed. of $1901$.

Linguistic Note

Don't you dare call it expectoration, you disgusting children.

Technical Note

The $\LaTeX$ code for $\expect {X}$ is \expect {X} .

When the argument is a single character, it is usual to omit the braces:

\expect X