Definition:Expectation

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Definition

The expectation of a random variable is the arithmetic mean of its values.


Its precise definition depends on the nature of the random variable, as follows:


Discrete Random Variable

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:

$\displaystyle \sum_{x \mathop \in \image X} \size {x \, \map \Pr {X = x} } < \infty$


Continuous Random Variable

Let $X$ be a continuous random variable over the probability space $\struct {\Omega, \Sigma, \Pr}$.

Let $F = \map \Pr {X < x}$ be the cumulative probability function of $X$.


The expectation of $X$ is written $\expect X$, and is defined over the probability measure as:

$\expect X := \displaystyle \int_{x \mathop \in \Omega} x \rd F$

whenever the integral is absolutely convergent, i.e. when:

$\displaystyle \int_{x \mathop \in \Omega} \size x \rd F < \infty$


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:

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

and so on.


Also see


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


Sources