# Definition:Expectation

## Definition

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

The expectation of an arbitrary integrable random variable can be handled with a single definition, the general definition given here suffices for this purpose.

Particular types of random variable give convenient formulas for computing their expectation.

In particular, familiar formulas for the expectation of integrable discrete random variables (in terms of their mass function) and integrable absolutely continuous random variables (in terms of their density function) can be obtained.

However, in elementary discussions of probability theory (say, of (early) undergraduate level), tools in measure theory are not usually accessible, so it is more usual to give these formulas as definitions instead.

On this page we present all three definitions, and then give proofs of consistency.

We also give a slightly less usual formula for the expectation of a general integrable continuous random variables, given as a Riemann-Stieltjes integral, and again prove consistency.

### General Definition

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $X$ be an integrable real-valued random variable on $\struct {\Omega, \Sigma, \Pr}$.

The expectation of $X$, written $\expect X$, is defined by:

$\ds \expect X = \int X \rd \Pr$

where the integral sign denotes the $\Pr$-integral of $X$.

### Discrete Random Variable

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:

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

### Absolutely Continuous Random Variable

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $X$ be an absolutely continuous random variable on $\struct {\Omega, \Sigma, \Pr}$.

Let $f_X$ be the probability density function of $X$.

The expectation of $X$, written $\expect X$, is defined by:

$\ds \expect X = \int_{-\infty}^\infty x \map {f_X} x \rd x$

whenever:

$\ds \int_{-\infty}^\infty \size x \map {f_X} x \rd x < \infty$

### Continuous Random Variable

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $X$ be a continuous random variable on $\struct {\Omega, \Sigma, \Pr}$.

Let $F_X$ be the cumulative distribution function of $X$.

The expectation of $X$, written $\expect X$, is defined by:

$\ds \expect X = \int_\R x \rd F_X$

whenever:

$\ds \int_\R \size x \rd F_X < \infty$

with the integrals being taken as Riemann-Stieltjes integrals.

## 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

• Results about expectation can be found here.

## 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