# Definition:Expectation

## Contents

## 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**:

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

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

- 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**mathematical expectation** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**mean**

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