# Definition:Expectation

## Contents

## Definition

Let $X$ be a discrete random variable.

The **expectation of $X$** is written $\operatorname E \paren X$, and is defined as:

- $\expect X := \displaystyle \sum_{x \mathop \in \image X} x \Pr \paren {X = x}$

whenever the sum is absolutely convergent, that is, when:

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

where $\Pr \paren {X = x}$ 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:

- $\forall x \in \R: x \notin \image X \implies \Pr \paren {X = x} = 0$

Hence we can express the expectation as:

- $\expect X := \displaystyle \sum_{x \mathop \in \R} x \Pr \paren {X = x}$

Also, from the definition of probability mass function, we see it can also be written:

- $\expect X:= \displaystyle \sum_{x \mathop \in \R} x p_X \paren x$

### Continuous Random Variable

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

Let $F = \Pr \paren {X < x}$ 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 $X$ is also called the **expected value of $X$** or the **mean of $X$**, and (for a given discrete 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**:

- $E \paren X$
- $\operatorname {\mathbb E} \left[{X}\right]$

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 $E \left({X}\right)$ 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

- 1986: Geoffrey Grimmett and Dominic Welsh:
*Probability: An Introduction*... (previous) ... (next): $\S 2.4$: Expectation: $(19)$