# 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

- Expectation of Real-Valued Discrete Random Variable shows that this definition is consistent with the general definition of expectation.

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`

## Sources

- 1986: Geoffrey Grimmett and Dominic Welsh:
*Probability: An Introduction*... (previous) ... (next): $\S 2.4$: Expectation: $(19)$ - 2001: Michael A. Bean:
*Probability: The Science of Uncertainty*: $\S 2.1$