Definition:Expectation/Discrete
Definition
Let $X$ be a discrete random variable.
The expectation of $X$ is written $\expect X$, and 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
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: $\S 2.1$