Definition:Expectation

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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