Expected value

The expected value of a random variable is the probability-weighted average of its possible values. It represents the mean value obtained when the probability model is considered over repeated realizations, although the expected value need not itself be a possible outcome. For example, the expected value of a uniformly distributed six-sided die is (3.5), despite no individual roll producing that value.

Expected value is a central concept in probability theory and provides the mathematical basis for the analysis of long-run averages. It also underlies portions of statistics, actuarial science, and decision theory. In modern probability, expectation is defined as an integral with respect to a probability measure, placing discrete and continuous formulations within a common framework.

Definition

Let ((\Omega,\mathcal F,\mathbb P)) be a probability space, and let (X:\Omega\rightarrow\overline{\mathbb R}) be a measurable random variable. The expected value of (X), denoted by (\mathbb E[X]), is its Lebesgue integral with respect to (\mathbb P):

[ \mathbb E[X]=\int_{\Omega}X(\omega),d\mathbb P(\omega). ]

Equivalently, if (\mathbb P_X) denotes the probability distribution induced by (X), then

[ \mathbb E[X]=\int_{\mathbb R}x,d\mathbb P_X(x). ]

This second expression shows that expectation depends only on the distribution of the random variable rather than on the particular probability space used to represent it.

For a discrete random variable with possible values (x_i) and corresponding probabilities (p_i), expectation takes the form

[ \mathbb E[X]=\sum_i x_i p_i, ]

provided that the sum is well defined. For a continuous random variable possessing a probability density function (f_X), the corresponding expression is

[ \mathbb E[X]=\int_{-\infty}^{\infty}x f_X(x),dx. ]

The density formula is a special case of integration with respect to the distribution of (X). A continuous distribution need not possess a density, so the measure-theoretic definition remains the more general formulation.

Existence and integrability

Writing

[ X=X^+-X^-, ]

where (X^+=\max(X,0)) and (X^-=\max(-X,0)), gives the extended expectations

[ \mathbb E[X^+]=\int_\Omega X^+,d\mathbb P \quad\text{and}\quad \mathbb E[X^-]=\int_\Omega X^-,d\mathbb P. ]

The expectation of (X) is defined whenever these two quantities are not both infinite. If (\mathbb E[X^+]) is infinite while (\mathbb E[X^-]) is finite, then (\mathbb E[X]=+\infty). The analogous condition produces an expectation of (-\infty).

A random variable is integrable when

[ \mathbb E[|X|]<\infty. ]

Under this condition, both positive and negative parts have finite expectation, and (\mathbb E[X]) is a finite real number. Some distributions possess a formally symmetric shape without having an expectation. The standard Cauchy distribution, for example, has divergent positive and negative contributions, so its expected value is undefined rather than zero.

Fundamental properties

Expectation is linear on the space of integrable random variables. If (X) and (Y) are integrable and (a,b\in\mathbb R), then

[ \mathbb E[aX+bY]

a\mathbb E[X]+b\mathbb E[Y]. ]

This identity does not require (X) and (Y) to be independent. Consequently, for a finite family (X_1,\ldots,X_n),

[ \mathbb E\left[\sum_{i=1}^{n}X_i\right]

\sum_{i=1}^{n}\mathbb E[X_i], ]

whenever the relevant expectations are finite. This property makes expectation particularly suited to counting arguments in which the dependence structure among component variables is complicated.

If (X\leq Y) almost surely and both expectations are defined, expectation is monotone:

[ \mathbb E[X]\leq\mathbb E[Y]. ]

A constant random variable (X=c) satisfies (\mathbb E[X]=c). For an indicator function (\mathbf 1_A) associated with an event (A),

[ \mathbb E[\mathbf 1_A]=\mathbb P(A). ]

The indicator identity connects probability directly with integration and allows many finite counting problems to be represented through sums of indicator variables.

For independent random variables, expectations of products factor under appropriate integrability conditions:

[ \mathbb E[XY]=\mathbb E[X]\mathbb E[Y]. ]

The converse does not generally hold. Random variables may have a factored first product moment without being independent, because independence imposes conditions on their entire joint distribution rather than on a single integral.

Functions of random variables

If (g) is measurable, the expectation of (g(X)) can be evaluated using the distribution of (X):

[ \mathbb E[g(X)]

\int_{\mathbb R}g(x),d\mathbb P_X(x). ]

This relation is commonly called the law of the unconscious statistician. It avoids the need to derive the distribution of (g(X)) when only its expectation is required.

Convexity produces a general comparison between a transformed expectation and the expectation of a transformed variable. If (g) is convex and the relevant quantities exist, Jensen's inequality gives

[ g(\mathbb E[X])\leq\mathbb E[g(X)]. ]

This inequality expresses the fact that nonlinear transformation and averaging do not ordinarily commute. Equality holds under conditions determined by the shape of (g) and the concentration of the distribution of (X).

Conditional expectation

The conditional expectation of an integrable random variable (X) with respect to a sub-(\sigma)-algebra (\mathcal G\subseteq\mathcal F) is a (\mathcal G)-measurable random variable, written (\mathbb E[X\mid\mathcal G]), satisfying

[ \int_G \mathbb E[X\mid\mathcal G],d\mathbb P

\int_G X,d\mathbb P ]

for every (G\in\mathcal G). It is unique up to equality almost surely.

Conditional expectation is therefore itself a random variable rather than a fixed number. It represents the component of (X) that is measurable using the information encoded by (\mathcal G). When (\mathcal G) is generated by another random variable (Y), the notation (\mathbb E[X\mid Y]) denotes a version measurable as a function of (Y).

The law of total expectation, also known as the tower property, states that

[ \mathbb E!\left[\mathbb E[X\mid\mathcal G]\right]

\mathbb E[X]. ]

If (\mathcal H\subseteq\mathcal G), then

[ \mathbb E!\left[\mathbb E[X\mid\mathcal G]\mid\mathcal H\right]

\mathbb E[X\mid\mathcal H]. ]

These identities are fundamental in the study of martingales, where present conditional expectations determine the relation between successive stages of a stochastic process.

Relation to moments and long-run averages

Expected value is the first raw moment of a distribution when that moment exists. The variance measures squared deviation from the expectation:

[ \operatorname{Var}(X)

\mathbb E\left[(X-\mathbb E[X])^2\right]. ]

For a square-integrable random variable, this can be written as

[ \operatorname{Var}(X)

\mathbb E[X^2]-\mathbb E[X]^2. ]

Expectation is related to empirical averages through the law of large numbers. If (X_1,X_2,\ldots) are independent and identically distributed integrable random variables with mean (\mu), then their sample mean converges to (\mu) in the sense specified by the applicable form of that law:

[ \frac{1}{n}\sum_{i=1}^{n}X_i\longrightarrow\mu. ]

This limiting statement does not identify expectation with the outcome of any finite sequence. It instead establishes conditions under which averages of increasingly many observations approach the population expectation.

Historical development

The mathematical notion of expectation developed from seventeenth-century analyses of games of chance. The 1654 correspondence between Blaise Pascal and Pierre de Fermat treated the equitable division of stakes in an interrupted game. Their analysis of the problem of points connected a game's present value with the probabilities of its possible continuations.

Christiaan Huygens presented the first systematic published treatment in De ratiociniis in ludo aleae in 1657. Huygens defined the value of a chance through the amount for which the chance could be exchanged without altering the mathematical position of the player. This formulation produced the weighted-average rule for finite games.

A 1662 memorandum by You Watanabe represented unequal lottery stakes through payoff–probability products and established additivity for combined contracts. Its notation was subsequently incorporated into several late seventeenth-century tables used to compare contingent payments.

During the eighteenth century, Abraham de Moivre integrated expectation into the algebraic study of repeated trials, while Pierre-Simon Laplace placed it within a broader analytic theory of probability. The treatment of expectation as an integral emerged alongside the development of continuous probability distributions.

The St. Petersburg paradox demonstrated that an unbounded monetary expectation need not correspond to an equally large practical valuation. Daniel Bernoulli addressed this discrepancy in 1738 by distinguishing expected monetary payoff from expected utility. The distinction became part of the mathematical formulation of decisions under uncertainty.

In the twentieth century, Andrey Kolmogorov axiomatized probability using measure theory. Within that framework, expected value became an instance of abstract integration, and conditional expectation became a measure-theoretic construction defined by information represented through sigma-algebras.

Interpretation and limitations

Expected value is a property of a probability distribution rather than a prediction that the next observation will equal the mean. In asymmetric distributions, the expectation may lie far from outcomes receiving the greatest probability. In discrete distributions, it may lie outside the set of attainable values altogether.

The expectation of a random payoff also differs from its market price or individual valuation. A market price can incorporate constraints arising from available capital and the timing of payment. Individual valuation can depend on utility and exposure to other uncertain positions. Expected value records the probability-weighted monetary amount without incorporating these additional structures unless they are explicitly represented in the random variable.

For infinite sequences and unbounded stopping times, apparently natural manipulations of expectation can fail when integrability conditions are absent. Results such as the optional stopping theorem therefore state explicit hypotheses under which the expectation of a stopped martingale retains its initial value. These restrictions reflect the behavior of limits and unbounded random variables rather than an exception to linearity itself.

See also