TheInfoList

OR:

In
probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the
weighted average The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The ...
. Informally, the expected value is the
arithmetic mean In mathematics and statistics, the arithmetic mean ( ) or arithmetic average, or just the '' mean'' or the '' average'' (when the context is clear), is the sum of a collection of numbers divided by the count of numbers in the collection. The col ...
of a large number of independently selected outcomes of a
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
. The expected value of a
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
with a finite number of outcomes is a
weighted average The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The ...
of all possible outcomes. In the case of a continuum of possible outcomes, the expectation is defined by
integration Integration may refer to: Biology *Multisensory integration * Path integration * Pre-integration complex, viral genetic material used to insert a viral genome into a host genome *DNA integration, by means of site-specific recombinase technolog ...
. In the axiomatic foundation for probability provided by
measure theory In mathematics, the concept of a measure is a generalization and formalization of geometrical measures ( length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simi ...
, the expectation is given by Lebesgue integration. The expected value of a random variable is often denoted by , , or , with also often stylized as or $\mathbb.$

# History

The idea of the expected value originated in the middle of the 17th century from the study of the so-called
problem of points The problem of points, also called the problem of division of the stakes, is a classical problem in probability theory. One of the famous problems that motivated the beginnings of modern probability theory in the 17th century, it led Blaise Pascal ...
, which seeks to divide the stakes ''in a fair way'' between two players, who have to end their game before it is properly finished. This problem had been debated for centuries. Many conflicting proposals and solutions had been suggested over the years when it was posed to
Blaise Pascal Blaise Pascal ( , , ; ; 19 June 1623 – 19 August 1662) was a French mathematician, physicist, inventor, philosopher, and Catholic writer. He was a child prodigy who was educated by his father, a tax collector in Rouen. Pascal's earliest m ...
by French writer and amateur mathematician Chevalier de Méré in 1654. Méré claimed that this problem couldn't be solved and that it showed just how flawed mathematics was when it came to its application to the real world. Pascal, being a mathematician, was provoked and determined to solve the problem once and for all. He began to discuss the problem in the famous series of letters to
Pierre de Fermat Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he ...
. Soon enough, they both independently came up with a solution. They solved the problem in different computational ways, but their results were identical because their computations were based on the same fundamental principle. The principle is that the value of a future gain should be directly proportional to the chance of getting it. This principle seemed to have come naturally to both of them. They were very pleased by the fact that they had found essentially the same solution, and this in turn made them absolutely convinced that they had solved the problem conclusively; however, they did not publish their findings. They only informed a small circle of mutual scientific friends in Paris about it. In Dutch mathematician Christiaan Huygens' book, he considered the problem of points, and presented a solution based on the same principle as the solutions of Pascal and Fermat. Huygens published his treatise in 1657, (see Huygens (1657)) "''De ratiociniis in ludo aleæ''" on probability theory just after visiting Paris. The book extended the concept of expectation by adding rules for how to calculate expectations in more complicated situations than the original problem (e.g., for three or more players), and can be seen as the first successful attempt at laying down the foundations of the theory of probability. In the foreword to his treatise, Huygens wrote: During his visit to France in 1655, Huygens learned about de Méré's Problem. From his correspondence with Carcavine a year later (in 1656), he realized his method was essentially the same as Pascal's. Therefore, he knew about Pascal's priority in this subject before his book went to press in 1657. In the mid-nineteenth century,
Pafnuty Chebyshev Pafnuty Lvovich Chebyshev ( rus, Пафну́тий Льво́вич Чебышёв, p=pɐfˈnutʲɪj ˈlʲvovʲɪtɕ tɕɪbɨˈʂof) ( – ) was a Russian mathematician and considered to be the founding father of Russian mathematics. Chebysh ...
became the first person to think systematically in terms of the expectations of
random variables A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
.

## Etymology

Neither Pascal nor Huygens used the term "expectation" in its modern sense. In particular, Huygens writes: More than a hundred years later, in 1814,
Pierre-Simon Laplace Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar and polymath whose work was important to the development of engineering, mathematics, statistics, physics, astronomy, and philosophy. He summarized ...
published his tract "''Théorie analytique des probabilités''", where the concept of expected value was defined explicitly:

# Notations

The use of the letter to denote expected value goes back to W. A. Whitworth in 1901. The symbol has become popular since then for English writers. In German, stands for "Erwartungswert", in Spanish for "Esperanza matemática", and in French for "Espérance mathématique". When "E" is used to denote expected value, authors use a variety of stylization: the expectation operator can be stylized as (upright), (italic), or $\mathbb$ (in
blackboard bold Blackboard bold is a typeface style that is often used for certain symbols in mathematical texts, in which certain lines of the symbol (usually vertical or near-vertical lines) are doubled. The symbols usually denote number sets. One way of pr ...
), while a variety of bracket notations (such as , , and ) are all used. Another popular notation is , whereas , , and $\overline$ are commonly used in physics, and in Russian-language literature.

# Definition

As discussed below, there are several context-dependent ways of defining the expected value. The simplest and original definition deals with the case of finitely many possible outcomes, such as in the flip of a coin. With the theory of infinite series, this can be extended to the case of countably many possible outcomes. It is also very common to consider the distinct case of random variables dictated by (piecewise-)continuous
probability density function In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) ...
s, as these arise in many natural contexts. All of these specific definitions may be viewed as special cases of the general definition based upon the mathematical tools of
measure theory In mathematics, the concept of a measure is a generalization and formalization of geometrical measures ( length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simi ...
and Lebesgue integration, which provide these different contexts with an axiomatic foundation and common language. Any definition of expected value may be extended to define an expected value of a multidimensional random variable, i.e. a
random vector In probability, and statistics, a multivariate random variable or random vector is a list of mathematical variables each of whose value is unknown, either because the value has not yet occurred or because there is imperfect knowledge of its value ...
. It is defined component by component, as . Similarly, one may define the expected value of a
random matrix In probability theory and mathematical physics, a random matrix is a matrix-valued random variable—that is, a matrix in which some or all elements are random variables. Many important properties of physical systems can be represented mathem ...
with components by .

## Random variables with finitely many outcomes

Consider a random variable with a ''finite'' list of possible outcomes, each of which (respectively) has probability of occurring. The expectation of is defined as : Since the probabilities must satisfy , it is natural to interpret as a
weighted average The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The ...
of the values, with weights given by their probabilities . In the special case that all possible outcomes are
equiprobable Equiprobability is a property for a collection of events that each have the same probability of occurring. In statistics and probability theory it is applied in the discrete uniform distribution and the equidistribution theorem for rational num ...
(that is, ), the weighted average is given by the standard
average In ordinary language, an average is a single number taken as representative of a list of numbers, usually the sum of the numbers divided by how many numbers are in the list (the arithmetic mean). For example, the average of the numbers 2, 3, 4, 7, ...
. In the general case, the expected value takes into account the fact that some outcomes are more likely than others.

### Examples

*Let $X$ represent the outcome of a roll of a fair six-sided . More specifically, $X$ will be the number of pips showing on the top face of the after the toss. The possible values for $X$ are 1, 2, 3, 4, 5, and 6, all of which are equally likely with a probability of . The expectation of $X$ is :: :If one rolls the $n$ times and computes the average (
arithmetic mean In mathematics and statistics, the arithmetic mean ( ) or arithmetic average, or just the '' mean'' or the '' average'' (when the context is clear), is the sum of a collection of numbers divided by the count of numbers in the collection. The col ...
) of the results, then as $n$ grows, the average will
almost surely In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (or Lebesgue measure 1). In other words, the set of possible exceptions may be non-empty, but it has probability 0 ...
converge Converge may refer to: * Converge (band), American hardcore punk band * Converge (Baptist denomination), American national evangelical Baptist body * Limit (mathematics) * Converge ICT Converge ICT Solutions Inc., commonly referred to as Con ...
to the expected value, a fact known as the
strong law of large numbers In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials shou ...
. *The roulette game consists of a small ball and a wheel with 38 numbered pockets around the edge. As the wheel is spun, the ball bounces around randomly until it settles down in one of the pockets. Suppose random variable $X$ represents the (monetary) outcome of a $1 bet on a single number ("straight up" bet). If the bet wins (which happens with probability in American roulette), the payoff is$35; otherwise the player loses the bet. The expected profit from such a bet will be :: :That is, the expected value to be won from a $1 bet is −$. Thus, in 190 bets, the net loss will probably be about \$10.

## Random variables with countably many outcomes

Informally, the expectation of a random variable with a
countable set In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural number ...
of possible outcomes is defined analogously as the weighted average of all possible outcomes, where the weights are given by the probabilities of realizing each given value. This is to say that : where are the possible outcomes of the random variable and are their corresponding probabilities. In many non-mathematical textbooks, this is presented as the full definition of expected values in this context. However, there are some subtleties with infinite summation, so the above formula is not suitable as a mathematical definition. In particular, the
Riemann series theorem In mathematics, the Riemann series theorem (also called the Riemann rearrangement theorem), named after 19th-century German mathematician Bernhard Riemann, says that if an infinite series of real numbers is conditionally convergent, then its terms ...
of
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions. These theories are usually studied ...
illustrates that the value of certain infinite sums involving positive and negative summands depends on the order in which the summands are given. Since the outcomes of a random variable have no naturally given order, this creates a difficulty in defining expected value precisely. For this reason, many mathematical textbooks only consider the case that the infinite sum given above converges absolutely, which implies that the infinite sum is a finite number independent of the ordering of summands. In the alternative case that the infinite sum does not converge absolutely, one says the random variable ''does not have finite expectation.''

### Examples

*Suppose $x_i = i$ and $p_i = \tfrac$ for $i = 1, 2, 3, \ldots,$ where $c = \tfrac$ is the scaling factor which makes the probabilities sum to 1. Then, using the direct definition for non-negative random variables, we have

## Random variables with density

Now consider a random variable which has a
probability density function In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) ...
given by a function on the
real number line In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a po ...
. This means that the probability of taking on a value in any given
open interval In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Othe ...
is given by the
integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
of over that interval. The expectation of is then given by the integral : A general and mathematically precise formulation of this definition uses
measure theory In mathematics, the concept of a measure is a generalization and formalization of geometrical measures ( length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simi ...
and Lebesgue integration, and the corresponding theory of ''absolutely continuous random variables'' is described in the next section. The density functions of many common distributions are piecewise continuous, and as such the theory is often developed in this restricted setting. For such functions, it is sufficient to only consider the standard Riemann integration. Sometimes ''continuous random variables'' are defined as those corresponding to this special class of densities, although the term is used differently by various authors. Analogously to the countably-infinite case above, there are subtleties with this expression due to the infinite region of integration. Such subtleties can be seen concretely if the distribution of is given by the
Cauchy distribution The Cauchy distribution, named after Augustin Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) funct ...
, so that . It is straightforward to compute in this case that :$\int_a^b xf\left(x\right)\,dx=\int_a^b \frac\,dx=\frac\ln\frac.$ The limit of this expression as and does not exist: if the limits are taken so that , then the limit is zero, while if the constraint is taken, then the limit is . To avoid such ambiguities, in mathematical textbooks it is common to require that the given integral converges absolutely, with left undefined otherwise. However, measure-theoretic notions as given below can be used to give a systematic definition of for more general random variables .

## Arbitrary real-valued random variables

All definitions of the expected value may be expressed in the language of
measure theory In mathematics, the concept of a measure is a generalization and formalization of geometrical measures ( length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simi ...
. In general, if is a real-valued
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
defined on a
probability space In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models t ...
, then the expected value of , denoted by , is defined as the
Lebesgue integral In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis. The Lebesgue integral, named after French mathematician Henri L ...
: Despite the newly abstract situation, this definition is extremely similar in nature to the very simplest definition of expected values, given above, as certain weighted averages. This is because, in measure theory, the value of the Lebesgue integral of is defined via weighted averages of ''approximations'' of which take on finitely many values. Moreover, if given a random variable with finitely or countably many possible values, the Lebesgue theory of expectation is identical with the summation formulas given above. However, the Lebesgue theory clarifies the scope of the theory of probability density functions. A random variable is said to be ''absolutely continuous'' if any of the following conditions are satisfied: * there is a nonnegative
measurable function In mathematics and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in d ...
on the real line such that ::$\text\left(X\in A\right)=\int_A f\left(x\right)\,dx,$ :for any
Borel set In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are named ...
, in which the integral is Lebesgue. * the
cumulative distribution function In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Ever ...
of is
absolutely continuous In calculus, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between the two central ...
. * for any Borel set of real numbers with
Lebesgue measure In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides ...
equal to zero, the probability of being valued in is also equal to zero * for any positive number there is a positive number such that: if is a Borel set with Lebesgue measure less than , then the probability of being valued in is less than . These conditions are all equivalent, although this is nontrivial to establish. In this definition, is called the ''probability density function'' of (relative to Lebesgue measure). According to the change-of-variables formula for Lebesgue integration, combined with the law of the unconscious statistician, it follows that : for any absolutely continuous random variable . The above discussion of continuous random variables is thus a special case of the general Lebesgue theory, due to the fact that every piecewise-continuous function is measurable.

## Infinite expected values

Expected values as defined above are automatically finite numbers. However, in many cases it is fundamental to be able to consider expected values of . This is intuitive, for example, in the case of the St. Petersburg paradox, in which one considers a random variable with possible outcomes , with associated probabilities , for ranging over all positive integers. According to the summation formula in the case of random variables with countably many outcomes, one has It is natural to say that the expected value equals . There is a rigorous mathematical theory underlying such ideas, which is often taken as part of the definition of the Lebesgue integral. The first fundamental observation is that, whichever of the above definitions are followed, any ''nonnegative'' random variable whatsoever can be given an unambiguous expected value; whenever absolute convergence fails, then the expected value can be defined as . The second fundamental observation is that any random variable can be written as the difference of two nonnegative random variables. Given a random variable , one defines the
positive and negative parts In mathematics, the positive part of a real or extended real-valued function is defined by the formula : f^+(x) = \max(f(x),0) = \begin f(x) & \mbox f(x) > 0 \\ 0 & \mbox \end Intuitively, the graph of f^+ is obtained by taking the graph of f ...
by and . These are nonnegative random variables, and it can be directly checked that . Since and are both then defined as either nonnegative numbers or , it is then natural to define: According to this definition, exists and is finite if and only if and are both finite. Due to the formula , this is the case if and only if is finite, and this is equivalent to the absolute convergence conditions in the definitions above. As such, the present considerations do not define finite expected values in any cases not previously considered; they are only useful for infinite expectations. *In the case of the St. Petersburg paradox, one has and so as desired. * Suppose the random variable takes values with respective probabilities . Then it follows that takes value with probability for each positive integer , and takes value with remaining probability. Similarly, takes value with probability for each positive integer and takes value with remaining probability. Using the definition for non-negative random variables, one can show that both and (see Harmonic series). Hence, in this case the expectation of is undefined. * Similarly, the Cauchy distribution, as discussed above, has undefined expectation.

# Expected values of common distributions

The following table gives the expected values of some commonly occurring probability distributions. The third column gives the expected values both in the form immediately given by the definition, as well as in the simplified form obtained by computation therefrom. The details of these computations, which are not always straightforward, can be found in the indicated references.

# Properties

The basic properties below (and their names in bold) replicate or follow immediately from those of
Lebesgue integral In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis. The Lebesgue integral, named after French mathematician Henri L ...
. Note that the letters "a.s." stand for "
almost surely In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (or Lebesgue measure 1). In other words, the set of possible exceptions may be non-empty, but it has probability 0 ...
"—a central property of the Lebesgue integral. Basically, one says that an inequality like $X \geq 0$ is true almost surely, when the probability measure attributes zero-mass to the complementary event $\left\$. *Non-negativity: If $X \geq 0$ (a.s.), then . *Linearity of expectation: The expected value operator (or expectation operator) $\operatorname$
cdot CDOT may refer to: *\cdot – the LaTeX input for the dot operator (⋅) *Cdot, a rapper from Sumter, South Carolina * Centre for Development of Telematics, India * Chicago Department of Transportation * Clustered Data ONTAP, an operating system f ...
/math> is
linear Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
in the sense that, for any random variables $X$ and $Y$, and a constant $a$, :whenever the right-hand side is well-defined. By induction, this means that the expected value of the sum of any finite number of random variables is the sum of the expected values of the individual random variables, and the expected value scales linearly with a multiplicative constant. Symbolically, for $N$ random variables $X_$ and constants $a_ \left(1\leq i \leq N\right)$, we have . If we think of the set of random variables with finite expected value as forming a vector space, then the linearity of expectation implies that the expected value is a
linear form In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers). If is a vector space over a field , the ...
on this vector space. *Monotonicity: If $X\leq Y$ (a.s.), and both
indicator_function In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , one has \mathbf_(x)=1 if x\ ...
_of_an_
event Event may refer to: Gatherings of people * Ceremony, an event of ritual significance, performed on a special occasion * Convention (meeting), a gathering of individuals engaged in some common interest * Event management, the organization of eve ...
_,_then__is_given_by_the_probability_of_._This_is_nothing_but_a_different_way_of_stating_the_expectation_of_a_ Bernoulli_random_variable,_as_calculated_in_the_table_above. *
• Formulas_in_terms_of_CDF:_If_$F\left(x\right)$_is_the_cumulative_distribution_function_ In_probability_theory_and_statistics,_the_cumulative_distribution_function_(CDF)_of_a_real-valued_random_variable_X,_or_just_distribution_function_of_X,_evaluated_at_x,_is_the_probability_that_X_will_take_a_value_less_than_or_equal_to_x. Ever_...
• _of_a_random_variable_,_then : \operatorname_=_\int_^\infty_x\,dF(x), _where_the_values_on_both_sides_are_well_defined_or_not_well_defined_simultaneously,_and_the_integral_is_taken_in_the_sense_of_ Lebesgue-Stieltjes._As_a_consequence_of_
integration_by_parts In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivati ...
_as_applied_to_this_representation_of_,_it_can_be_proved_that__\operatorname_=_\int_0^\infty_(1-F(x))\,dx_-__\int^0__F(x)\,dx,_with_the_integrals_taken_in_the_sense_of_Lebesgue._As_a_special_case,_for_any_random_variable__valued_in_the_nonnegative_integers_,_one_has__\operatorname_\sum__^\infty_\operatorname(X>n), :where__denotes_the_underlying_probability_measure. *Non-multiplicativity:_In_general,_the_expected_value_is_not_multiplicative,_i.e._
independent Independent or Independents may refer to: Arts, entertainment, and media Artist groups * Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s * Independe ...
,_then_one_can_show_that_
inner_product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
_of_$f$_and_$g$:_\operatorname
(X) An emoticon (, , rarely , ), short for "emotion icon", also known simply as an emote, is a pictorial representation of a facial expression using characters—usually punctuation marks, numbers, and letters—to express a person's feelings, ...
=_\int__g(x)_f(x)\,_dx_._This_formula_also_holds_in_multidimensional_case,_when_$g$_is_a_function_of_several_random_variables,_and_$f$_is_their_ joint_density.

## __Inequalities_

Concentration_inequalities_control_the_likelihood_of_a_random_variable_taking_on_large_values._
Markov's_inequality In probability theory, Markov's inequality gives an upper bound for the probability that a non-negative function of a random variable is greater than or equal to some positive constant. It is named after the Russian mathematician Andrey Markov, ...
_is_among_the_best-known_and_simplest_to_prove:_for_a_''nonnegative''_random_variable__and_any_positive_number_,_it_states_that_ \operatorname(X\geq_a)\leq\frac. If__is_any_random_variable_with_finite_expectation,_then_Markov's_inequality_may_be_applied_to_the_random_variable__to_obtain_
Chebyshev's_inequality In probability theory, Chebyshev's inequality (also called the Bienaymé–Chebyshev inequality) guarantees that, for a wide class of probability distributions, no more than a certain fraction of values can be more than a certain distance from t ...
_ \operatorname(, X-\text \geq_a)\leq\frac, where__is_the_
variance In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers ...
._These_inequalities_are_significant_for_their_nearly_complete_lack_of_conditional_assumptions._For_example,_for_any_random_variable_with_finite_expectation,_the_Chebyshev_inequality_implies_that_there_is_at_least_a_75%_probability_of_an_outcome_being_within_two_
standard_deviation In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, whil ...
s_of_the_expected_value._However,_in_special_cases_the_Markov_and_Chebyshev_inequalities_often_give_much_weaker_information_than_is_otherwise_available._For_example,_in_the_case_of_an_unweighted_dice,_Chebyshev's_inequality_says_that_odds_of_rolling_between_1_and_6_is_at_least_53%;_in_reality,_the_odds_are_of_course_100%._The_ Kolmogorov_inequality_extends_the_Chebyshev_inequality_to_the_context_of_sums_of_random_variables. The_following_three_inequalities_are_of_fundamental_importance_in_the_field_of_mathematical_analysis_ Analysis_is_the_branch_of_mathematics_dealing_with__continuous_functions,__limits,_and_related_theories,_such_as__differentiation,__integration,__measure,__infinite_sequences,__series,_and__analytic_functions. These_theories_are_usually_studied_...
_and_its_applications_to_probability_theory. *
Jensen's_inequality In mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function. It was proved by Jensen in 1906, building on an earlier p ...
:_Let__be_a_
convex_function In mathematics, a real-valued function is called convex if the line segment between any two points on the graph of the function lies above the graph between the two points. Equivalently, a function is convex if its epigraph (the set of poin ...
_and__a_random_variable_with_finite_expectation._Then_ ____f(\operatorname(X))_\leq_\operatorname_(f(X)). :Part_of_the_assertion_is_that_the_ negative_part_of__has_finite_expectation,_so_that_the_right-hand_side_is_well-defined_(possibly_infinite)._Convexity_of__can_be_phrased_as_saying_that_the_output_of_the_weighted_average_of_''two''_inputs_under-estimates_the_same_weighted_average_of_the_two_outputs;_Jensen's_inequality_extends_this_to_the_setting_of_completely_general_weighted_averages,_as_represented_by_the_expectation._In_the_special_case_that__for_positive_numbers_,_one_obtains_the_Lyapunov_inequality_ \left(\operatorname, X, ^s\right)^\leq\left(\operatorname, X, ^t\right)^. _ :This_can_also_be_proved_by_the_Hölder_inequality._In_measure_theory,_this_is_particularly_notable_for_proving_the_inclusion__of_ ,_in_the_special_case_of_probability_space_ In_probability_theory,_a_probability_space_or_a_probability_triple_(\Omega,_\mathcal,_P)_is_a__mathematical_construct_that_provides_a_formal_model_of_a_random_process_or_"experiment"._For_example,_one_can_define_a_probability_space_which_models_t_...
s. *_ Hölder's_inequality:_if__and__are_numbers_satisfying_,_then_ \operatorname, XY, \leq(\operatorname, X, ^p)^(\operatorname, Y, ^q)^. :_for_any_random_variables__and_._The_special_case_of__is_called_the_
Cauchy–Schwarz_inequality The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is considered one of the most important and widely used inequalities in mathematics. The inequality for sums was published by . The corresponding inequality fo ...
,_and_is_particularly_well-known. *_ Minkowski_inequality:_given_any_number_,_for_any_random_variables__and__with__and__both_finite,_it_follows_that__is_also_finite_and_ \Bigl(\operatorname, X+Y, ^p\Bigr)^\leq\Bigl(\operatorname, X, ^p\Bigr)^+\Bigl(\operatorname, Y, ^p\Bigr)^. The_Hölder_and_Minkowski_inequalities_can_be_extended_to_general_
measure_space A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the -algebra) and the method that ...
s,_and_are_often_given_in_that_context._By_contrast,_the_Jensen_inequality_is_special_to_the_case_of_probability_spaces.

## __Expectations_under_convergence_of_random_variables_

In_general,_it_is_not_the_case_that_
Monotone_convergence_theorem In the mathematical field of real analysis, the monotone convergence theorem is any of a number of related theorems proving the convergence of monotonic sequences (sequences that are non-decreasing or non-increasing) that are also bounded. In ...
:_Let_$\$_be_a_sequence_of_random_variables,_with_$0_\leq_X_n_\leq_X_$_(a.s)_for_each_$_n_\geq_0$._Furthermore,_let_$_X_n_\to_X_$_pointwise._Then,_the_monotone_convergence_theorem_states_that___Using_the_monotone_convergence_theorem,_one_can_show_that_expectation_indeed_satisfies_countable_additivity_for_non-negative_random_variables._In_particular,_let_$\^\infty_$_be_non-negative_random_variables.__It_follows_from_
monotone_convergence_theorem In the mathematical field of real analysis, the monotone convergence theorem is any of a number of related theorems proving the convergence of monotonic sequences (sequences that are non-decreasing or non-increasing) that are also bounded. In ...
_that_ \operatorname\left sum^\infty_X_i\right=_\sum^\infty_\operatorname _i * Fatou's_lemma:_Let_$\$_be_a_sequence_of_non-negative_random_variables._Fatou's_lemma_states_that_\operatorname liminf_n_X_n\leq_\liminf_n_\operatorname _n___Corollary._Let_$_X_n_\geq_0$_with__for_all_$_n_\geq_0$._If_$X_n_\to_X$_(a.s),_then___Proof_is_by_observing_that__X_=_\liminf_n_X_n_(a.s.)_and_applying_Fatou's_lemma. * Dominated_convergence_theorem:_Let_$\$_be_a_sequence_of_random_variables._If_$X_n\to_X$_
pointwise In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f(x) of some function f. An important class of pointwise concepts are the ''pointwise operations'', that is, operations defined ...
_(a.s.),_$, X_n, \leq_Y_\leq_+\infty$_(a.s.),_and_._Then,_according_to_the_dominated_convergence_theorem, **; **

## _Relationship_with_characteristic_function

The_probability_density_function_$f_X$_of_a_scalar_random_variable_$X$_is_related_to_its_
characteristic_function In mathematics, the term "characteristic function" can refer to any of several distinct concepts: * The indicator function of a subset, that is the function ::\mathbf_A\colon X \to \, :which for a given subset ''A'' of ''X'', has value 1 at points ...
_$\varphi_X$_by_the_inversion_formula: :_$f_X\left(x\right)_=_\frac\int__e^\varphi_X\left(t\right)_\,_\mathrmt.$ For_the_expected_value_of_$g\left(X\right)$_(where_$g:\to$_is_a_ Borel_function),_we_can_use_this_inversion_formula_to_obtain :$_\operatorname$
(X) An emoticon (, , rarely , ), short for "emotion icon", also known simply as an emote, is a pictorial representation of a facial expression using characters—usually punctuation marks, numbers, and letters—to express a person's feelings, ...
=_\frac_\int__g(x)\left \int__e^\varphi_X(t)_\,_\mathrmt_\right,\mathrmx. If_$\operatorname$
(X) An emoticon (, , rarely , ), short for "emotion icon", also known simply as an emote, is a pictorial representation of a facial expression using characters—usually punctuation marks, numbers, and letters—to express a person's feelings, ...
/math>_is_finite,_changing_the_order_of_integration,_we_get,_in_accordance_with_ Fubini–Tonelli_theorem, :$_\operatorname$
(X) An emoticon (, , rarely , ), short for "emotion icon", also known simply as an emote, is a pictorial representation of a facial expression using characters—usually punctuation marks, numbers, and letters—to express a person's feelings, ...
=_\frac_\int__G(t)_\varphi_X(t)_\,_\mathrmt, where :$G\left(t\right)_=_\int__g\left(x\right)_e^_\,_\mathrmx$ is_the_Fourier_transform_of_$_g\left(x\right)._$_The_expression_for_$\operatorname$
(X) An emoticon (, , rarely , ), short for "emotion icon", also known simply as an emote, is a pictorial representation of a facial expression using characters—usually punctuation marks, numbers, and letters—to express a person's feelings, ...
/math>_also_follows_directly_from_
Plancherel_theorem In mathematics, the Plancherel theorem (sometimes called the Parseval–Plancherel identity) is a result in harmonic analysis, proven by Michel Plancherel in 1910. It states that the integral of a function's squared modulus is equal to the inte ...
.

# __Uses_and_applications_

The_expectation_of_a_random_variable_plays_an_important_role_in_a_variety_of_contexts._For_example,_in_
decision_theory Decision theory (or the theory of choice; not to be confused with choice theory) is a branch of applied probability theory concerned with the theory of making decisions based on assigning probabilities to various factors and assigning numerical ...
,_an_agent_making_an_optimal_choice_in_the_context_of_incomplete_information_is_often_assumed_to_maximize_the_expected_value_of_their_
utility_function As a topic of economics, utility is used to model worth or value. Its usage has evolved significantly over time. The term was introduced initially as a measure of pleasure or happiness as part of the theory of utilitarianism by moral philosoph ...
. For_a_different_example,_in_
statistics Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, indus ...
,_where_one_seeks_estimates_for_unknown_parameters_based_on_available_data,_the_estimate_itself_is_a_random_variable._In_such_settings,_a_desirable_criterion_for_a_"good"_estimator_is_that_it_is_'' unbiased'';_that_is,_the_expected_value_of_the_estimate_is_equal_to_the_ true_value_of_the_underlying_parameter. It_is_possible_to_construct_an_expected_value_equal_to_the_probability_of_an_event,_by_taking_the_expectation_of_an_
indicator_function In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , one has \mathbf_(x)=1 if x\ ...
_that_is_one_if_the_event_has_occurred_and_zero_otherwise._This_relationship_can_be_used_to_translate_properties_of_expected_values_into_properties_of_probabilities,_e.g._using_the_
law_of_large_numbers In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials s ...
_to_justify_estimating_probabilities_by_
frequencies Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from '' angular frequency''. Frequency is measured in hertz (Hz) which is e ...
. The_expected_values_of_the_powers_of_''X''_are_called_the_ moments_of_''X'';_the_ moments_about_the_mean_of_''X''_are_expected_values_of_powers_of_._The_moments_of_some_random_variables_can_be_used_to_specify_their_distributions,_via_their_ moment_generating_functions. To_empirically_
estimate Estimation (or estimating) is the process of finding an estimate or approximation, which is a value that is usable for some purpose even if input data may be incomplete, uncertain, or unstable. The value is nonetheless usable because it is de ...
_the_expected_value_of_a_random_variable,_one_repeatedly_measures_observations_of_the_variable_and_computes_the_arithmetic_mean_ In_mathematics_and_statistics,_the_arithmetic_mean_(_)_or_arithmetic_average,_or_just_the_''_mean''_or_the_''_average''_(when_the_context_is_clear),_is_the_sum_of_a_collection_of_numbers_divided_by_the_count_of_numbers_in_the_collection._The_col_...
_of_the_results._If_the_expected_value_exists,_this_procedure_estimates_the_true_expected_value_in_an_ unbiased_manner_and_has_the_property_of_minimizing_the_sum_of_the_squares_of_the_ residuals_(the_sum_of_the_squared_differences_between_the_observations_and_the_
estimate Estimation (or estimating) is the process of finding an estimate or approximation, which is a value that is usable for some purpose even if input data may be incomplete, uncertain, or unstable. The value is nonetheless usable because it is de ...
)._The_
law_of_large_numbers In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials s ...
_demonstrates_(under_fairly_mild_conditions)_that,_as_the_
size Size in general is the magnitude or dimensions of a thing. More specifically, ''geometrical size'' (or ''spatial size'') can refer to linear dimensions ( length, width, height, diameter, perimeter), area, or volume. Size can also b ...
_of_the_ sample_gets_larger,_the_
variance In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers ...
_of_this_
estimate Estimation (or estimating) is the process of finding an estimate or approximation, which is a value that is usable for some purpose even if input data may be incomplete, uncertain, or unstable. The value is nonetheless usable because it is de ...
_gets_smaller. This_property_is_often_exploited_in_a_wide_variety_of_applications,_including_general_problems_of_ statistical_estimation_and_
machine_learning Machine learning (ML) is a field of inquiry devoted to understanding and building methods that 'learn', that is, methods that leverage data to improve performance on some set of tasks. It is seen as a part of artificial intelligence. Machine ...
,_to_estimate_(probabilistic)_quantities_of_interest_via_
Monte_Carlo_methods Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be determini ...
,_since_most_quantities_of_interest_can_be_written_in_terms_of_expectation,_e.g._
classical_mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by class ...
,_the_
center_of_mass In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force m ...
_is_an_analogous_concept_to_expectation._For_example,_suppose_''X''_is_a_discrete_random_variable_with_values_''xi''_and_corresponding_probabilities_''pi''._Now_consider_a_weightless_rod_on_which_are_placed_weights,_at_locations_''xi''_along_the_rod_and_having_masses_''pi''_(whose_sum_is_one)._The_point_at_which_the_rod_balances_is_E 'X'' Expected_values_can_also_be_used_to_compute_the_
variance In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers ...
,_by_means_of_the_computational_formula_for_the_variance : A_very_important_application_of_the_expectation_value_is_in_the_field_of_
quantum_mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, qua ...
._The_expectation_value_of_a_quantum_mechanical_operator_$\hat$_operating_on_a_
quantum_state In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution i ...
_vector_$, \psi\rangle$_is_written_as_$\langle\hat\rangle_=_\langle\psi, A, \psi\rangle$._The_
uncertainty Uncertainty refers to epistemic situations involving imperfect or unknown information. It applies to predictions of future events, to physical measurements that are already made, or to the unknown. Uncertainty arises in partially observable o ...
_in_$\hat$_can_be_calculated_by_the_formula_$\left(\Delta_A\right)^2_=_\langle\hat^2\rangle_-_\langle_\hat_\rangle^2_$.

# _See_also

*
Center_of_mass In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force m ...
*
Central_tendency In statistics, a central tendency (or measure of central tendency) is a central or typical value for a probability distribution.Weisberg H.F (1992) ''Central Tendency and Variability'', Sage University Paper Series on Quantitative Applications in ...
*
Chebyshev's_inequality In probability theory, Chebyshev's inequality (also called the Bienaymé–Chebyshev inequality) guarantees that, for a wide class of probability distributions, no more than a certain fraction of values can be more than a certain distance from t ...
_(an_inequality_on_location_and_scale_parameters) *
Conditional_expectation In probability theory, the conditional expectation, conditional expected value, or conditional mean of a random variable is its expected value – the value it would take “on average” over an arbitrarily large number of occurrences – given ...
*
Expectation Expectation or Expectations may refer to: Science * Expectation (epistemic) * Expected value, in mathematical probability theory * Expectation value (quantum mechanics) * Expectation–maximization algorithm, in statistics Music * ''Expectatio ...
_(the_general_term) * Expectation_value_(quantum_mechanics) * Law_of_total_expectation—the_expected_value_of_the_conditional_expected_value_of_''X''_given_''Y''_is_the_same_as_the_expected_value_of_''X''. *
Moment_(mathematics) In mathematics, the moments of a function are certain quantitative measures related to the shape of the function's graph. If the function represents mass density, then the zeroth moment is the total mass, the first moment (normalized by total ma ...
* Nonlinear_expectation_(a_generalization_of_the_expected_value) *
Sample_mean The sample mean (or "empirical mean") and the sample covariance are statistics computed from a sample of data on one or more random variables. The sample mean is the average value (or mean value) of a sample of numbers taken from a larger ...
*
Population_mean In statistics, a population is a set of similar items or events which is of interest for some question or experiment. A statistical population can be a group of existing objects (e.g. the set of all stars within the Milky Way galaxy) or a h ...
* Wald's_equation—an_equation_for_calculating_the_expected_value_of_a_random_number_of_random_variables

# _Literature

*_ *_ * * * * * *_ *

{{DEFAULTSORT:Expected_Value Theory_of_probability_distributions Gambling_terminology Articles_containing_proofshtml" ;"title="X, ]=0, then $X=0$ (a.s.). * If $X = Y$ (a.s.), then
indicator function In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , one has \mathbf_(x)=1 if x\ ...
of an
event Event may refer to: Gatherings of people * Ceremony, an event of ritual significance, performed on a special occasion * Convention (meeting), a gathering of individuals engaged in some common interest * Event management, the organization of eve ...
, then is given by the probability of . This is nothing but a different way of stating the expectation of a Bernoulli random variable, as calculated in the table above. *
• Formulas in terms of CDF: If $F\left(x\right)$ is the
cumulative distribution function In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Ever ...
of a random variable , then : where the values on both sides are well defined or not well defined simultaneously, and the integral is taken in the sense of Lebesgue-Stieltjes. As a consequence of
integration by parts In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivati ...
as applied to this representation of , it can be proved that with the integrals taken in the sense of Lebesgue. As a special case, for any random variable valued in the nonnegative integers , one has :where denotes the underlying probability measure. *Non-multiplicativity: In general, the expected value is not multiplicative, i.e.
independent Independent or Independents may refer to: Arts, entertainment, and media Artist groups * Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s * Independe ...
, then one can show that
inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
of $f$ and $g$: $\operatorname$
(X) An emoticon (, , rarely , ), short for "emotion icon", also known simply as an emote, is a pictorial representation of a facial expression using characters—usually punctuation marks, numbers, and letters—to express a person's feelings, ...
= \int_ g(x) f(x)\, dx . This formula also holds in multidimensional case, when $g$ is a function of several random variables, and $f$ is their joint density.

## Inequalities

Concentration inequalities control the likelihood of a random variable taking on large values.
Markov's inequality In probability theory, Markov's inequality gives an upper bound for the probability that a non-negative function of a random variable is greater than or equal to some positive constant. It is named after the Russian mathematician Andrey Markov, ...
is among the best-known and simplest to prove: for a ''nonnegative'' random variable and any positive number , it states that $\operatorname(X\geq a)\leq\frac.$ If is any random variable with finite expectation, then Markov's inequality may be applied to the random variable to obtain
Chebyshev's inequality In probability theory, Chebyshev's inequality (also called the Bienaymé–Chebyshev inequality) guarantees that, for a wide class of probability distributions, no more than a certain fraction of values can be more than a certain distance from t ...
where is the
variance In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers ...
. These inequalities are significant for their nearly complete lack of conditional assumptions. For example, for any random variable with finite expectation, the Chebyshev inequality implies that there is at least a 75% probability of an outcome being within two
standard deviation In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, whil ...
s of the expected value. However, in special cases the Markov and Chebyshev inequalities often give much weaker information than is otherwise available. For example, in the case of an unweighted dice, Chebyshev's inequality says that odds of rolling between 1 and 6 is at least 53%; in reality, the odds are of course 100%. The Kolmogorov inequality extends the Chebyshev inequality to the context of sums of random variables. The following three inequalities are of fundamental importance in the field of
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions. These theories are usually studied ...
and its applications to probability theory. *
Jensen's inequality In mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function. It was proved by Jensen in 1906, building on an earlier p ...
: Let be a
convex function In mathematics, a real-valued function is called convex if the line segment between any two points on the graph of the function lies above the graph between the two points. Equivalently, a function is convex if its epigraph (the set of poin ...
and a random variable with finite expectation. Then $f(\operatorname(X)) \leq \operatorname (f(X)).$ :Part of the assertion is that the negative part of has finite expectation, so that the right-hand side is well-defined (possibly infinite). Convexity of can be phrased as saying that the output of the weighted average of ''two'' inputs under-estimates the same weighted average of the two outputs; Jensen's inequality extends this to the setting of completely general weighted averages, as represented by the expectation. In the special case that for positive numbers , one obtains the Lyapunov inequality $\left(\operatorname, X, ^s\right)^\leq\left(\operatorname, X, ^t\right)^.$ :This can also be proved by the Hölder inequality. In measure theory, this is particularly notable for proving the inclusion of , in the special case of
probability space In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models t ...
s. * Hölder's inequality: if and are numbers satisfying , then $\operatorname, XY, \leq(\operatorname, X, ^p)^(\operatorname, Y, ^q)^.$ : for any random variables and . The special case of is called the
Cauchy–Schwarz inequality The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is considered one of the most important and widely used inequalities in mathematics. The inequality for sums was published by . The corresponding inequality fo ...
, and is particularly well-known. * Minkowski inequality: given any number , for any random variables and with and both finite, it follows that is also finite and $\Bigl(\operatorname, X+Y, ^p\Bigr)^\leq\Bigl(\operatorname, X, ^p\Bigr)^+\Bigl(\operatorname, Y, ^p\Bigr)^.$ The Hölder and Minkowski inequalities can be extended to general
measure space A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the -algebra) and the method that ...
s, and are often given in that context. By contrast, the Jensen inequality is special to the case of probability spaces.

## Expectations under convergence of random variables

In general, it is not the case that
Monotone convergence theorem In the mathematical field of real analysis, the monotone convergence theorem is any of a number of related theorems proving the convergence of monotonic sequences (sequences that are non-decreasing or non-increasing) that are also bounded. In ...
: Let $\$ be a sequence of random variables, with $0 \leq X_n \leq X_$ (a.s) for each $n \geq 0$. Furthermore, let $X_n \to X$ pointwise. Then, the monotone convergence theorem states that Using the monotone convergence theorem, one can show that expectation indeed satisfies countable additivity for non-negative random variables. In particular, let $\^\infty_$ be non-negative random variables. It follows from
monotone convergence theorem In the mathematical field of real analysis, the monotone convergence theorem is any of a number of related theorems proving the convergence of monotonic sequences (sequences that are non-decreasing or non-increasing) that are also bounded. In ...
that * Fatou's lemma: Let $\$ be a sequence of non-negative random variables. Fatou's lemma states that Corollary. Let $X_n \geq 0$ with for all $n \geq 0$. If $X_n \to X$ (a.s), then Proof is by observing that $X = \liminf_n X_n$ (a.s.) and applying Fatou's lemma. * Dominated convergence theorem: Let $\$ be a sequence of random variables. If $X_n\to X$
pointwise In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f(x) of some function f. An important class of pointwise concepts are the ''pointwise operations'', that is, operations defined ...
(a.s.), $, X_n, \leq Y \leq +\infty$ (a.s.), and . Then, according to the dominated convergence theorem, **; **

## Relationship with characteristic function

The probability density function $f_X$ of a scalar random variable $X$ is related to its
characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts: * The indicator function of a subset, that is the function ::\mathbf_A\colon X \to \, :which for a given subset ''A'' of ''X'', has value 1 at points ...
$\varphi_X$ by the inversion formula: : $f_X\left(x\right) = \frac\int_ e^\varphi_X\left(t\right) \, \mathrmt.$ For the expected value of $g\left(X\right)$ (where $g:\to$ is a Borel function), we can use this inversion formula to obtain :$\operatorname$
(X) An emoticon (, , rarely , ), short for "emotion icon", also known simply as an emote, is a pictorial representation of a facial expression using characters—usually punctuation marks, numbers, and letters—to express a person's feelings, ...
= \frac \int_ g(x)\left \int_ e^\varphi_X(t) \, \mathrmt \right,\mathrmx. If $\operatorname$
(X) An emoticon (, , rarely , ), short for "emotion icon", also known simply as an emote, is a pictorial representation of a facial expression using characters—usually punctuation marks, numbers, and letters—to express a person's feelings, ...
/math> is finite, changing the order of integration, we get, in accordance with Fubini–Tonelli theorem, :$\operatorname$
(X) An emoticon (, , rarely , ), short for "emotion icon", also known simply as an emote, is a pictorial representation of a facial expression using characters—usually punctuation marks, numbers, and letters—to express a person's feelings, ...
= \frac \int_ G(t) \varphi_X(t) \, \mathrmt, where :$G\left(t\right) = \int_ g\left(x\right) e^ \, \mathrmx$ is the Fourier transform of $g\left(x\right).$ The expression for $\operatorname$
(X) An emoticon (, , rarely , ), short for "emotion icon", also known simply as an emote, is a pictorial representation of a facial expression using characters—usually punctuation marks, numbers, and letters—to express a person's feelings, ...
/math> also follows directly from
Plancherel theorem In mathematics, the Plancherel theorem (sometimes called the Parseval–Plancherel identity) is a result in harmonic analysis, proven by Michel Plancherel in 1910. It states that the integral of a function's squared modulus is equal to the inte ...
.

# Uses and applications

The expectation of a random variable plays an important role in a variety of contexts. For example, in
decision theory Decision theory (or the theory of choice; not to be confused with choice theory) is a branch of applied probability theory concerned with the theory of making decisions based on assigning probabilities to various factors and assigning numerical ...
, an agent making an optimal choice in the context of incomplete information is often assumed to maximize the expected value of their
utility function As a topic of economics, utility is used to model worth or value. Its usage has evolved significantly over time. The term was introduced initially as a measure of pleasure or happiness as part of the theory of utilitarianism by moral philosoph ...
. For a different example, in
statistics Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, indus ...
, where one seeks estimates for unknown parameters based on available data, the estimate itself is a random variable. In such settings, a desirable criterion for a "good" estimator is that it is '' unbiased''; that is, the expected value of the estimate is equal to the true value of the underlying parameter. It is possible to construct an expected value equal to the probability of an event, by taking the expectation of an
indicator function In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , one has \mathbf_(x)=1 if x\ ...
that is one if the event has occurred and zero otherwise. This relationship can be used to translate properties of expected values into properties of probabilities, e.g. using the
law of large numbers In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials s ...
to justify estimating probabilities by
frequencies Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from '' angular frequency''. Frequency is measured in hertz (Hz) which is e ...
. The expected values of the powers of ''X'' are called the moments of ''X''; the moments about the mean of ''X'' are expected values of powers of . The moments of some random variables can be used to specify their distributions, via their moment generating functions. To empirically
estimate Estimation (or estimating) is the process of finding an estimate or approximation, which is a value that is usable for some purpose even if input data may be incomplete, uncertain, or unstable. The value is nonetheless usable because it is de ...
the expected value of a random variable, one repeatedly measures observations of the variable and computes the
arithmetic mean In mathematics and statistics, the arithmetic mean ( ) or arithmetic average, or just the '' mean'' or the '' average'' (when the context is clear), is the sum of a collection of numbers divided by the count of numbers in the collection. The col ...
of the results. If the expected value exists, this procedure estimates the true expected value in an unbiased manner and has the property of minimizing the sum of the squares of the residuals (the sum of the squared differences between the observations and the
estimate Estimation (or estimating) is the process of finding an estimate or approximation, which is a value that is usable for some purpose even if input data may be incomplete, uncertain, or unstable. The value is nonetheless usable because it is de ...
). The
law of large numbers In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials s ...
demonstrates (under fairly mild conditions) that, as the
size Size in general is the magnitude or dimensions of a thing. More specifically, ''geometrical size'' (or ''spatial size'') can refer to linear dimensions ( length, width, height, diameter, perimeter), area, or volume. Size can also b ...
of the sample gets larger, the
variance In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers ...
of this
estimate Estimation (or estimating) is the process of finding an estimate or approximation, which is a value that is usable for some purpose even if input data may be incomplete, uncertain, or unstable. The value is nonetheless usable because it is de ...
gets smaller. This property is often exploited in a wide variety of applications, including general problems of statistical estimation and
machine learning Machine learning (ML) is a field of inquiry devoted to understanding and building methods that 'learn', that is, methods that leverage data to improve performance on some set of tasks. It is seen as a part of artificial intelligence. Machine ...
, to estimate (probabilistic) quantities of interest via
Monte Carlo methods Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be determini ...
, since most quantities of interest can be written in terms of expectation, e.g. In
classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by class ...
, the
center of mass In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force m ...
is an analogous concept to expectation. For example, suppose ''X'' is a discrete random variable with values ''xi'' and corresponding probabilities ''pi''. Now consider a weightless rod on which are placed weights, at locations ''xi'' along the rod and having masses ''pi'' (whose sum is one). The point at which the rod balances is E 'X'' Expected values can also be used to compute the
variance In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers ...
, by means of the computational formula for the variance : A very important application of the expectation value is in the field of
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, qua ...
. The expectation value of a quantum mechanical operator $\hat$ operating on a
quantum state In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution i ...
vector $, \psi\rangle$ is written as $\langle\hat\rangle = \langle\psi, A, \psi\rangle$. The
uncertainty Uncertainty refers to epistemic situations involving imperfect or unknown information. It applies to predictions of future events, to physical measurements that are already made, or to the unknown. Uncertainty arises in partially observable o ...
in $\hat$ can be calculated by the formula $\left(\Delta A\right)^2 = \langle\hat^2\rangle - \langle \hat \rangle^2$.

*
Center of mass In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force m ...
*
Central tendency In statistics, a central tendency (or measure of central tendency) is a central or typical value for a probability distribution.Weisberg H.F (1992) ''Central Tendency and Variability'', Sage University Paper Series on Quantitative Applications in ...
*
Chebyshev's inequality In probability theory, Chebyshev's inequality (also called the Bienaymé–Chebyshev inequality) guarantees that, for a wide class of probability distributions, no more than a certain fraction of values can be more than a certain distance from t ...
(an inequality on location and scale parameters) *
Conditional expectation In probability theory, the conditional expectation, conditional expected value, or conditional mean of a random variable is its expected value – the value it would take “on average” over an arbitrarily large number of occurrences – given ...
*
Expectation Expectation or Expectations may refer to: Science * Expectation (epistemic) * Expected value, in mathematical probability theory * Expectation value (quantum mechanics) * Expectation–maximization algorithm, in statistics Music * ''Expectatio ...
(the general term) * Expectation value (quantum mechanics) * Law of total expectation—the expected value of the conditional expected value of ''X'' given ''Y'' is the same as the expected value of ''X''. *
Moment (mathematics) In mathematics, the moments of a function are certain quantitative measures related to the shape of the function's graph. If the function represents mass density, then the zeroth moment is the total mass, the first moment (normalized by total ma ...
* Nonlinear expectation (a generalization of the expected value) *
Sample mean The sample mean (or "empirical mean") and the sample covariance are statistics computed from a sample of data on one or more random variables. The sample mean is the average value (or mean value) of a sample of numbers taken from a larger ...
*
Population mean In statistics, a population is a set of similar items or events which is of interest for some question or experiment. A statistical population can be a group of existing objects (e.g. the set of all stars within the Milky Way galaxy) or a h ...
* Wald's equation—an equation for calculating the expected value of a random number of random variables

# Literature

* * * * * * * * *

{{DEFAULTSORT:Expected Value Theory of probability distributions Gambling terminology Articles containing proofs>X, 0, then $X=0$ (a.s.). * If $X = Y$ (a.s.), then
indicator function In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , one has \mathbf_(x)=1 if x\ ...
of an
event Event may refer to: Gatherings of people * Ceremony, an event of ritual significance, performed on a special occasion * Convention (meeting), a gathering of individuals engaged in some common interest * Event management, the organization of eve ...
, then is given by the probability of . This is nothing but a different way of stating the expectation of a Bernoulli random variable, as calculated in the table above. *
• Formulas in terms of CDF: If $F\left(x\right)$ is the
cumulative distribution function In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Ever ...
of a random variable , then : where the values on both sides are well defined or not well defined simultaneously, and the integral is taken in the sense of Lebesgue-Stieltjes. As a consequence of
integration by parts In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivati ...
as applied to this representation of , it can be proved that with the integrals taken in the sense of Lebesgue. As a special case, for any random variable valued in the nonnegative integers , one has :where denotes the underlying probability measure. *Non-multiplicativity: In general, the expected value is not multiplicative, i.e.
independent Independent or Independents may refer to: Arts, entertainment, and media Artist groups * Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s * Independe ...
, then one can show that
inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
of $f$ and $g$: $\operatorname$
(X) An emoticon (, , rarely , ), short for "emotion icon", also known simply as an emote, is a pictorial representation of a facial expression using characters—usually punctuation marks, numbers, and letters—to express a person's feelings, ...
= \int_ g(x) f(x)\, dx . This formula also holds in multidimensional case, when $g$ is a function of several random variables, and $f$ is their joint density.

## Inequalities

Concentration inequalities control the likelihood of a random variable taking on large values.
Markov's inequality In probability theory, Markov's inequality gives an upper bound for the probability that a non-negative function of a random variable is greater than or equal to some positive constant. It is named after the Russian mathematician Andrey Markov, ...
is among the best-known and simplest to prove: for a ''nonnegative'' random variable and any positive number , it states that $\operatorname(X\geq a)\leq\frac.$ If is any random variable with finite expectation, then Markov's inequality may be applied to the random variable to obtain
Chebyshev's inequality In probability theory, Chebyshev's inequality (also called the Bienaymé–Chebyshev inequality) guarantees that, for a wide class of probability distributions, no more than a certain fraction of values can be more than a certain distance from t ...
where is the
variance In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers ...
. These inequalities are significant for their nearly complete lack of conditional assumptions. For example, for any random variable with finite expectation, the Chebyshev inequality implies that there is at least a 75% probability of an outcome being within two
standard deviation In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, whil ...
s of the expected value. However, in special cases the Markov and Chebyshev inequalities often give much weaker information than is otherwise available. For example, in the case of an unweighted dice, Chebyshev's inequality says that odds of rolling between 1 and 6 is at least 53%; in reality, the odds are of course 100%. The Kolmogorov inequality extends the Chebyshev inequality to the context of sums of random variables. The following three inequalities are of fundamental importance in the field of
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions. These theories are usually studied ...
and its applications to probability theory. *
Jensen's inequality In mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function. It was proved by Jensen in 1906, building on an earlier p ...
: Let be a
convex function In mathematics, a real-valued function is called convex if the line segment between any two points on the graph of the function lies above the graph between the two points. Equivalently, a function is convex if its epigraph (the set of poin ...
and a random variable with finite expectation. Then $f(\operatorname(X)) \leq \operatorname (f(X)).$ :Part of the assertion is that the negative part of has finite expectation, so that the right-hand side is well-defined (possibly infinite). Convexity of can be phrased as saying that the output of the weighted average of ''two'' inputs under-estimates the same weighted average of the two outputs; Jensen's inequality extends this to the setting of completely general weighted averages, as represented by the expectation. In the special case that for positive numbers , one obtains the Lyapunov inequality $\left(\operatorname, X, ^s\right)^\leq\left(\operatorname, X, ^t\right)^.$ :This can also be proved by the Hölder inequality. In measure theory, this is particularly notable for proving the inclusion of , in the special case of
probability space In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models t ...
s. * Hölder's inequality: if and are numbers satisfying , then $\operatorname, XY, \leq(\operatorname, X, ^p)^(\operatorname, Y, ^q)^.$ : for any random variables and . The special case of is called the
Cauchy–Schwarz inequality The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is considered one of the most important and widely used inequalities in mathematics. The inequality for sums was published by . The corresponding inequality fo ...
, and is particularly well-known. * Minkowski inequality: given any number , for any random variables and with and both finite, it follows that is also finite and $\Bigl(\operatorname, X+Y, ^p\Bigr)^\leq\Bigl(\operatorname, X, ^p\Bigr)^+\Bigl(\operatorname, Y, ^p\Bigr)^.$ The Hölder and Minkowski inequalities can be extended to general
measure space A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the -algebra) and the method that ...
s, and are often given in that context. By contrast, the Jensen inequality is special to the case of probability spaces.

## Expectations under convergence of random variables

In general, it is not the case that
Monotone convergence theorem In the mathematical field of real analysis, the monotone convergence theorem is any of a number of related theorems proving the convergence of monotonic sequences (sequences that are non-decreasing or non-increasing) that are also bounded. In ...
: Let $\$ be a sequence of random variables, with $0 \leq X_n \leq X_$ (a.s) for each $n \geq 0$. Furthermore, let $X_n \to X$ pointwise. Then, the monotone convergence theorem states that Using the monotone convergence theorem, one can show that expectation indeed satisfies countable additivity for non-negative random variables. In particular, let $\^\infty_$ be non-negative random variables. It follows from
monotone convergence theorem In the mathematical field of real analysis, the monotone convergence theorem is any of a number of related theorems proving the convergence of monotonic sequences (sequences that are non-decreasing or non-increasing) that are also bounded. In ...
that * Fatou's lemma: Let $\$ be a sequence of non-negative random variables. Fatou's lemma states that Corollary. Let $X_n \geq 0$ with for all $n \geq 0$. If $X_n \to X$ (a.s), then Proof is by observing that $X = \liminf_n X_n$ (a.s.) and applying Fatou's lemma. * Dominated convergence theorem: Let $\$ be a sequence of random variables. If $X_n\to X$
pointwise In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f(x) of some function f. An important class of pointwise concepts are the ''pointwise operations'', that is, operations defined ...
(a.s.), $, X_n, \leq Y \leq +\infty$ (a.s.), and . Then, according to the dominated convergence theorem, **; **

## Relationship with characteristic function

The probability density function $f_X$ of a scalar random variable $X$ is related to its
characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts: * The indicator function of a subset, that is the function ::\mathbf_A\colon X \to \, :which for a given subset ''A'' of ''X'', has value 1 at points ...
$\varphi_X$ by the inversion formula: : $f_X\left(x\right) = \frac\int_ e^\varphi_X\left(t\right) \, \mathrmt.$ For the expected value of $g\left(X\right)$ (where $g:\to$ is a Borel function), we can use this inversion formula to obtain :$\operatorname$
(X) An emoticon (, , rarely , ), short for "emotion icon", also known simply as an emote, is a pictorial representation of a facial expression using characters—usually punctuation marks, numbers, and letters—to express a person's feelings, ...
= \frac \int_ g(x)\left \int_ e^\varphi_X(t) \, \mathrmt \right,\mathrmx. If $\operatorname$
(X) An emoticon (, , rarely , ), short for "emotion icon", also known simply as an emote, is a pictorial representation of a facial expression using characters—usually punctuation marks, numbers, and letters—to express a person's feelings, ...
/math> is finite, changing the order of integration, we get, in accordance with Fubini–Tonelli theorem, :$\operatorname$
(X) An emoticon (, , rarely , ), short for "emotion icon", also known simply as an emote, is a pictorial representation of a facial expression using characters—usually punctuation marks, numbers, and letters—to express a person's feelings, ...
= \frac \int_ G(t) \varphi_X(t) \, \mathrmt, where :$G\left(t\right) = \int_ g\left(x\right) e^ \, \mathrmx$ is the Fourier transform of $g\left(x\right).$ The expression for $\operatorname$
(X) An emoticon (, , rarely , ), short for "emotion icon", also known simply as an emote, is a pictorial representation of a facial expression using characters—usually punctuation marks, numbers, and letters—to express a person's feelings, ...
/math> also follows directly from
Plancherel theorem In mathematics, the Plancherel theorem (sometimes called the Parseval–Plancherel identity) is a result in harmonic analysis, proven by Michel Plancherel in 1910. It states that the integral of a function's squared modulus is equal to the inte ...
.

# Uses and applications

The expectation of a random variable plays an important role in a variety of contexts. For example, in
decision theory Decision theory (or the theory of choice; not to be confused with choice theory) is a branch of applied probability theory concerned with the theory of making decisions based on assigning probabilities to various factors and assigning numerical ...
, an agent making an optimal choice in the context of incomplete information is often assumed to maximize the expected value of their
utility function As a topic of economics, utility is used to model worth or value. Its usage has evolved significantly over time. The term was introduced initially as a measure of pleasure or happiness as part of the theory of utilitarianism by moral philosoph ...
. For a different example, in
statistics Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, indus ...
, where one seeks estimates for unknown parameters based on available data, the estimate itself is a random variable. In such settings, a desirable criterion for a "good" estimator is that it is '' unbiased''; that is, the expected value of the estimate is equal to the true value of the underlying parameter. It is possible to construct an expected value equal to the probability of an event, by taking the expectation of an
indicator function In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , one has \mathbf_(x)=1 if x\ ...
that is one if the event has occurred and zero otherwise. This relationship can be used to translate properties of expected values into properties of probabilities, e.g. using the
law of large numbers In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials s ...
to justify estimating probabilities by
frequencies Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from '' angular frequency''. Frequency is measured in hertz (Hz) which is e ...
. The expected values of the powers of ''X'' are called the moments of ''X''; the moments about the mean of ''X'' are expected values of powers of . The moments of some random variables can be used to specify their distributions, via their moment generating functions. To empirically
estimate Estimation (or estimating) is the process of finding an estimate or approximation, which is a value that is usable for some purpose even if input data may be incomplete, uncertain, or unstable. The value is nonetheless usable because it is de ...
the expected value of a random variable, one repeatedly measures observations of the variable and computes the
arithmetic mean In mathematics and statistics, the arithmetic mean ( ) or arithmetic average, or just the '' mean'' or the '' average'' (when the context is clear), is the sum of a collection of numbers divided by the count of numbers in the collection. The col ...
of the results. If the expected value exists, this procedure estimates the true expected value in an unbiased manner and has the property of minimizing the sum of the squares of the residuals (the sum of the squared differences between the observations and the
estimate Estimation (or estimating) is the process of finding an estimate or approximation, which is a value that is usable for some purpose even if input data may be incomplete, uncertain, or unstable. The value is nonetheless usable because it is de ...
). The
law of large numbers In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials s ...
demonstrates (under fairly mild conditions) that, as the
size Size in general is the magnitude or dimensions of a thing. More specifically, ''geometrical size'' (or ''spatial size'') can refer to linear dimensions ( length, width, height, diameter, perimeter), area, or volume. Size can also b ...
of the sample gets larger, the
variance In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers ...
of this
estimate Estimation (or estimating) is the process of finding an estimate or approximation, which is a value that is usable for some purpose even if input data may be incomplete, uncertain, or unstable. The value is nonetheless usable because it is de ...
gets smaller. This property is often exploited in a wide variety of applications, including general problems of statistical estimation and
machine learning Machine learning (ML) is a field of inquiry devoted to understanding and building methods that 'learn', that is, methods that leverage data to improve performance on some set of tasks. It is seen as a part of artificial intelligence. Machine ...
, to estimate (probabilistic) quantities of interest via
Monte Carlo methods Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be determini ...
, since most quantities of interest can be written in terms of expectation, e.g. In
classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by class ...
, the
center of mass In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force m ...
is an analogous concept to expectation. For example, suppose ''X'' is a discrete random variable with values ''xi'' and corresponding probabilities ''pi''. Now consider a weightless rod on which are placed weights, at locations ''xi'' along the rod and having masses ''pi'' (whose sum is one). The point at which the rod balances is E 'X'' Expected values can also be used to compute the
variance In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers ...
, by means of the computational formula for the variance : A very important application of the expectation value is in the field of
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, qua ...
. The expectation value of a quantum mechanical operator $\hat$ operating on a
quantum state In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution i ...
vector $, \psi\rangle$ is written as $\langle\hat\rangle = \langle\psi, A, \psi\rangle$. The
uncertainty Uncertainty refers to epistemic situations involving imperfect or unknown information. It applies to predictions of future events, to physical measurements that are already made, or to the unknown. Uncertainty arises in partially observable o ...
in $\hat$ can be calculated by the formula $\left(\Delta A\right)^2 = \langle\hat^2\rangle - \langle \hat \rangle^2$.

*
Center of mass In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force m ...
*
Central tendency In statistics, a central tendency (or measure of central tendency) is a central or typical value for a probability distribution.Weisberg H.F (1992) ''Central Tendency and Variability'', Sage University Paper Series on Quantitative Applications in ...
*
Chebyshev's inequality In probability theory, Chebyshev's inequality (also called the Bienaymé–Chebyshev inequality) guarantees that, for a wide class of probability distributions, no more than a certain fraction of values can be more than a certain distance from t ...
(an inequality on location and scale parameters) *
Conditional expectation In probability theory, the conditional expectation, conditional expected value, or conditional mean of a random variable is its expected value – the value it would take “on average” over an arbitrarily large number of occurrences – given ...
*
Expectation Expectation or Expectations may refer to: Science * Expectation (epistemic) * Expected value, in mathematical probability theory * Expectation value (quantum mechanics) * Expectation–maximization algorithm, in statistics Music * ''Expectatio ...
(the general term) * Expectation value (quantum mechanics) * Law of total expectation—the expected value of the conditional expected value of ''X'' given ''Y'' is the same as the expected value of ''X''. *
Moment (mathematics) In mathematics, the moments of a function are certain quantitative measures related to the shape of the function's graph. If the function represents mass density, then the zeroth moment is the total mass, the first moment (normalized by total ma ...
* Nonlinear expectation (a generalization of the expected value) *
Sample mean The sample mean (or "empirical mean") and the sample covariance are statistics computed from a sample of data on one or more random variables. The sample mean is the average value (or mean value) of a sample of numbers taken from a larger ...
*
Population mean In statistics, a population is a set of similar items or events which is of interest for some question or experiment. A statistical population can be a group of existing objects (e.g. the set of all stars within the Milky Way galaxy) or a h ...
* Wald's equation—an equation for calculating the expected value of a random number of random variables

# Literature

* * * * * * * * *