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{{About|the term used in probability theory and statistics}}
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In [[probability theory]], the '''expected value''' of a [[random variable]] is a key aspect of its [[probability distribution]]. Intuitively, a random variable's expected value represents the average of a large number of independent realizations of the random variable. For example, the expected value of rolling a six-sided [[dice|<!--THERE IS-->d<!--NO LETTER C-->i<!--IN THE SINGULAR-->e<!--OF 'DICE'-->]] is 3.5, because the average of all the numbers that come up converges to 3.5 as the number of rolls [[law of large numbers| approaches]] infinity (see {{section link||Examples}} for details). The expected value is also known as the '''expectation''', '''mathematical expectation''', '''mean''', or '''first moment'''.

A bit more formally, the expected value of a [[discrete random variable]] is the [[Weighted average|probability-weighted average]] of all its possible values. In other words, each possible value the random variable can assume is multiplied by its probability of occurring, and the resulting products are summed to produce the expected value. The same principle applies to an [[absolute continuity#Absolute continuity of measures|absolutely continuous]] random variable, except that an [[Lebesgue integral|integral]] of the variable with respect to its [[Probability density function|probability density]] replaces the sum. The formal definition subsumes both of these and also works for distributions which are  neither discrete nor absolutely continuous; the expected value of a random variable is the integral of the random variable with respect to its [[probability measure]].<ref name=Ross>{{cite book |title=Introduction to probability models |author=Sheldon M Ross |edition=9th |publisher=Academic Press |isbn=0-12-598062-0 |year=2007 |chapter=§2.4 Expectation of a random variable |page=38 ''ff'' |url=https://books.google.com/books?id=12Pk5zZFirEC&pg=PA38 }}</ref><ref name=Hamming>{{cite book |title=The art of probability for scientists and engineers |author=[[Richard W Hamming]] |chapter=§2.5 Random variables, mean and the expected value |page=64 ''ff'' |publisher=Addison–Wesley |year=1991 |isbn=0-201-40686-1 |url=https://books.google.com/books?id=jX_F-77TA3gC&pg=PA64 }}</ref>

The expectation of a random variable plays an important role in a variety of contexts. For example, in [[decision theory]], an agent making an optimal choice in the context of incomplete information is often assumed to [[expected utility maximization|maximize]] the expected value of their [[von Neumann–Morgenstern utility function| utility function]].  
For a different example, in [[statistics]], 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 estimator| unbiased]] – that is, the expected value of the estimate is equal to the true value of the underlying parameter. 

{{TOC limit|3}}

==History==
{{refimprove section|date=October 2019}}
The idea of the expected value originated in the middle of the 17th century from the study of the so-called [[problem of points]], which seeks to divide the stakes ''in a fair way'' between two players who have to end their game before it's properly finished. This problem had been debated for centuries, and many conflicting proposals and solutions had been suggested over the years, when it was posed in 1654 to [[Blaise Pascal]] by French writer and amateur mathematician [[Antoine Gombaud|Chevalier de Méré]]. 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 a now famous series of letters to [[Pierre de Fermat]]. 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 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.<ref>{{cite journal |title=Ore, Pascal and the Invention of Probability Theory |journal=The American Mathematical Monthly |volume=67 |issue=5 |year=1960 |pages=409–419 |doi=10.2307/2309286}}</ref>

Three years later, in 1657, a Dutch mathematician [[Christiaan Huygens]], who had just visited Paris, published a treatise (see {{harvtxt|Huygens|1657}}) "''De ratiociniis in ludo aleæ''" on probability theory. In this book he considered the problem of points and presented a solution based on the same principle as the solutions of Pascal and Fermat. Huygens also 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). In this sense this book can be seen as the first successful attempt at laying down the foundations of the [[theory of probability]].

In the foreword to his book, Huygens wrote: "It should be said, also, that for some time some of the best mathematicians of France have occupied themselves with this kind of calculus so that no one should attribute to me the honour of the first invention. This does not belong to me. But these savants, although they put each other to the test by proposing to each other many questions difficult to solve, have hidden their methods. I have had therefore to examine and go deeply for myself into this matter by beginning with the elements, and it is impossible for me for this reason to affirm that I have even started from the same principle. But finally I have found that my answers in many cases do not differ from theirs." (cited by {{harvtxt|Edwards|2002}}). Thus, Huygens learned about [[de Méré's Problem]] in 1655 during his visit to France; later on in 1656 from his correspondence with Carcavi he learned that his method was essentially the same as Pascal's; so that before his book went to press in 1657 he knew about Pascal's priority in this subject.

===Etymology===
Neither Pascal nor Huygens used the term "expectation" in its modern sense. In particular, Huygens writes: "That my Chance or Expectation to win any thing is worth just such a Sum, as wou'd procure me in the same Chance and Expectation at a fair Lay. ... If I expect a or b, and have an equal Chance of gaining them, my Expectation is worth {{frac2|a+b|2}}." More than a hundred years later, in 1814, [[Pierre-Simon Laplace]] published his tract "''Théorie analytique des probabilités''", where the concept of expected value was defined explicitly:

{{quote|… this advantage in the theory of chance is the product of the sum hoped for by the probability of obtaining it; it is the partial sum which ought to result when we do not wish to run the risks of the event in supposing that the division is made proportional to the probabilities. This division is the only equitable one when all strange circumstances are eliminated; because an equal degree of probability gives an equal right for the sum hoped for. We will call this advantage ''mathematical hope''.}}

The use of the letter ''E'' to denote expected value goes back to [[William Allen Whitworth|W. A. Whitworth]] in 1901,<ref>Whitworth, W.A. (1901) ''Choice and Chance with One Thousand Exercises''. Fifth edition. Deighton Bell, Cambridge. [Reprinted by Hafner Publishing Co., New York, 1959.]</ref> who used a script E. The symbol has become popular since for English writers it meant "Expectation", for Germans "Erwartungswert", for Spanish "Esperanza matemática" and for French "Espérance mathématique".<ref>{{cite web|title=Earliest uses of symbols in probability and statistics|url=http://jeff560.tripod.com/stat.html}}</ref>

==Definition==

===Finite case===
Let <math>X</math> be a random variable with a finite number of finite outcomes <math>x_1, x_2, \ldots, x_k</math> occurring with probabilities <math>p_1, p_2, \ldots, p_k,</math> respectively. The '''expectation''' of <math>X</math> is defined as

:<math>\operatorname{E}[X] =\sum_{i=1}^k x_i\,p_i=x_1p_1 + x_2p_2 + \cdots + x_kp_k.</math>

Since all probabilities <math>p_i</math> add up to 1 (<math>p_1 + p_2 + \cdots + p_k = 1</math>), the expected value is the [[weighted average]], with <math>p_i</math>’s being the weights.

If all outcomes <math>x_i</math> are [[equiprobable]] (that is, <math>p_1 = p_2 = \cdots = p_k</math>), then the weighted average turns into the simple [[arithmetic mean|average]]. If the outcomes <math>x_i</math> are not equiprobable, then the simple average must be replaced with the weighted average, which takes into account the fact that some outcomes are more likely than the others. 

[[File:Largenumbers.svg|An illustration of the convergence of sequence averages of rolls of a {{dice}} to the expected value of 3.5 as the number of rolls (trials) grows.|thumb|right|400 px]]

====Examples====
*Let <math>X</math> represent the outcome of a roll of a fair six-sided {{dice}}. More specifically, <math>X</math> will be the number of [[Pip (counting)|pips]] showing on the top face of the {{dice}} after the toss. The possible values for <math>X</math> are 1, 2, 3, 4, 5, and 6, all of which are equally likely with a probability of {{frac2|1|6}}. The expectation of <math>X</math> is
:: <math>\operatorname{E}[X] = 1\cdot\frac16 + 2\cdot\frac16 + 3\cdot\frac16 + 4\cdot\frac16 + 5\cdot\frac16 + 6\cdot\frac16 = 3.5.</math>

:If one rolls the {{dice}} <math>n</math> times and computes the average ([[arithmetic mean]]) of the results, then as <math>n</math> grows, the average will [[almost surely]] [[Convergent sequence|converge]] to the expected value, a fact known as the [[strong law of large numbers]]. 

*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 <math>X</math> represents the (monetary) outcome of a $1 bet on a single number ("straight up" bet). If the bet wins (which happens with probability {{frac2|1|38}} in American roulette), the payoff is $35; otherwise the player loses the bet. The expected profit from such a bet will be
:: <math>   \operatorname{E}[\,\text{gain from }\$1\text{ bet}\,] = -\$1 \cdot \frac{37}{38} + \$35 \cdot \frac{1}{38} = -\$\frac{1}{19}.</math>

:That is, the bet of $1 stands to lose <math>-\$\frac{1}{19}</math>, so its expected value is <math>-\$\frac{1}{19}.</math>

===Countably infinite case===
Intuitively, the expectation of a random variable taking values in a countable set of outcomes is defined analogously as the weighted sum of the outcome values, where the weights correspond to the probabilities of realizing that value. However, convergence issues associated with the infinite sum necessitate a more careful definition. A rigorous definition first defines expectation of a non-negative random variable, and then adapts it to general random variables. 

Let <math>X</math> be a non-negative random variable with a countable set of outcomes <math>x_1, x_2, \ldots,</math> occurring with probabilities <math>p_1, p_2, \ldots,</math> respectively. Analogous to the discrete case, the expected value of <math>X</math> is then defined as the series
: <math>    \operatorname{E}[X] = \sum_{i=1}^\infty x_i\, p_i.</math>
Note that since <math>x_i p_i \geq 0</math>, the infinite sum is well-defined and does not depend on the [[absolute convergence|order]] in which it is computed. Unlike the discrete case, the expectation here can be equal to infinity, if the infinite sum above increases without bound. 

For a general random variable <math>X</math> that need not be non-negative, first define <math> X^+ = \max\{X, 0\}</math> and <math> X^- = \max\{-X, 0\}</math>. Observe that <math> X = X^+ - X^- </math>, and both <math> X^+ </math> and <math> X^- </math> are non-negative random variables. Hence, <math> \operatorname{E}[X^+] </math> and <math> \operatorname{E}[X^-] </math> are well-defined (using either the definition for finite discrete random variables or non-negative countable random variables). Then, we define <math> \operatorname{E}[X] </math> as follows:
:<math>
\operatorname{E}[X] = \begin{cases}  \operatorname{E}[X^+] -  \operatorname{E}[X^-] & \text{if }  \operatorname{E}[X^+] < \infty \text{ and } \operatorname{E}[X^-] < \infty;\\
\infty  & \text{if }  \operatorname{E}[X^+] = \infty \text{ and } \operatorname{E}[X^-] < \infty;\\
-\infty &   \text{if }  \operatorname{E}[X^+] < \infty \text{ and } \operatorname{E}[X^-] = \infty;\\
\text{undefined} &  \text{if }  \operatorname{E}[X^+] = \infty \text{ and } \operatorname{E}[X^-] = \infty.\\
\end{cases}
</math>

====Examples====

*Suppose <math>x_i = i</math> and <math> p_i = \frac{k}{i2^i},</math> for <math>i = 1, 2, 3, \ldots</math>, where <math>k = \frac{1}{\ln 2}</math> (with <math>\ln</math> being the [[natural logarithm]]) is the scale factor such that the probabilities sum to 1. Then, using the direct definition for non-negative random variables, we have
::<math>\operatorname{E}[X]= \sum_i x_i p_i = 1\left(\frac{k}{2}\right) + 2\left(\frac{k}{8}\right) + 3\left(\frac{k}{24}\right) + \dots = \frac{k}{2}+\frac{k}{4}+\frac{k}{8}+\dots = k.</math> 

*An example where the expectation is infinite arises in the context of the [[St. Petersburg paradox]]. Let <math>x_i=2^i</math> and <math>p_i=\frac{1}{2^i}</math> for <math>i = 1, 2, 3, \ldots</math>. Once again, since the random variable is non-negative, the expected value calculation gives
::<math> \operatorname{E}[X]= \sum_{i=1}^\infty x_i\,p_i  =2\cdot \frac{1}{2}+4\cdot\frac{1}{4} + 8\cdot\frac{1}{8}+ 16\cdot\frac{1}{16}+ \cdots = 1 + 1 + 1 + 1 + \cdots  \, = \infty. </math>

*For an example where the expectation is not well-defined, suppose the random variable <math>X</math> takes values 1, −2, 3, −4, ..., with respective probabilities <math>\frac{c}{1^2},\frac{c}{2^2},\frac{c}{3^2},\frac{c}{4^2}</math>, ..., where <math>c=\frac{6}{\pi^2}</math> is a normalizing constant that ensures the probabilities sum up to one. 
:Then, it follows that <math>X^+</math> takes value <math>2k-1</math> with probability <math>c/(2k-1)^2</math> for <math> k = 1,2,3,\cdots</math> and takes value <math>0</math> with remaining probability. Similarly, <math>X^-</math> takes value <math>2k</math> with probability <math>c/(2k)^2</math> for <math> k = 1,2,3,\cdots</math> and takes value <math>0</math> with remaining probability. Using the definition for non-negative random variables, one can show that both <math>\operatorname{E}[X^+] = \infty </math> and <math>\operatorname{E}[X^-] = \infty </math> (see [[harmonic series (mathematics)| Harmonic series]]). Hence, the expectation of <math>X</math> is not well-defined.

===Absolutely continuous case===
If <math>X</math> is a random variable whose [[cumulative distribution function]] admits a [[probability density function|density]] <math>f(x)</math>, then the expected value is defined as the following [[Lebesgue integral]], if the integral exists:
: <math>    \operatorname{E}[X] = \int_{\R} x f(x)\, dx.</math>

The expected value of a random variable may be undefined, if the integral does not exist. An example of such a random variable is one with the [[Cauchy distribution]]<ref name=Hamming2>{{cite book |title=The art of probability for scientists and engineers |author=[[Richard W Hamming]] |chapter=Example 8.7–1 The Cauchy distribution |page=290 ''ff'' |publisher=Addison-Wesley |year=1991 |isbn=0-201-40686-1 |url=https://books.google.com/books?id=jX_F-77TA3gC&printsec=frontcover#v=onepage&q=Cauchy |quote=Sampling from the Cauchy distribution and averaging gets you nowhere — one sample has the same distribution as the average of 1000 samples!}}</ref>, due to its [[Heavy-tailed distribution|large "tails"]].

===General case===
In general, if <math>X</math> is a non-negative [[random variable]] defined on a [[probability space]] <math>(\Omega,\Sigma,\operatorname{P})</math>, then the expected value of <math>X</math>, denoted by <math>\operatorname{E}[X]</math>, is defined as the [[Lebesgue integration|Lebesgue integral]] 
:<math>\operatorname{E} [X]  = \int_\Omega X(\omega)\,d\operatorname{P}(\omega).</math>

For a general random variable <math>X</math>, define as before <math>X^+(\omega)=\max(X(\omega),0)</math> and <math>X^-(\omega)=-\min(X(\omega),0)</math>, and note that <math>X=X^+-X^-</math>, with both <math>X^+</math> and <math>X^-</math> nonnegative. Then, the expected value of <math>X</math> is defined as 
:<math>
\operatorname{E}[X] = \begin{cases}  \operatorname{E}[X^+] -  \operatorname{E}[X^-] & \text{if }  \operatorname{E}[X^+] < \infty \text{ and } \operatorname{E}[X^-] < \infty;\\
\infty  & \text{if }  \operatorname{E}[X^+] = \infty \text{ and } \operatorname{E}[X^-] < \infty;\\
-\infty &   \text{if }  \operatorname{E}[X^+] < \infty \text{ and } \operatorname{E}[X^-] = \infty;\\
\text{undefined} &  \text{if }  \operatorname{E}[X^+] = \infty \text{ and } \operatorname{E}[X^-] = \infty.
\end{cases}
</math>

For multidimensional random variables, their expected value is defined per component, i.e.

:<math>\operatorname{E}[(X_1,\ldots,X_n)]=(\operatorname{E}[X_1],\ldots,\operatorname{E}[X_n])</math>

and, for a random matrix <math>X</math> with elements <math>X_{ij}</math>, <math>(\operatorname{E}[X])_{ij}=\operatorname{E}[X_{ij}].</math>

==Uses and applications==
It is possible to construct an expected value equal to the probability of an event by taking the expectation of an [[indicator function]] 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 to justify estimating probabilities by [[Statistical frequency|frequencies]].

The expected values of the powers of ''X'' are called the [[moment (mathematics)|moments]] of ''X''; the [[moment about the mean|moments about the mean]] of ''X'' are expected values of powers of ''X'' − E[''X'']. The moments of some random variables can be used to specify their distributions, via their [[moment generating function]]s.

To empirically [[Estimation theory|estimate]] the expected value of a random variable, one repeatedly measures observations of the variable and computes the [[arithmetic mean]] of the results. If the expected value exists, this procedure estimates the true expected value in an [[estimator bias|unbiased]] manner and has the property of minimizing the sum of the squares of the [[errors and residuals in statistics|residuals]] (the sum of the squared differences between the observations and the [[estimator|estimate]]). The [[law of large numbers]] demonstrates (under fairly mild conditions) that, as the [[Sample size|size]] of the [[statistical sample|sample]] gets larger, the [[variance]] of this [[estimator|estimate]] gets smaller.

This property is often exploited in a wide variety of applications, including general problems of [[statistical estimation]] and [[machine learning]], to estimate (probabilistic) quantities of interest via [[Monte Carlo methods]], since most quantities of interest can be written in terms of expectation, e.g. <math>\operatorname{P}({X \in \mathcal{A}}) = \operatorname{E}[{\mathbf 1}_{\mathcal{A}}]</math>, where <math> {\mathbf 1}_{\mathcal{A}}</math> is the indicator function of the set <math>\mathcal{A}</math>.

[[File:Beta first moment.svg|thumb|The mass of probability distribution is balanced at the expected value, here a Beta(α,β) distribution with expected value α/(α+β).]] In [[classical mechanics]], the [[center of mass]] is an analogous concept to expectation. For example, suppose ''X'' is a discrete random variable with values ''x<sub>i</sub>'' and corresponding probabilities ''p<sub>i</sub>''. Now consider a weightless rod on which are placed weights, at locations ''x<sub>i</sub>'' along the rod and having masses ''p<sub>i</sub>'' (whose sum is one). The point at which the rod balances is E[''X''].

Expected values can also be used to compute the [[variance]], by means of the [[computational formula for the variance]]

:<math>\operatorname{Var}(X)=  \operatorname{E}[X^2] - (\operatorname{E}[X])^2.</math>

A very important application of the expectation value is in the field of [[quantum mechanics]]. The expectation value of a quantum mechanical operator <math>\hat{A}</math> operating on a [[quantum state]] vector <math>|\psi\rangle</math> is written as <math>\langle\hat{A}\rangle = \langle\psi|A|\psi\rangle</math>. The [[uncertainty principle|uncertainty]] in <math>\hat{A}</math> can be calculated using the formula <math>(\Delta A)^2 = \langle\hat{A}^2\rangle - \langle \hat{A} \rangle^2 </math>.

==Basic properties==
{{Technical|section|date=February 2019}}
The basic properties below replicate or follow immediately from those of [[Lebesgue integral]].

* Let  <math> {\mathbf 1}_A </math> denote the [[indicator function]] of an [[Event (probability theory)|event]] <math>A</math>. Then <math> \operatorname{E}[{\mathbf 1}_A] =1\cdot\operatorname{P}(A)+0\cdot\operatorname{P}(\Omega\setminus A) = \operatorname{P}(A). </math> 

* If <math>X = Y</math>  [[almost surely| (a.s.)]],  then <math> \operatorname{E}[ X] = \operatorname{E}[ Y]</math>.

* If <math>X=c</math> (a.s.) for some constant <math>c\in [-\infty,+\infty]</math>, then <math>\operatorname{E}[X]=c</math>. In particular, for a random variable <math>X</math> with well-defined expectation, <math>\operatorname{E}[\operatorname{E}[X]]=\operatorname{E}[X]</math>.

* ''' Non-negativity:''' If <math>X \geq 0 </math>  (a.s.),  then <math> \operatorname{E}[ X] \geq 0</math>.

* '''Linearity of expectation:''' The expected value operator (or '''expectation operator''') <math>\operatorname{E}[\cdot]</math> is [[linear operator|linear]] in the sense that, for any random variables <math>X</math> and <math>Y</math>, and a constant <math>a</math>,
::<math>\begin{align}
  \operatorname{E}[X + Y] &=   \operatorname{E}[X] + \operatorname{E}[Y], \\
  \operatorname{E}[aX]    &= a \operatorname{E}[X],
\end{align}
</math>
:whenever the right-hand side is well-defined. 

* The following statements regarding a random variable <math>X</math> are equivalent:
** <math>\operatorname{E}[X]</math> exists and is finite.
** Both <math>\operatorname{E}[X^+]</math> and <math>\operatorname{E}[X^-]</math> are finite.
** <math>\operatorname{E}[|X|]</math> is finite.
:Sketch of proof: Indeed, <math> |X| = X^+ + X^- </math>. By linearity, <math> \operatorname{E}[|X|] = \operatorname{E}[X^+] + \operatorname{E}[X^-] </math>.
:For the reasons above, the expressions "<math>X</math> is integrable" and "the expected value of <math>X</math> is finite" are used interchangeably throughout this article.

* For a random variable <math>X</math> with well-defined expectation: <math> |\operatorname{E}[X]| \leq \operatorname{E}|X|</math>. 
:The proof follows from the [[triangle inequality]] and the  linearity of expectation, as follows:
::<math>
\begin{align}
|\operatorname{E}[X]| &= \Bigl|\operatorname{E}[X^+]-\operatorname{E}[X^-]\Bigr|\leq\Bigl|\operatorname{E}[X^+]\Bigr|+\Bigl|\operatorname{E}[X^-]\Bigr|
                     &=\operatorname{E}[X^+]+\operatorname{E}[X^-]=\operatorname{E}[X^+ + X^-]
                     &=\operatorname{E}|X|.
\end{align}
</math>
:This result is a special case of the more general [[Jensen's inequality|Jensen's inequality]].

* '''Monotonicity:''' If <math>X\leq Y</math> (a.s.), and both <math>\operatorname{E}[X]</math> and <math>\operatorname{E}[Y]</math> exist, then <math>\operatorname{E}[X]\leq\operatorname{E}[Y]</math>.
: Proof follows from the linearity and the non-negativity property for <math>Z=Y-X</math>, since <math>Z\geq 0</math> (a.s.).

* '''Non-degeneracy:''' If <math>\operatorname{E}|X|=0</math>, then <math>X=0</math> (a.s.).

* If <math>\operatorname{E}[X]<+\infty</math> then <math>X<+\infty</math> (a.s.). Similarly, if <math>\operatorname{E}[X]>-\infty</math> then <math>X>-\infty</math> (a.s.). 

* '''Non-multiplicativity:''' In general, the expected value operator is not multiplicative, i.e. <math>\operatorname{E}[XY]</math> is not necessarily equal to <math>\operatorname{E}[X]\cdot \operatorname{E}[Y]</math>. Indeed, let <math>X</math> assume the values of 1 and −1 with probability 0.5 each. Then
::<math>\left(\operatorname{E}[X]\right)^2=\left(\frac{1}{2}\cdot (-1)+\frac{1}{2}\cdot 1\right)^2=0,</math>
:and
::<math>\operatorname{E}[X^2]=\frac{1}{2}\cdot (-1)^2+\frac{1}{2}\cdot 1^2=1, \text{ so } \operatorname{E}[X^2]\neq(\operatorname{E}[X])^2.</math>
:However, if <math>X</math> and <math>Y</math> are [[independent random variables|independent]], then indeed one can show that <math>\operatorname{E}[XY]=\operatorname{E}[X] \operatorname{E}[Y]</math>.

* '''[[Law of the unconscious statistician]]:''' The expected value of a measurable function of <math>X</math>, <math>g(X)</math>, given that <math>X</math> has a probability density function <math>f(x)</math>, is given by the [[inner product]] of <math>f</math> and <math>g</math>:
::<math>\operatorname{E}[g(X)] = \int_{\R} g(x) f(x)\, dx .</math>
:This formula also holds in multidimensional case, when <math>g</math> is a function of several random variables, and <math>f</math> is their [[Probability density function#Densities associated with multiple variables|joint density]].<ref name=wolframMultipleContinuousVariables>{{Citation
| url   = http://mathworld.wolfram.com/ExpectationValue.html
| title = Expectation Value
| work  = 
| publisher  = 
| accessdate = August 8, 2017
}}</ref><ref name=Pap84>{{Citation
| last       = Papoulis
| first      = A.
| year       = 1984
| title      = Probability, Random Variables, and Stochastic Processes
| publisher  = New York: McGraw–Hill
| publication-place =
| pages       = 139–152
| url        = 
| accessdate =
}}</ref>

==Interchanging limits and expectation==

In general, it is not the case that <math>\operatorname{E}[X_n] \to \operatorname{E}[X]</math> despite <math>X_n\to X</math> pointwise. Thus, one cannot interchange limits and expectation, without additional conditions on the random variables. To see this, let <math>U</math> be a random variable distributed uniformly on <math>[0,1]</math>. For <math>n\geq 1,</math> define a sequence of random variables
:<math>
X_n = n \cdot \mathbf{1}\left\{ U \in  \left[0,\tfrac{1}{n}\right]\right\},
</math>
with <math>{\mathbf 1}\{A\}</math> being the indicator function of the event <math>A</math>. Then, it follows that <math>X_n \to 0</math> (a.s). But, <math>\operatorname{E}[X_n] = n \cdot \operatorname{P}\left(U \in \left[ 0, \tfrac{1}{n}\right] \right) = n \cdot \tfrac{1}{n} = 1</math> for each <math>n</math>. Hence, <math> \lim_{n \to \infty} \operatorname{E}[X_n] = 1 \neq 0 = \operatorname{E}\left[ \lim_{n \to \infty} X_n \right].</math>

Analogously, for general sequence of random variables <math>\{ Y_n : n \geq 0\}</math>, the expected value operator is not <math>\sigma</math>-additive, i.e.
:<math>
\operatorname{E}\left[\sum^\infty_{n=0} Y_n\right] \neq \sum^\infty_{n=0}\operatorname{E}[Y_n].
</math>
An example is easily obtained by setting <math>Y_0 = X_1</math> and <math>Y_n = X_{n+1} - X_n</math> for <math>n \geq 1</math>, where <math>X_n</math> is as in the previous example.

A number of convergence results specify exact conditions which allow one to interchange limits and expectations, as specified below. 

* [[Monotone convergence theorem]]: Let <math>\{X_n : n \geq 0\}</math> be a sequence of random variables, with <math>0 \leq X_n \leq X_{n+1}</math> (a.s) for each <math> n \geq 0</math>. Furthermore, let <math> X_n \to X </math> pointwise. Then, the monotone convergence theorem states that <math>\lim_n\operatorname{E}[X_n]=\operatorname{E}[X].</math>
:Using the monotone convergence theorem, one can show that expectation indeed satisfies countable additivity for non-negative random variables. In particular, let <math>\{X_i\}^\infty_{i=0}</math> be non-negative random variables.  It follows from [[#Monotone convergence theorem|monotone convergence theorem]] that
::<math>
\operatorname{E}\left[\sum^\infty_{i=0}X_i\right] = \sum^\infty_{i=0}\operatorname{E}[X_i].
</math>
* [[Fatou's lemma]]: Let <math>\{ X_n \geq 0 : n \geq 0\}</math> be a sequence of non-negative random variables. Fatou's lemma states that
:<math>\operatorname{E}[\liminf_n X_n] \leq \liminf_n \operatorname{E}[X_n]. </math>

:'''Corollary.''' Let <math> X_n \geq 0</math> with <math>\operatorname{E}[X_n] \leq C </math> for all <math> n \geq 0</math>. If <math>X_n \to X</math> (a.s), then <math>\operatorname{E}[X] \leq C. </math>
:'''Proof''' is by observing that <math>\textstyle X = \liminf_n X_n</math> (a.s.) and applying Fatou's lemma.

* [[Dominated convergence theorem]]: Let <math>\{X_n : n \geq 0 \}</math> be a sequence of random variables. If <math>X_n\to X</math> [[pointwise convergence|pointwise]] (a.s.), <math>|X_n|\leq Y \leq +\infty</math> (a.s.), and <math>\operatorname{E}[Y]<\infty</math>. Then, according to the dominated convergence theorem,
**<math>\operatorname{E}|X| \leq \operatorname{E}[Y] <\infty</math>;
**<math>\lim_n\operatorname{E}[X_n]=\operatorname{E}[X]</math>
** <math>\lim_n\operatorname{E}|X_n - X| = 0. </math>

* [[Uniform integrability]]: In some cases, the equality <math>\displaystyle\lim_n\operatorname{E}[X_n]=\operatorname{E}[\lim_n X_n]</math> holds when the sequence <math>\{X_n\}</math> is ''uniformly integrable''.

==Inequalities==
There are a number of inequalities involving the expected values of functions of random variables. The following list includes some of the more basic ones.

* [[Markov's inequality]]: For a ''nonnegative'' random variable <math>X</math> and <math>a>0</math>, Markov's inequality states that
:<math>
\operatorname{P}(X\geq a)\leq\frac{\operatorname{E}[X]}{a}.
</math>

* [[Bienaymé-Chebyshev inequality]]: Let <math>X</math> be an arbitrary random variable with finite expected value <math>\operatorname{E}[X]</math> and finite [[variance]] <math>\operatorname{Var}[X]\neq 0</math>. The Bienaymé-Chebyshev inequality states that, for any real number <math>k>0</math>,
:<math>
\operatorname{P}\Bigl(\Bigl|X-\operatorname{E}[X]\Bigr|\geq k\sqrt{\operatorname{Var}[X]}\Bigr)\leq\frac{1}{k^2}.
</math>

* [[Jensen's inequality]]: Let <math>f:{\mathbb R}\to{\mathbb R}</math> be a [[Measurable function|measurable]] [[convex function]] and <math>X</math> a random variable such that <math>\operatorname{E}|X|<\infty</math>. Jensen's inequality states that
:<math>
    f(\operatorname{E}(X)) \leq \operatorname{E} (f(X)).
</math>
: For example, Jensen's inequality implies that <math>|\operatorname{E}[X]|\leq \operatorname{E}|X|</math> since the absolute value function is convex.

* [[Lyapunov's inequality]]: Let <math>0<s<t</math>. Lyapunov's inequality states that
:<math>
\left(\operatorname{E}|X|^s\right)^{1/s}\leq\left(\operatorname{E}|X|^t\right)^{1/t}.
</math>
:'''Proof.''' Applying [[#Jensen's inequality|Jensen's inequality]] to <math>|X|^s</math> and <math>g(x)=|x|^{t/s}</math>, obtain <math>\Bigl|\operatorname{E}|X^s|\Bigr|^{t/s}\leq\operatorname{E}|X^s|^{t/s}=\operatorname{E}|X|^t</math>. Taking the <math>t^{th}</math> root of each side completes the proof.

* [[Cauchy–Bunyakovsky–Schwarz inequality]]: The Cauchy–Bunyakovsky–Schwarz inequality states that
:<math>
(\operatorname{E}[XY])^2\leq\operatorname{E}[X^2]\cdot\operatorname{E}[Y^2].
</math>

* [[Hölder's inequality]]: Let <math>p</math> and <math>q</math> satisfy <math>1\leq p\leq\infty</math>, <math>1\leq q\leq\infty</math>, and <math>1/p+1/q=1</math>. The Hölder's inequality states that
:<math>
\operatorname{E}|XY|\leq(\operatorname{E}|X|^p)^{1/p}(\operatorname{E}|Y|^q)^{1/q}.
</math>

* [[Minkowski inequality]]: Let <math>p</math> be an integer satisfying <math>1\leq p\leq\infty</math>. Let, in addition, <math>\operatorname{E}|X|^p<\infty</math> and <math>\operatorname{E}|Y|^p<\infty</math>. Then, according to the Minkowski inequality, <math>\operatorname{E}|X+Y|^p<\infty</math> and
:<math>
\Bigl(\operatorname{E}|X+Y|^p\Bigr)^{1/p}\leq\Bigl(\operatorname{E}|X|^p\Bigr)^{1/p}+\Bigl(\operatorname{E}|Y|^p\Bigr)^{1/p}.
</math>

==Relationship with characteristic function==
The probability density function <math>f_X</math> of a scalar random variable <math>X</math> is related to its [[characteristic function (probability)|characteristic function]] <math>\varphi_X</math> by the inversion formula:
: <math>f_X(x) = \frac{1}{2\pi}\int_{\mathbb{R}} e^{-itx}\varphi_X(t) \, dt.</math>

For the expected value of <math>g(X)</math> (where <math>g:{\mathbb R}\to{\mathbb R}</math> is a Borel function), we can use this inversion formula to obtain

:<math> \operatorname{E}[g(X)] = \frac{1}{2\pi} \int_{\mathbb R} g(x)\left[ \int_{\mathbb R} e^{-itx}\varphi_X(t) \, dt \right]\,dx.</math>

If <math>\operatorname{E}[g(X)]</math> is finite, changing the order of integration, we get, in accordance with [[Fubini theorem|Fubini–Tonelli theorem]],

:<math> \operatorname{E}[g(X)] = \frac{1}{2\pi} \int_{\mathbb R} G(t) \varphi_X(t) \, dt,</math>

where

:<math>G(t) = \int_{\mathbb R} g(x) e^{-itx} \, dx</math>

is the Fourier transform of <math> g(x). </math> The expression for <math>\operatorname{E}[g(X)]</math> also follows directly from [[Plancherel theorem]].

==Alternative formula for expected value==
For non-negative random variables, one can compute the expected value using an alternative formula involving only the [[cumulative distribution function]] of the random variable. Similar formulas for general random variables can be obtained by using the relation <math>X = X^{+} - X^{-}</math> and noting that both <math>X^{+}</math> and  <math>X^{-}</math> are non-negative, for which the following applies.   

====Finite and countably infinite case====
For a non-negative integer-valued random variable <math>X:\Omega\to\{0,1,2,3,\ldots\}\cup\{+\infty\}, </math>

:<math> \operatorname{E}[X]=\sum _{n=0}^\infty \operatorname{P}(X> n) = \sum_{n=0}^\infty \bar{F}(n),</math>
where <math>\bar{F}(x) = 1 -F(x)</math>, and <math>F(x) = P(X \leq x)</math> is the cdf of <math>X</math>.

{| role="presentation" class="wikitable mw-collapsible mw-collapsed"
| '''Proof.'''
|-
|
If <math> \operatorname{P}(X=+\infty) > 0, </math> then <math>\operatorname{E}[X] = +\infty.</math> On the other hand,

:<math>\operatorname{P}(X > n) \geq \operatorname{P}( X=+\infty ) > 0,</math>

so the series on the right diverges to <math>+\infty,</math> and the equality holds.

If <math> \operatorname{P}(X=+\infty) = 0, </math> then

:<math> \sum _{n=0}^\infty \operatorname{P} (X > n) = \sum _{n=0}^\infty \sum _{j=n+1}^\infty \operatorname{P}(X = j).</math>

Let
:<math>
M=
  \begin{bmatrix}
    \operatorname{P}(X=1)&\operatorname{P}(X=2)&\operatorname{P}(X=3)&\cdots&\operatorname{P}(X=n)& \cdots \\
                         &\operatorname{P}(X=2)&\operatorname{P}(X=3)&\cdots&\operatorname{P}(X=n)& \cdots \\
                         &                     &\operatorname{P}(X=3)&\cdots&\operatorname{P}(X=n)& \cdots \\
                         &                     &                     &\ddots&\vdots               &        \\
                         &                     &                     &      &\operatorname{P}(X=n)& \cdots \\
                         &                     &                     &      &                     & \ddots
  \end{bmatrix}
</math>
be an infinite upper triangular matrix. The double series <math>\textstyle \sum _{i=1}^\infty \sum _{j=i}^\infty \operatorname{P}(X = j)</math> is the sum of <math>M</math>'s elements if summation is done row by row.
Since every summand is non-negative, the series either converges absolutely or diverges to <math>+\infty.</math> In both cases, changing summation order does not affect the sum.  Changing summation order, from row-by-row to column-by-column, gives us

:<math>\begin{align}
\sum _{n=0}^\infty \sum _{j=n+1}^\infty \operatorname{P}(X = j) &=\sum _{j=1}^\infty \sum _{n=0}^{j-1} \operatorname{P}(X = j)\\
                   &=\sum _{j=1}^\infty j \operatorname{P}(X = j)\\
                   &=\sum _{j=0}^\infty j \operatorname{P}(X = j)\\
                   &=\operatorname{E}[X].
\end{align}</math>

=====Example=====
In a [[coin tossing]] experiment, let the probability of heads be <math>p</math>. Including the final attempt, how many tosses can we expect until the first head?

'''Solution.''' If <math>N</math> is the random variable indicating the numbers of coin tosses before and including the first head, then, for <math>n\geq 0</math>,
:<math>
\begin{align}
\operatorname{P}(N > n)&=1-\operatorname{P}(N\leq n)\\[1pt]
                         &=1-\sum\limits_{j=1}^{n}\operatorname{P}(N=j)\\[1pt]
                         &=1-\sum\limits_{j=1}^{n}(1-p)^{j-1}p\\[1pt]
                         &=1-\frac{1-(1-p)^{n}}{p}\cdot p\\[1pt]
                         &=(1-p)^{n},
\end{align}
</math>
where we took into account the [[geometric series]] summation formula. We now compute
:<math>
\begin{align}
\operatorname{E}[N]&=\sum\limits_{n=0}^{\infty}\operatorname{P}(N > n)\\
                   &=\sum\limits_{n=0}^\infty (1-p)^{n}\\
                   &=\frac{1}{p}.
\end{align}
</math>
|}

====General case====
If <math>X:\Omega\to [0,+\infty]</math> is a non-negative random variable, then

:<math>  \operatorname{E}[X]=\int\limits_{[0,+\infty)} \operatorname{P}(X > x)\,dx = \int\limits_{[0,+\infty)} \bar{F}(x)\,dx, </math>
where <math>\bar{F}(x) = 1 -F(x)</math>, and <math>F</math> is the cdf of <math>X</math>.
{| role="presentation" class="wikitable mw-collapsible mw-collapsed"
| '''Proof.'''
|-
|
1. For every <math>\omega\in\Omega</math>,
:<math>
X(\omega)=\int\limits_{(0,X(\omega))} dx=\int\limits_{[0,+\infty)} {\mathbf 1}_{(0,X(\omega))}(x)\,dx,
</math>
where <math>{\mathbf 1}_{(0,X(\omega))}</math> is the indicator function of <math>(0,X(\omega))</math>. Substituting this into the definition of <math>\operatorname{E}[X]</math>, obtain

:<math>
\begin{align}
\operatorname{E}[X]&=\int\limits_\Omega Xd\operatorname{P}\\
                   &=\int\limits_\Omega\int\limits_{[0,+\infty)} {\mathbf 1}_{(0,X(\omega)]}(x)\,dx\,d\operatorname{P}(\omega).
\end{align}
</math>

Since <math>X(\omega)\geq 0</math> and <math>{\mathbf 1}_{(0,X(\omega)]}(x)\geq 0, </math> this integral (finite or infinite) meets the requirements of Tonelli's theorem. Changing the order of integration gives us

:<math>
\begin{align}
&\int\limits_{[0,+\infty)} \int\limits_\Omega {\mathbf 1}_{(0,X(\omega)]}(x)\,d\operatorname{P}(\omega)\,dx\\
&=\int\limits_{[0,+\infty)} \operatorname{P}(X> x)\,dx.
\end{align}
</math>
|}

==See also==
*[[Center of mass]]
*[[Central tendency]]
*[[Chebyshev's inequality]] (an inequality on location and scale parameters)
*[[Conditional expectation]]
*Expected value is also a key concept in [[economics]], [[finance]], and many other subjects
*The general term [[Expectation (epistemic)|expectation]]
*[[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)]]
*[[Nonlinear expectation]] (a generalization of the expected value)
*[[Wald's equation]] for calculating the expected value of a random number of random variables

== References ==
{{Reflist}}

==Literature==
{{refbegin}}
* {{cite book
  | last = Edwards | first = A.W.F
  | title = Pascal's arithmetical triangle: the story of a mathematical idea
  | year = 2002
  | edition = 2nd
  | publisher = JHU Press
  | isbn = 0-8018-6946-3
  | ref = harv
  }}
* {{cite book
  | last = Huygens | first = Christiaan
  | title = De ratiociniis in ludo aleæ
  | format  = English translation, published in 1714: | url = http://www.york.ac.uk/depts/maths/histstat/huygens.pdf
  | year = 1657
  | ref = harv
  }}{{refend}}

{{Theory of probability distributions}}

{{DEFAULTSORT:Expected Value}}
[[Category:Theory of probability distributions]]
[[Category:Gambling terminology]]
[[Category:Articles containing proofs]]