In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of the possible values a random variable can take, weighted by the probability of those outcomes. Since it is obtained through arithmetic, the expected value sometimes may not even be included in the sample data set; it is not the value you would "expect" to get in reality.
The expected value of a random variable with a finite number of outcomes is a weighted average of all possible outcomes. In the case of a continuum of possible outcomes, the expectation is defined by integration. In the axiomatic foundation for probability provided by measure theory, the expectation is given by Lebesgue integration.
The expected value of a random variable is often denoted by,, or, with also often stylized as
E
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 is properly finished.[4] 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 by French writer and amateur mathematician Chevalier de Méré in 1654. Méré claimed that this problem could not 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. 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.[5]
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:
In the mid-nineteenth century, Pafnuty Chebyshev became the first person to think systematically in terms of the expectations of random variables.[6]
Neither Pascal nor Huygens used the term "expectation" in its modern sense. In particular, Huygens writes:[7]
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:[8]
The use of the letter to denote "expected value" goes back to W. A. Whitworth in 1901.[9] The symbol has since become popular for English writers. In German, stands for Erwartungswert, in Spanish for esperanza matemática, and in French for espérance mathématique.[10]
When "E" is used to denote "expected value", authors use a variety of stylizations: the expectation operator can be stylized as (upright), (italic), or
E
Another popular notation is, whereas,, and
\overline{X}
As discussed above, 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 functions, 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 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 . It is defined component by component, as . Similarly, one may define the expected value of a random matrix with components by .
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 of the values, with weights given by their probabilities .
In the special case that all possible outcomes are equiprobable (that is,), the weighted average is given by the standard average. In the general case, the expected value takes into account the fact that some outcomes are more likely than others.
X
X
X
X
n
n
X
Informally, the expectation of a random variable with a countably infinite set 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 thatwhere 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 of mathematical analysis 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.
xi=i
pi=\tfrac{c}{i ⋅ 2i}
i=1,2,3,\ldots,
c=\tfrac{1}{ln2}
Now consider a random variable which has a probability density function given by a function on the real number line. This means that the probability of taking on a value in any given open interval is given by the integral of over that interval. The expectation of is then given by the integralA general and mathematically precise formulation of this definition uses measure theory 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, so that . It is straightforward to compute in this case thatThe 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 .
All definitions of the expected value may be expressed in the language of measure theory. In general, if is a real-valued random variable defined on a probability space, then the expected value of, denoted by, is defined as the Lebesgue integralDespite 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 to 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:
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 thatfor 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.
The expected value of any real-valued random variable
X
F
\operatorname{E}[X]=\mu
\mu
x
y
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 hasIt 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 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.
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.
Distribution | Notation | Mean E(X) | |||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Bernoulli | X\sim~b(1,p) | 0 ⋅ (1-p)+1 ⋅ p=p | |||||||||||||||||||||
Binomial | X\simB(n,p) |
i{n\choosei}pi(1-p)n-i=np | |||||||||||||||||||||
Poisson | X\simPo(λ) |
=λ | |||||||||||||||||||||
Geometric | X\simGeometric(p) |
ip(1-p)i-1=
| |||||||||||||||||||||
Uniform | X\simU(a,b) |
| |||||||||||||||||||||
Exponential | X\sim\exp(λ) |
λxe-λdx=
| |||||||||||||||||||||
Normal | X\simN(\mu,\sigma2) |
| |||||||||||||||||||||
Standard Normal | X\simN(0,1) |
| |||||||||||||||||||||
Pareto | X\simPar(\alpha,k) |
k\alphax-\alphadx=\begin{cases}
&if\alpha>1\ infty&if0<\alpha\leq1\end{cases} | |||||||||||||||||||||
Cauchy | X\simCauchy(x0,\gamma) |
dx |
The basic properties below (and their names in bold) replicate or follow immediately from those of Lebesgue integral. Note that the letters "a.s." stand for "almost surely"—a central property of the Lebesgue integral. Basically, one says that an inequality like
X\geq0
\left\{X<0\right\}.
X\geq0
\operatorname{E}[X]\geq0.
\operatorname{E}[ ⋅ ]
X
Y,
a,
N
Xi
ai(1\leqi\leqN),
X\leqY
\operatorname{E}[X]
\operatorname{E}[Y]
\operatorname{E}[X]\leq\operatorname{E}[Y].
Z=Y-X,
Z\geq0
\operatorname{E}[|X|]=0,
X=0
X=Y
\operatorname{E}[X]=\operatorname{E}[Y].
X=c
\operatorname{E}[X]=c.
X
\operatorname{E}[\operatorname{E}[X]]=\operatorname{E}[X].
X
|\operatorname{E}[X]|\leq\operatorname{E}|X|.
F(x)
\operatorname{E}[XY]
\operatorname{E}[X] ⋅ \operatorname{E}[Y].
X
Y
\operatorname{E}[XY]=\operatorname{E}[X]\operatorname{E}[Y].
\operatorname{E}[XY] ≠ \operatorname{E}[X]\operatorname{E}[Y],
X,
g(X),
X
f(x),
f
g
g
f
Concentration inequalities control the likelihood of a random variable taking on large values. Markov's inequality is among the best-known and simplest to prove: for a nonnegative random variable and any positive number, it states that
If is any random variable with finite expectation, then Markov's inequality may be applied to the random variable to obtain Chebyshev's inequality where is the variance. 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 deviations 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 and its applications to probability theory.
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 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 spaces.
\operatorname|XY|\leq(\operatorname|X|^p)^(\operatorname|Y|^q)^. for any random variables and . The special case of is called the Cauchy–Schwarz inequality, and is particularly well-known.
\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 spaces, and are often given in that context. By contrast, the Jensen inequality is special to the case of probability spaces.
In general, it is not the case that
\operatorname{E}[Xn]\to\operatorname{E}[X]
Xn\toX
U
[0,1].
n\geq1,
1\{A\}
A.
Xn\to0
\operatorname{E}[Xn]=n ⋅ \Pr\left(U\in\left[0,\tfrac{1}{n}\right]\right)=n ⋅ \tfrac{1}{n}=1
n.
\limn\operatorname{E}[Xn]=1 ≠ 0=\operatorname{E}\left[\limnXn\right].
Analogously, for general sequence of random variables
\{Yn:n\geq0\},
\sigma
An example is easily obtained by setting
Y0=X1
Yn=Xn+1-Xn
n\geq1,
Xn
A number of convergence results specify exact conditions which allow one to interchange limits and expectations, as specified below.
\{Xn:n\geq0\}
0\leqXn\leqXn+1
n\geq0.
Xn\toX
\limn\operatorname{E}[Xn]=\operatorname{E}[X].
\{Xi\}
infty | |
i=0 |
\{Xn\geq0:n\geq0\}
Xn\geq0
\operatorname{E}[Xn]\leqC
n\geq0.
Xn\toX
\operatorname{E}[X]\leqC.
\{Xn:n\geq0\}
Xn\toX
|Xn|\leqY\leq+infty
\operatorname{E}[Y]<infty.
\operatorname{E}|X|\leq\operatorname{E}[Y]<infty
\limn\operatorname{E}[Xn]=\operatorname{E}[X]
\limn\operatorname{E}|Xn-X|=0.
\limn\operatorname{E}[Xn]=\operatorname{E}[\limnXn]
\{Xn\}
The probability density function
fX
X
\varphiX
For the expected value of
g(X)
g:{R}\to{R}
If
\operatorname{E}[g(X)]
g(x).
\operatorname{E}[g(X)]
The expectation of a random variable plays an important role in a variety of contexts.
In statistics, where one seeks estimates for unknown parameters based on available data gained from samples, the sample mean serves as an estimate for the expectation, and is itself a random variable. In such settings, the sample mean is considered to meet the desirable criterion for a "good" estimator in being unbiased; that is, the expected value of the estimate is equal to the true value of the underlying parameter.
See also: Estimation theory.
For a different example, in decision theory, an agent making an optimal choice in the context of incomplete information is often assumed to maximize the expected value of their utility function.
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 frequencies.
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 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 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). The law of large numbers demonstrates (under fairly mild conditions) that, as the size of the sample gets larger, the variance of this 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.
\operatorname{P}({X\inl{A}})=\operatorname{E}[{1}l{A
{1}l{A
l{A}.
In classical mechanics, the center of mass 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, by means of the computational formula for the variance
\hat{A}
|\psi\rangle
\langle\hat{A}\rangle=\langle\psi|\hat{A}|\psi\rangle.
\hat{A}
(\DeltaA)2=\langle\hat{A}2\rangle-\langle\hat{A}\rangle2
m