Expected value

In probability theory, the expected value of a random variable is a generalization of the weighted average and intuitively is the arithmetic mean of a large number of independent realizations of that variable. The expected value is also known as the expectation, mathematical expectation, mean, average, or first moment.
By definition, the expected value of a constant random variable is. The expected value of a random variable with equiprobable outcomes is defined as the arithmetic mean of the terms If some of the probabilities of an individual outcome are unequal, then the expected value is defined to be the probability-weighted average of the s, i.e. the sum of the products.
Expected value of a general random variable involves integration in the sense of Lebesgue.


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 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.
Three years later, in 1657, a Dutch mathematician Christiaan Huygens, who had just visited Paris, published a treatise "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. 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:
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.


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 published his tract "Théorie analytique des probabilités", where the concept of expected value was defined explicitly:


The use of the letter to denote expected value goes back to W. A. Whitworth in 1901. The symbol has become popular since for English writers. In German, stands for "Erwartungswert", in Spanish for "Esperanza matemática", and in French for "Espérance mathématique".
Another popular notation is, whereas is commonly used in physics, and in Russian-language literature.


Finite case

Let be a random variable with a finite number of finite outcomes occurring with probabilities respectively. The expectation of is defined as
Since the sum of all probabilities is 1, the expected value is the weighted sum of the values, with the values being the weights.
If all outcomes are equiprobable, then the weighted average turns into the simple average. If the outcomes 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.


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 be a non-negative random variable with a countable set of outcomes occurring with probabilities respectively. Analogous to the discrete case, the expected value of is then defined as the series
Note that since, the infinite sum is well-defined and does not depend on the order in which it is computed. Unlike the finite case, the expectation here can be equal to infinity, if the infinite sum above increases without bound.
For a general random variable with a countable number of outcomes, set and. By definition,
Like with non-negative random variables, can, once again, be finite or infinite. The third option here is that is no longer guaranteed to be well defined at all. The latter happens whenever.


If is a random variable with a probability density function of, then the expected value is defined as the Lebesgue integral
where the values on both sides are well defined or not well defined simultaneously.
Example. A random variable that has the Cauchy distribution has a density function, but the expected value is undefined since the distribution has large "tails".

General case

In general, if is a random variable defined on a probability space, then the expected value of, denoted by, is defined as the Lebesgue integral
For multidimensional random variables, their expected value is defined per component, i.e.
and, for a random matrix with elements,

Basic properties

The basic properties below replicate or follow immediately from those of Lebesgue integral. Note that the letters "a.s." stand for "almost surely" that is a central property of the Lebesgue integral. Basically, one says that an inequality like is true almost surely when the probability measure attributes zero-mass to the complementary event.
For an arbitrary
The last equality holds because the fact that where implies that and hence Conversely, if where then
The integrand in the above expression for is non-negative, so Tonelli's theorem applies, and the order of integration may be switched without altering the result. We have
Arguing as above,
Recalling that completes the proof.
If then On the other hand,
so the series on the right diverges to and the equality holds.
If then
Define the infinite upper-triangular matrix
The double series is the sum of 's elements if summation is done row by row.
Since every summand is non-negative, the series either converges absolutely or diverges to In both cases, changing summation order does not affect the sum. Changing summation order, from row-by-row to column-by-column, gives us

Uses and applications

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 maximize the expected value of their 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 – 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 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 X − E. 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 law of large numbers demonstrates 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 quantities of interest via Monte Carlo methods, since most quantities of interest can be written in terms of expectation, e.g., where is the indicator function of the set.
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. The point at which the rod balances is E.
Expected values can also be used to compute the variance, by means of the computational formula for the variance
A very important application of the expectation value is in the field of quantum mechanics. The expectation value of a quantum mechanical operator operating on a quantum state vector is written as. The uncertainty in can be calculated using the formula.

Interchanging limits and expectation

In general, it is not the case that despite pointwise. Thus, one cannot interchange limits and expectation, without additional conditions on the random variables. To see this, let be a random variable distributed uniformly on. For define a sequence of random variables
with being the indicator function of the event. Then, it follows that . But, for each. Hence,
Analogously, for general sequence of random variables, the expected value operator is not -additive, i.e.
An example is easily obtained by setting and for, where 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.
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.

Relationship with characteristic function

The probability density function of a scalar random variable is related to its characteristic function by the inversion formula:
For the expected value of , we can use this inversion formula to obtain
If is finite, changing the order of integration, we get, in accordance with Fubini–Tonelli theorem,
is the Fourier transform of The expression for also follows directly from Plancherel theorem.