In probability theory, the expected value of a random variable, intuitively, is the long-run average value of repetitions of the same experiment it represents. For example, the expected value in rolling a six-sided die is 3.5, because the average of all the numbers that come up is 3.5 as the number of rolls approaches infinity (see § Examples for details). In other words, the law of large numbers states that the arithmetic mean of the values almost surely converges to the expected value as the number of repetitions approaches infinity. The expected value is also known as the expectation, mathematical expectation, EV, average, mean value, mean, or first moment.
More practically, the expected value of a discrete random variable is the probability-weighted average of all 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 absolutely continuous random variable, except that an integral of the variable with respect to its 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.
The expected value is a key aspect of how one characterizes a probability distribution; it is one type of location parameter. By contrast, the variance is a measure of dispersion of the possible values of the random variable around the expected value. The variance itself is defined in terms of two expectations: it is the expected value of the squared deviation of the variable's value from the variable's expected value (var(X) = E[(X – E[X])2] = E(X2) – [E(X)]2).
The expected value plays important roles in a variety of contexts. In regression analysis, one desires a formula in terms of observed data that will give a "good" estimate of the parameter giving the effect of some explanatory variable upon a dependent variable. The formula will give different estimates using different samples of data, so the estimate it gives is itself a random variable. A formula is typically considered good in this context if it is an unbiased estimator—that is if the expected value of the estimate (the average value it would give over an arbitrarily large number of separate samples) can be shown to equal the true value of the desired parameter.
In decision theory, and in particular in choice under uncertainty, an agent is described as making an optimal choice in the context of incomplete information. For risk neutral agents, the choice involves using the expected values of uncertain quantities, while for risk averse agents it involves maximizing the expected value of some objective function such as a von Neumann–Morgenstern utility function. One example of using expected value in reaching optimal decisions is the Gordon–Loeb model of information security investment. According to the model, one can conclude that the amount a firm spends to protect information should generally be only a small fraction of the expected loss (i.e., the expected value of the loss resulting from a cyber or information security breach).
- If one rolls thedietimes and computes the average (arithmetic mean) of the results, then asgrows, the average willalmost surelyconvergeto the expected value, a fact known as thestrong law of large numbers. One example sequence of ten rolls of thedieis 2, 3, 1, 2, 5, 6, 2, 2, 2, 6, which has the average of 3.1, with the distance of 0.4 from the expected value of 3.5. The convergence is relatively slow: the probability that the average falls within the range3.5 ± 0.1is 21.6% for ten rolls, 46.1% for a hundred rolls and 93.7% for a thousand rolls. See the figure for an illustration of the averages of longer sequences of rolls of thedieand how they converge to the expected value of 3.5. More generally, the rate of convergence can be roughly quantified by e.g.Chebyshev's inequalityand theBerry–Esseen theorem.
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 represents the (monetary) outcome of a $1 bet on a single number ("straight up" bet). If the bet wins (which happens with probability 1/38 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 bet of $1 stands to lose $0.0526, so its expected value is -$0.0526.
Countably infinite case
Remark 2. Due to absolute convergence, the expected value does not depend on the order in which the outcomes are presented. By contrast, a conditionally convergent series can be made to converge or diverge arbitrarily, via the Riemann rearrangement theorem.
Suppose and for , where (with being the natural logarithm) is the scale factor such that the probabilities sum to 1. Then
For an example that is not absolutely convergent, suppose random variable takes values 1, −2, 3, −4, ..., with respective probabilities , ..., where is a normalizing constant that ensures the probabilities sum up to one. Then the infinite sum
An example that diverges arises in the context of the St. Petersburg paradox. Let and for . The expected value calculation gives
Absolutely continuous case
then the values (whether finite or infinite) of both integrals agree.
The following scenarios are possible:
is finite, i.e.
is infinite, i.e. and
is neither finite nor infinite, i.e.
where the integral is interpreted in the sense of Lebesgue–Stieltjes.
Remark 3. An example of a distribution for which there is no expected value is Cauchy distribution.
Remark 4. For multidimensional random variables, their expected value is defined per component, i.e.
The properties below replicate or follow immediately from those of Lebesgue integral.
If X = Y (a.s.) then E[X] = E[Y]
Expected value of a constant
If is also defined (i.e. differs from ), then
Let be finite, and be a finite scalar. Then
E[X] exists and is finite if and only if E[|X|] is finite
exists and is finite.
Both and are finite.
If X ≥ 0 (a.s.) then E[X] ≥ 0
If (a.s.) and is finite then so is
If and then
Counterexample for infinite measure
for every ,
equality holds if and only if
whence the extremal property follows.
If then (a.s.)
Corollary: if then (a.s.)
Corollary: if then (a.s.)
Proof. By definition of Lebesgue integral,
This result can also be proved based on Jensen's inequality.
The amount by which the multiplicativity fails is called the covariance:
Counterexample: despite pointwise
and a random variable
By finite additivity,
Countable additivity for non-negative random variables
The Cauchy–Bunyakovsky–Schwarz inequality states that
Taking limits under the sign
Monotone convergence theorem
all the expected values and are defined (differ from );
is the pointwise limit of (a.s.), i.e. (a.s.).
The monotone convergence theorem states that
all the expected values and are defined (differ from );
(a.s.), for every
Fatou's lemma states that
for some constant (independent from );
(a.s.), for every
Dominated convergence theorem
the function is measurable (hence a random variable);
all the expected values and are defined (do not have the form );
(both sides may be infinite);
Relationship with characteristic function
Uses and applications
The mass of probability distribution is balanced at the expected value, here a Beta(α,β) distribution with expected value α/(α+β).
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[X]. 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.
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
The law of the unconscious statistician
Alternative formula for expected value
Formula for non-negative random variables
Finite and countably infinite case
Formula for non-positive random variables
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 (see 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 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.
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 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:
… 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.
Chebyshev's inequality (an inequality on location and scale parameters)
Expected value is also a key concept in economics, finance, and many other subjects
The general term 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.
Nonlinear expectation (a generalization of the expected value)
Wald's equation for calculating the expected value of a random number of random variables