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In probability (and especially gambling), the expected value (or expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff ("value"). Thus, it represents the average amount one "expects" to win per bet if bets with identical odds are repeated many times. Note that the value itself may not be expected in the general sense, it may be unlikely or even impossible. The word probability derives from the Latin probare (to prove, or to test). ...
Gambling (or betting) is any behavior involving risking money or valuables (making a wager or placing a stake) on the outcome of a game, contest, or other event in which the outcome of that activity depends partially or totally upon chance or upon ones ability to do something. ...
A random variable can be thought of as the numeric result of operating a non-deterministic mechanism or performing a non-deterministic experiment to generate a random result. ...
expectation in the context of probability theory and statistics, see expected value. ...
For example, an American roulette wheel has 38 equally possible outcomes. A bet placed on a single number pays 35-to-1 (this means that you are paid 35 times your bet and your bet is returned, so you get 36 times your bet). So the expected value of the profit resulting from a $1 bet on a single number is, considering all 38 possible outcomes: , which is about -$0.0263. Therefore one expects, on average, to lose over two cents for every dollar bet. This page is about the game. ...
Mathematical definition
In general, if X is a random variable defined on a probability space (Ω,P), then the expected value of X (denoted E(X) or sometimes or ) is defined as A random variable can be thought of as the numeric result of operating a non-deterministic mechanism or performing a non-deterministic experiment to generate a random result. ...
In mathematics, a probability space is a set S, together with a σ-algebra X on S and a measure P on that σ-algebra such that P(S) = 1. ...
where the Lebesgue integral is employed. Note that not all random variables have an expected value, since the integral may not exist (e.g., Cauchy distribution). Two variables with the same probability distribution will have the same expected value, if it is defined. In mathematics, the integral of a function of one real variable can be regarded as the area of a plane region bounded by the graph of that function. ...
Standard Cauchy_Lorentz probability distribution function The Cauchy_Lorentz distribution is a probability distribution with probability density function where x0 is the location parameter, specifying the location of the peak of the distribution, and γ is the scale parameter which specifies the half_width at half_maximum (HWHM). ...
In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ...
If X is a discrete random variable with values x1, x2, ... and corresponding probabilities p1, p2, ... which add up to 1, then E(X) can be computed as the sum or series In mathematics, a random variable is discrete if its probability distribution is discrete; a discrete probability distribution is one that is fully characterized by a probability mass function. ...
In mathematics, a series is a sum of a sequence of terms. ...
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as in the gambling example mentioned above.
If the probability distribution of X admits a probability density function f(x), then the expected value can be computed as In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ...
In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ...
It follows directly from the discrete case definition that if X is a constant random variable, i.e. X = b for some fixed real number b, then the expected value of X is also b. In probability theory, a constant random variable is a discrete random variable that takes a constant value, regardless of any event that occurs. ...
The text or formatting below is generated by a template which has been proposed for deletion. ...
Properties
Linearity The expected value operator (or expectation operator) E is linear in the sense that In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are...
- E(aX + bY) = aE(X) + bE(Y)
for any two random variables X and Y (which need to be defined on the same probability space) and any real numbers a and b.
Functional non-invariance In general, the expectation operator and functions of random variables do not commute; that is In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ...
In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ...
except as noted above.
Non-multiplicativity In general, the expected value operator is not multiplicative, i.e. E(XY) is not necessarily equal to E(X)E(Y), except if X and Y are independent or uncorrelated. This lack of multiplicativity gives rise to study of covariance and correlation. In probability theory and statistics, to call two real-valued random variables X and Y uncorrelated means that their correlation is zero, or, equivalently, their covariance is zero. ...
In probability theory and statistics, the covariance between two real_valued random variables X and Y, with expected values and is defined as: where E is the expected value. ...
In probability theory and statistics, correlation, also called correlation coefficient, is a numeric measure of the strength of linear relationship between two random variables. ...
Iterated expectation For any two random variables X,Y one may define the conditional expectation: In probability theory, a conditional expectation is the expected value of a real random variable with respect to a conditional probability distribution. ...
Then the expectation of X satisfies The right hand side of this equation is referred to as the iterated expectation. This proposition is treated in law of total expectation. The proposition in probability theory known as the law of total expectation, or the law of iterated expectations, or perhaps by any of a variety of other names, states that if X is an integrable random variable (i. ...
Uses and applications of the expected value 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. See also moment (physics). ...
In probability theory and statistics, the kth moment about the mean (or kth central moment) of a real-valued random variable X is the quantity E[(X − E[X])k], where E is the expectation operator. ...
In probability theory and statistics, the moment-generating function of a random variable X is The moment-generating function generates the moments of the probability distribution, as follows: If X has a continuous probability density function f(x) then the moment generating function is given by where is the ith...
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. This 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 that (under fairly mild conditions) as the size of the sample gets larger, the variance of this estimate gets smaller. In mathematics and statistics, the arithmetic mean of a set of numbers is the sum of all the members of the set divided by the number of items in the set. ...
In statistics, a biased estimator is one that for some reason on average over_ or underestimates what is being estimated. ...
In statistics, the concepts of error and residual are easily confused with each other. ...
In a statistical context, laws of large numbers implies that the average of a random sample from a large population is likely to be close to the mean of the whole population. ...
A sample is that part of a population which is actually observed. ...
In probability theory and statistics, the variance of a random variable is a measure of its statistical dispersion, indicating how far from the expected value its values typically are. ...
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 (its center of gravity) is E(X). In physics, Classical mechanics is one of the two major sub-fields of study in the science of mechanics, which is concerned with the motions of bodies, and the forces that cause them. ...
The center of mass or center of inertia of an object is a point at which the objects mass can be assumed, for many purposes, to be concentrated. ...
In physics, the center of gravity (CoG) of an object is the average location of its weight. ...
Expectation of matrices If X is an matrix, then the expected value of the matrix is a matrix of expected values: For the square matrix section, see square matrix. ...
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This property is utilized in covariance matrices. In statistics and probability theory, the covariance matrix is a matrix of covariances between elements of a vector. ...
See also For probability distributions having an expected value and a median, the mean (i. ...
Economics (deriving from the Greek words οίκω [oeko], house, and νέμω [nemo], distribute) is the social science that studies the allocation of scarce resources. ...
Finance studies and addresses the ways in which individuals, businesses and organizations raise, allocate and use monetary resources over time, taking into account the risks entailed in their projects. ...
expectation in the context of probability theory and statistics, see expected value. ...
External links - Expectation (http://planetmath.org/?op=getobj&from=objects&id=505) on PlanetMath.
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