NationMaster - Encyclopedia: Expected value

In probability theory the expected value (or mathematical 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" as the outcome of the random trial when 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. For example, the expected value from the roll of an ordinary six-sided die is 3.5, which is not one of the possible outcomes. Probability theory is the mathematical study of phenomena characterized by randomness or uncertainty. ... A random variable is a mathematical function that maps outcomes of random experiments to numbers. ... expectation in the context of probability theory and statistics, see expected value. ... Two standard six-sided pipped dice with rounded corners. ...

A common application of expected value is in gambling. For example, an American roulette wheel has 38 equally likely outcomes. A winning 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 considering all 38 possible outcomes, the expected value of the profit resulting from a $1 bet on a single number is: Gambling has had many different meanings depending on the cultural and historical context in which it is used. ... Please wikify (format) this article or section as suggested in the Guide to layout and the Manual of Style. ...

$left( -$1 times frac{37}{38} right) + left( $35 times frac{1}{38} right),$

which is about −$0.0526. Therefore one expects, on average, to lose over five cents for every dollar bet, and the expected value of a one dollar bet is $0.9474. In gambling or betting, a game or situation in which the expected value for the player is zero (no net gain nor loss) is called a "fair game."

1 Value expected concept
2 Mathematical definition
3 Conventional terminology
4 Properties
5 Uses and applications of the expected value
6 Expectation of matrices
7 See also
8 External links

Value expected concept

The concept of mathematical expectation is commonly used to refer to the value expected as the outcome any strategy; common examples are bets and other type of gambits. These strategies may involve betting on games of chance, attempts at medical therapies, or solving problems in general as in passing laws to address specific needs.

The mathematical expectation is defined as the value of the sum of all possible gains and losses multiplied by the probabilities of each gain and loss. The units used to express this value depend on the situation. In games such as card games or lotteries, the units are commonly monetary, while in medical therapies, the units might be quality of health.

The symbolic expression of this concept is simple:

MV = (GV x GP) + (BV x BP)

MV = mathematical expectation (value) for a strategy

GV = value of good/desirable result

GP = probability of a good/desirable result

BV = value of bad/undesirable result

BP = probability of an bad/undesirable result

The units of the variables MV, GV, and BV are identical and depend on the strategy.

The variables GP and BP are numbers (scalars) without units. The concept of a scalar is used in mathematics and physics. ...

Multiple, or even an infinite number, of terms like (GV x GP) or (BV x BP) may be required for accurate expression of the mathematical expectation. Since the concept requires that every possible result be considered, the equation will generate an erroneous result unless a term is present for every possible result.

More comprehensive and rigorous definitions are found below.

Mathematical definition

In general, if $X,$ is a random variable defined on a probability space $(Omega, P),$ , then the expected value of $X,$ (denoted $operatorname{E}(X),$ or sometimes $langle X rangle$ or $mathbb{E}(X)$ ) is defined as A random variable is a mathematical function that maps outcomes of random experiments to numbers. ... In mathematics, a probability space or probability measure is a set S, together with a Ïƒ-algebra X on S and a measure P on that Ïƒ-algebra such that P(S) = 1. ...

$operatorname{E}(X) = int_Omega X, operatorname{d}P$

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. The integral can be interpreted as the area under a curve. ... The Cauchy-Lorentz distribution, named after Augustin Cauchy, is a continuous probability distribution with probability density function where x0 is the location parameter, specifying the location of the peak of the distribution, and Î³ is the scale parameter which specifies the half-width at half-maximum (HWHM). ... In mathematics and statistics, a probability distribution, more properly called a probability density, 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 $x 1$ , $x 2$ , ... and corresponding probabilities $p 1$ , $p 2$ , ... which add up to 1, then $operatorname{E}(X)$ 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. ...

$operatorname{E}(X) = sum_i p_i x_i,$

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 and statistics, a probability distribution, more properly called a probability density, 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. ...

$operatorname{E}(X) = int_{-infty}^infty x f(x), operatorname{d}x .$

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. ... In mathematics, the real numbers may be described informally in several different ways. ...

The expected value of an arbitrary function of X, g(X), with respect to the probability density function f(x) is given by:

$operatorname{E}(g(X)) = int_{-infty}^infty g(x) f(x), operatorname{d}x .$

Conventional terminology

When one speaks of the "expected price", one means the expected value of a random variable that is a price.
When one speaks of the "expected height", one means the expected value of a random variable that is a height.
When one speaks of the "expected number of attempts needed to get one successful attempt," one might conservatively approximate it as the reciprocal of the probability of success for such an attempt.

And so on.

Properties

Constants

Expected value of a constant is equal to that constant or If c is a constant, E(c) = c

Monotonicity

If X and Y are random variables so that $X le Y$ almost surely, then $operatorname{E}(X) le operatorname{E}(Y)$ . In mathematics, specifically, in probability theory, the phrase almost surely is a concise, precise way to state except on a set or event of probability measure zero. ...

Linearity

The expected value operator (or expectation operator) $operatorname{E}$ 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...

$operatorname{E}(X + c)= operatorname{E}(X) + c,$

$operatorname{E}(aX)= a operatorname{E}(X),$

Combining the results from previous two equations, we can see that -

$operatorname{E}(aX + b)= a operatorname{E}(X) + b,$

$operatorname{E}(a X + b Y) = a operatorname{E}(X) + b operatorname{E}(Y),$

for any two random variables $X$ and $Y$ (which need to be defined on the same probability space) and any real numbers $a$ and $b$ .

Iterated expectation

Iterated expectation for discrete random variables

For any two discrete random variables $X, Y$ one may define the conditional expectation: In mathematics, a probability distribution is called discrete, if it is fully characterized by a probability mass function. ... In probability theory, a conditional expectation is the expected value of a real random variable with respect to a conditional probability distribution. ...

$operatorname{E}(X|Y)(y) = operatorname{E}(X|Y=y) = sumlimits_x x cdot operatorname{P}(X=x|Y=y).$

Then the expectation of $X$ satisfies

$operatorname{E} left( operatorname{E}(X|Y) right)= sumlimits_y operatorname{E}(X|Y=y) cdot operatorname{P}(Y=y) ,$

$=sumlimits_y left( sumlimits_x x cdot operatorname{P}(X=x|Y=y) right) cdot operatorname{P}(Y=y),$

$=sumlimits_y sumlimits_x x cdot operatorname{P}(X=x|Y=y) cdot operatorname{P}(Y=y),$

$=sumlimits_y sumlimits_x x cdot operatorname{P}(Y=y|X=x) cdot operatorname{P}(X=x) ,$

$=sumlimits_x x cdot operatorname{P}(X=x) cdot left( sumlimits_y operatorname{P}(Y=y|X=x) right) ,$

$=sumlimits_x x cdot operatorname{P}(X=x) ,$

$=operatorname{E}(X).,$

Hence, the following equation holds:

$operatorname{E}(X) = operatorname{E} left( operatorname{E}(X|Y) right).$

The right hand side of this equation is referred to as the iterated expectation and is also sometimes called the tower rule. 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. ...

Iterated expectation for continuous random variables

In the continuous case, the results are completely analogous. The definition of conditional expectation would use inequalities, density functions, and integrals to replace equalities, mass functions, and summations, respectively. However, the main result still holds: In mathematics and statistics, a probability distribution, more properly called a probability density, assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ...

$operatorname{E}(X) = operatorname{E} left( operatorname{E}(X|Y) right).$

Inequality

If a random variable X is always less than or equal to another random variable Y, the expectation of X is less than or equal to that of Y:

If $X leq Y$ , then $operatorname{E}(X) leq operatorname{E}(Y)$ .

In particular, since $X leq |X|$ and $-X leq |X|$ , the absolute value of expectation of a random variable is less or equal to the expectation of its absolute value:

$|operatorname{E}(X)| leq operatorname{E}(|X|)$

Representation

The following formula holds for any nonnegative real-valued random variable $X$ (such that $operatorname{E}(X) < infty$ ), and positive real number $α$ :

$operatorname{E}(X^alpha) = alpha int_{0}^{infty} t^{alpha -1}operatorname{P}(X>t) , operatorname{d}t.$

Non-multiplicativity

In general, the expected value operator is not multiplicative, i.e. $operatorname{E}(X Y)$ is not necessarily equal to $operatorname{E}(X) operatorname{E}(Y)$ , 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. ... Positive linear correlations between 1000 pairs of numbers. ...

Functional non-invariance

In general, the expectation operator and functions of random variables do not commute; that is Partial plot of a function f. ... Mathematical meaning A map or binary operation is said to be commutative when, for any x in A and any y in B . ...

$operatorname{E}(g(X)) = int_{Omega} g(X), operatorname{d}P neq g(operatorname{E}(X)),$

except as noted above,

Law of the Unconscious Statistician

Given that $f(x) < infty$ then law of the unconscious statistician states that

$operatorname{E} (f(x)) = sumlimits_x f(x)cdot operatorname{P}(X=x)$

Where x is an element of X(Ω).

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 $X - operatorname{E}(X)$ . The moments of some random variables can be used to specify their distributions, via their moment generating functions.-1... 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 (under fairly mild conditions) that, as the size of the sample gets larger, the variance of this estimate gets smaller. Estimation is approximate or uncertain calculation of a result, often based on approximate, uncertain, incomplete, or noisy inputs. ... In mathematics and statistics, the arithmetic mean (or simply the mean) of a list of numbers is the sum of all the members of the list divided by the number of items in the list. ... Estimation is approximate or uncertain calculation of a result, often based on approximate, uncertain, incomplete, or noisy inputs. ... In statistics, the difference between an estimators expected value and the true value of the parameter being estimated is called the bias. ... In statistics and optimization, the concepts of error and residual are easily confused with each other. ... In statistics, an estimator is a function of the observable sample data that is used to estimate an unknown population parameter; an estimate is the result from the actual application of the function to a particular set of data. ... The law of large numbers is a fundamental concept in statistics and probability that describes how the average of a randomly selected sample from a large population is likely to be close to the average of the whole population. ... Sample size, usually designated N, is the number of repeated measurements in a statistical sample. ... A sample is that part of a population which is actually observed. ... In probability theory and statistics, the variance of a random variable (or equivalently, of a probability distribution) is a measure of its statistical dispersion, indicating how its possible values are spread around the expected value. ... In statistics, an estimator is a function of the observable sample data that is used to estimate an unknown population parameter; an estimate is the result from the actual application of the function to a particular set of data. ...

In classical mechanics, the center of mass is an analogous concept to expectation. For example, suppose $X$ is a discrete random variable with values $x i$ and corresponding probabilities $p i$ . Now consider a weightless rod on which are placed weights, at locations $x i$ along the rod and having masses $p i$ (whose sum is one). The point at which the rod balances is $operatorname{E}(X)$ . Classical mechanics is a branch of physics which studies the deterministic motion of objects. ... In physics, the center of mass of a system of particles is a specific point at which, for many purposes, the systems mass behaves as if it were concentrated. ...

Expected values can also be used to compute variance. In probability theory and statistics, the variance of a random variable (or equivalently, of a probability distribution) is a measure of its statistical dispersion, indicating how its possible values are spread around the expected value. ...

$operatorname{Var}(X)= operatorname{E}(X^2) - (operatorname{E}(X))^2$

Expectation of matrices

If $X$ is an $m times n$ matrix, then the expected value of the matrix is defined as the matrix of expected values: In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ...

$operatorname{E}(X) = operatorname{E} begin{pmatrix} x_{1,1} & x_{1,2} & cdots & x_{1,n} x_{2,1} & x_{2,2} & cdots & x_{2,n} vdots x_{m,1} & x_{m,2} & cdots & x_{m,n} end{pmatrix} = begin{pmatrix} operatorname{E}(x_{1,1}) & operatorname{E}(x_{1,2}) & cdots & operatorname{E}(x_{1,n}) operatorname{E}(x_{2,1}) & operatorname{E}(x_{2,2}) & cdots & operatorname{E}(x_{2,n}) vdots operatorname{E}(x_{m,1}) & operatorname{E}(x_{m,2}) & cdots & operatorname{E}(x_{m,n}) end{pmatrix}$

This is utilized in covariance matrices. In statistics and probability theory, the covariance matrix is a matrix of covariances between elements of a vector. ...

External links

Expectation on PlanetMath

Categories: Probability theory | Gambling terminology PlanetMath is a free, collaborative, online mathematics encyclopedia. ...

Results from FactBites:

Expected value - definition of Expected value in Encyclopedia (573 words)
	The expected value operator (or expectation operator) E is linear in the sense that
	To empirically estimate the expected value of a random variable, one repeatedly measures values of the variable and computes the arithmetic mean of the results.
	This estimates the true expected value and has the property of minimizing the sum of the squares of the errors away from the expected value.

Encyclopedia: Expected value (505 words)
	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 expected value of a statistic is therefore the mean of the sampling distribution of the statistic.
	Expected values of variables are indicated by an "E" with the variable enclosed in brackets.

More results at FactBites »

COMMENTARY

Share your thoughts, questions and commentary here

Want to know more?
Search encyclopedia, statistics and forums:

Feeds | Contact
The Wikipedia article included on this page is licensed under the GFDL.
Images may be subject to relevant owners' copyright.
All other elements are (c) copyright NationMaster.com 2003-5. All Rights Reserved.
Usage implies agreement with terms.

Contents

Value expected concept GA_googleFillSlot("encyclopedia_square");