Expected value

"Expectation value" redirects here. For the expectation value in quantum mechanics, see expectation value (quantum mechanics). In Quantum mechanics, the expectation value is the predicted mean value of the result of an experiment

In probability theory the expected value (or mathematical expectation, or mean) of a discrete random variable is the sum of the probability of each possible outcome of the experiment multiplied by the outcome value (or payoff). Probability theory is the branch of Mathematics concerned with analysis of random phenomena A random variable is a rigorously defined mathematical entity used mainly to describe Chance and Probability in a mathematical way 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 - the "expected value" itself may be unlikely or even impossible. In the case of Uncertainty, expectation is what is considered the most likely to happen

Examples

The expected value from the roll of an ordinary six-sided die is 3. For other uses see either Die or Dice (disambiguation. Dice (the Plural of Die, from Old French 5, which is not among the possible outcomes:

$\operatorname{E}(X) = \frac{1 + 2 + 3 + 4 + 5 + 6}{6} = 3.5$

A common application of expected value is to gambling. For example, an American roulette wheel has 38 places where the ball may land, all equally likely. Roulette is a Casino and Gambling game named after the French word meaning "small wheel" A winning bet on a single number pays 35-to-1, meaning that the original stake is not lost, and 35 times that amount is won, so you receive 36 times what you've bet. Considering all 38 possible outcomes, the expected value of the profit resulting from a dollar bet on a single number is the sum of what you may lose times the odds of losing and what you will win times the odds of winning:

$\operatorname{E}(X) = \left( -$1 \times \frac{37}{38} \right) + \left( $35 \times \frac{1}{38} \right) \approx -$0.0526$

The change in your financial holdings is −$1 when you lose, and $35 when you win. Thus one may expect, on average, to lose about five cents for every dollar bet, and the expected value of a one-dollar bet is $0. 9474. In gambling, an event of which the expected value equals the stake (of which the bettor's expected profit is zero) is called a "fair game. "

Mathematical definition

In general, if $X\,$ is a random variable defined on a probability space $(\Omega, \Sigma, F)\,$ (where $Ω$ is the sample space and F is the cumulative distribution function of probability, ( $F(x) := P(X \leq x)\,$ )), then the expected value of $X\,$ (denoted $\operatorname{E}(X)\,$ or sometimes $\langle X \rangle$ or $\mathbb{E}(X)$ ) is defined as

$\operatorname{E}(X) = \int_\Omega X\, \operatorname{d}F$

where the Lebesgue integral is employed. A random variable is a rigorously defined mathematical entity used mainly to describe Chance and Probability in a mathematical way A probability space, in Probability theory, is the conventional Mathematical model of Randomness. In Probability theory and Statistics, the cumulative distribution function (CDF, also probability distribution function or just distribution function In Mathematics, the Integral of a non-negative function can be regarded in the simplest case as the Area between the graph of Note that not all random variables have an expected value, since the integral may not exist (e. g. , Cauchy distribution). The Cauchy–Lorentz distribution, named after Augustin Cauchy and Hendrik Lorentz, is a continuous Probability distribution. Two variables with the same probability distribution will have the same expected value, if it is defined. In Probability theory and Statistics, a probability distribution identifies either the probability of each value of an unidentified Random variable

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

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

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

It follows directly from the discrete case definition that if $X$ is a constant random variable, i. In Probability theory and Statistics, a probability distribution identifies either the probability of each value of an unidentified Random variable In Mathematics, a probability density function (pdf is a function that represents a Probability distribution in terms of Integrals Formally a probability WikipediaWikiProject Probability#Standards for a discussionof standards used for probability distribution articles such as this one e. $X = b$ for some fixed real number $b$ , then the expected value of $X$ is also $b$ . 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", "expected height", etc. one means the expected value of a random variable that is a price, a height, etc.
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. Cf. expected value of the geometric distribution. In Probability theory and Statistics, the geometric distribution is either of two Discrete probability distributions the probability

Properties

Constants

The expected value of a constant is equal to the constant itself; i. e. , if 'c' is a constant, then 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 Probability theory, one says that an event happens almost surely (a

Linearity

The expected value operator (or expectation operator) $\operatorname{E}$ is linear in the sense that

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

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

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

Combining the results from previous three 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 $b$ . In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that

Iterated expectation

Iterated expectation for discrete random variables

For any two discrete random variables $X, Y$ one may define the conditional expectation:

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

which means that $\operatorname{E}(X|Y)(y)$ is a function on $y$ . In Probability theory, a Probability distribution is called discrete if it is characterized by a Probability mass function. In Probability theory, a conditional expectation (also known as conditional expected value or conditional mean is the Expected value of a real random variable with

Then the expectation of $X$ satisfies

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

$=\sum\limits_y \left( \sum\limits_x x \cdot \operatorname{P}(X=x|Y=y) \right) \cdot \operatorname{P}(Y=y)\,$

$=\sum\limits_y \sum\limits_x x \cdot \operatorname{P}(X=x|Y=y) \cdot \operatorname{P}(Y=y)\,$

$=\sum\limits_y \sum\limits_x x \cdot \operatorname{P}(Y=y|X=x) \cdot \operatorname{P}(X=x) \,$

$=\sum\limits_x x \cdot \operatorname{P}(X=x) \cdot \left( \sum\limits_y \operatorname{P}(Y=y|X=x) \right) \,$

$=\sum\limits_x x \cdot \operatorname{P}(X=x) \,$

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

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, the law of iterated expectations, the tower rule, the smoothing theorem

Iterated expectation for continuous random variables

In the continuous case, the results are completely analogous. In Probability theory and Statistics, a probability distribution identifies either the probability of each value of an unidentified Random variable 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:

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

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 than 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 $α$ :

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

In particular, this reduces to:

$\operatorname{E}(X) = \int_{0}^{\infty} \lbrace 1-F(t) \rbrace \, \operatorname{d}t.$

Non-multiplicativity

In general, the expected value operator is not multiplicative, i. e. $\operatorname{E}(X Y)$ is not necessarily equal to $\operatorname{E}(X) \operatorname{E}(Y)$ . If multiplicativity occurs, the $X$ and $Y$ variables are said to be uncorrelated (independent variables are a notable case of uncorrelated variables). In Probability theory and Statistics, two real-valued Random variables are said to be uncorrelated if their Covariance is zero In Probability theory, to say that two events are independent intuitively means that the occurrence of one event makes it neither more nor less probable that the other The lack of multiplicativity gives rise to study of covariance and correlation. In Probability theory and Statistics, covariance is a measure of how much two variables change together (the Variance is a special case of the covariance In Probability theory and Statistics, correlation, (often measured as a correlation coefficient) indicates the strength and direction of a linear

Functional non-invariance

In general, the expectation operator and functions of random variables do not commute; that is

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

A notable inequality concerning this topic is Jensen's inequality, involving expected values of convex (or concave) functions. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In Mathematics, commutativity is the ability to change the order of something without changing the end result In Mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value of a Convex function of an Integral

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 $X - \operatorname{E}(X)$ . In Probability theory and Statistics, the k th moment about the Mean (or k th central moment The moments of some random variables can be used to specify their distributions, via their moment generating functions. In Probability theory and Statistics, the moment-generating function of a Random variable X is M_X(t=\operatorname{E}\left(e^{tX}\right

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. Estimation is the calculated Approximation of a result which is usable even if Input data may be incomplete or uncertain. 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 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). Estimation is the calculated Approximation of a result which is usable even if Input data may be incomplete or uncertain. In Statistics, the difference between an Estimator 's Expected value and the true value of the parameter being estimated is called the bias. In Statistics and optimization, the concepts of statistical 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 (which is called the 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. The law of large numbers (LLN is a theorem in Probability that describes the long-term stability of the mean of a Random variable. The sample size of a Statistical sample is the number of observations that constitute it In Statistics, a sample is a Subset of a population. Typically the population is very large making a Census or a complete Enumeration In Probability theory and Statistics, the variance of a Random variable, Probability distribution, or sample is one measure of In Statistics, an estimator is a function of the observable sample data that is used to estimate an unknown population Parameter (which is called the

In classical mechanics, the center of mass is an analogous concept to expectation. Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects 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 $\operatorname{E}(X)$ .

Expected values can also be used to compute the variance, by means of the computational formula for the variance

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

A very important application of the expectation value is in the field of quantum mechanics. In Probability theory and Statistics, the variance of a Random variable, Probability distribution, or sample is one measure of In Probability theory, the computational formula for the Variance Var( X) of a Random variable X is the formula \operatorname{Var}(X Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons The expectation value of a quantum mechanical operator $\hat{A}$ operating on a quantum state vector $|\psi\rangle$ is written as $\langle\hat{A}\rangle = \langle\psi|A|\psi\rangle$ . In Quantum physics, a quantum state is a mathematical object that fully describes a quantum system. The uncertainty in $\hat{A}$ can be calculated using the formula $(\Delta A)^2 = \langle\hat{A}^2\rangle - \langle\hat{A}\rangle^2$ . In Quantum physics, the Heisenberg uncertainty principle states that locating a particle in a small region of space makes the Momentum of the particle uncertain

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:

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

This is utilized in covariance matrices. In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally In Statistics and Probability theory, the covariance matrix is a matrix of Covariances between elements of a vector

Computation

It is often useful to update a computed expected value as new data comes in. This can be done as follows, where $n e w_v a l u e$ is the $c o u n t$ -th value, and we use the previous estimate $\operatorname{E}_\mathrm{prev}(X)$ to compute $\operatorname{E}_\mathrm{new}(X)$ :

$\operatorname{E}_\mathrm{new}(X) = [(count-1) \cdot \operatorname{E}_\mathrm{prev}(X) + new\_value]/count$

Formula for non-negative integral values

When a random variable takes only values in ${0,1,2,3,. . . }$ we can use the following formula for computing its expectation:

$\operatorname{E}(X)=\sum\limits_{i=1}^\infty P(X\geq i)$

For example, suppose we toss a coin where the probability of heads is $p$ . How many tosses can we expect until the first heads? Let $X$ be this number. Note that we are counting only the tails and not the heads which ends the experiment; in particular, we can have $X = 0$ . The expectation of $X$ may be computed by $\sum_{i= 1}^\infty (1-p)^i=\frac{1-p}{p}$ . This is because the number of tosses is at least $i$ exactly when the first $i$ tosses yielded tails. This matches the expectation of a random variable with an Exponential distribution. WikipediaWikiProject Probability#Standards for a discussionof standards used for probability distribution articles such as this one We used the formula for Geometric progression: $\sum_{k=1}^\infty r^k=\frac{r}{(1-r)}$ . In Mathematics, a geometric progression, also known as a geometric sequence, is a Sequence of Numbers where each term after the first is found

External links

Expectation on PlanetMath

In Probability theory, a conditional expectation (also known as conditional expected value or conditional mean is the Expected value of a real random variable with For Probability distributions having an Expected value and a Median, the mean (i Economics is the social science that studies the production distribution, and consumption of goods and services. The field of finance refers to the concepts of Time, Money and Risk and how they are interrelated In the case of Uncertainty, expectation is what is considered the most likely to happen Pascal's Wager (or Pascal's Gambit) is a suggestion posed by the French Philosopher Blaise Pascal that even though the Existence of God In Quantum mechanics, the expectation value is the predicted mean value of the result of an experiment In Economics, the St Petersburg paradox is a Paradox related to Probability theory and Decision theory. PlanetMath is a free, collaborative online Mathematics Encyclopedia.

Dictionary

-noun

(probability theory) Of a discrete random variable, the sum of the probability of each possible outcome of the experiment multiplied by the outcome value (or payoff).

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