In probability theory (and especially gambling), 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" 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. A game or situation in which the expected value for the player is zero (no net gain nor loss) is called a "fair game."

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:

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

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.

Mathematical definition

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In general, if $X\backslash ,$ is a random variable defined on a probability space $(\backslash Omega,\; P)\backslash ,$, then the expected value of $X\backslash ,$ (denoted $\backslash mathrm\{E\}(X)\backslash ,$ or sometimes $\backslash langle\; X\; \backslash rangle$ or $\backslash mathbb\{E\}(X)$) is defined as

$\backslash mathrm\{E\}(X)\; =\; \backslash int\_\backslash Omega\; X\backslash ,\; dP$

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.

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 $\backslash mathrm\{E\}(X)$ can be computed as the sum or series

$\backslash mathrm\{E\}(X)\; =\; \backslash sum\_i\; p\_i\; x\_i\backslash ,$

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

$\backslash mathrm\{E\}(X)\; =\; \backslash int\_\{-\backslash infty\}^\backslash infty\; x\; f(x)\backslash ,\; \backslash mathrm\; d\; x.$

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 $b$.

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

$\backslash mathrm\{E\}(g(X))\; =\; \backslash int\_\{-\backslash infty\}^\backslash infty\; g(x)\; f(x)\backslash ,\; \backslash mathrm\; d\; x.$

Conventional terminology

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• 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 means the expected value of a random variable that is the number of such attempts.

And so on.

Properties

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Linearity

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

$\backslash mathrm\{E\}(a\; X\; +\; b\; Y)\; =\; a\; \backslash mathrm\{E\}(X)\; +\; b\; \backslash mathrm\{E\}(Y)\backslash ,$

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

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

$\backslash mathrm\{E\}=\; \backslash mathrm\{E\}[X|Y=y=\; \backslash sum\_x\; x\; \backslash cdot\; \backslash mathrm\{P\}(X=x|Y=y).$

Then the expectation of $X$ satisfies

$$

\begin{matrix} \mathrm{E} \left( \mathrm{E}\right) & = & \sum_y \mathrm{E}[X|Y=y \cdot \mathrm{P}(Y=y) \\ & = & \sum_y \left( \sum_x x \cdot \mathrm{P}(X=x|Y=y) \right) \cdot \mathrm{P}(Y=y) \\ & = & \sum_y \sum_x x \cdot \mathrm{P}(X=x|Y=y) \cdot \mathrm{P}(Y=y) \\ & = & \sum_y \sum_x x \cdot \mathrm{P}(Y=y|X=x) \cdot \mathrm{P}(X=x) \\ & = & \sum_x x \cdot \mathrm{P}(X=x) \cdot \left( \sum_y \mathrm{P}(Y=y|X=x) \right) \\ & = & \sum_x x \cdot \mathrm{P}(X=x) \\ & = & \mathrm{E}^*. \end{matrix}

Hence, the following equation holds:

$\backslash mathrm\{E\}=\; \backslash mathrm\{E\}\; \backslash left(\; \backslash mathrm\{E\}[X|Y\backslash right).$

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.

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 $X\; \backslash leq\; Y$, then $\backslash mathrm\{E\}\backslash leq\; \backslash mathrm\{E\}[Y$.

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

$|\backslash mathrm\{E\}\backslash leq\; \backslash mathrm\{E\}[|X|$

Representation

The following formula holds for any nonnegative real--valued random variable $X$ (such that ), and positive real number $\backslash alpha$:

$\backslash mathrm\{E\}*=\; \backslash alpha\; \backslash int\_\{0\}^\{\backslash infty\}\; t^\{\backslash alpha\; -1\}\backslash mathrm\{P\}(X>t)\; \backslash mathrm\; d\; t.$

Non-multiplicativity

In general, the expected value operator is not multiplicative, i.e. $\backslash mathrm\{E\}(X\; Y)$ is not necessarily equal to $\backslash mathrm\{E\}(X)\; \backslash mathrm\{E\}(Y)$, except if $X$ and $Y$ are independent or uncorrelated. This lack of multiplicativity gives rise to study of covariance and correlation.

Functional non-invariance

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

$\backslash mathrm\{E\}(g(X))\; =\; \backslash int\_\{\backslash Omega\}\; g(X)\backslash ,\; \backslash mathrm\; d\; P\; \backslash neq\; g(\backslash operatorname\{E\}X),$

except as noted above.

Uses and applications of the expected value

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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 $X\; -\; \backslash mathrm\{E\}(X)$. 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. 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 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 $\backslash mathrm\{E\}(X)$.

Expectation of matrices

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If $X$ is an $m\; \backslash times\; n$ matrix, then the expected value of the matrix is a matrix of expected values:

$$

\mathrm{E}^* = \mathrm{E} \begin{bmatrix} 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{bmatrix} = \begin{bmatrix} \mathrm{E}(x_{1,1}) & \mathrm{E}(x_{1,2}) & \cdots & \mathrm{E}(x_{1,n}) \\ \mathrm{E}(x_{2,1}) & \mathrm{E}(x_{2,2}) & \cdots & \mathrm{E}(x_{2,n}) \\ \vdots \\ \mathrm{E}(x_{m,1}) & \mathrm{E}(x_{m,2}) & \cdots & \mathrm{E}(x_{m,n}) \end{bmatrix}

This property is utilized in covariance matrices.

See also

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• Conditional expectation
• An inequality on location and scale parameters.
• Expected value is also a key concept in economics and finance.
• The general term expectation.

External links

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Probability theory | Gambling terminology

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