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# Expected value

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.

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, while also your bet is returned, together you get 36 times his bet). So the expected value of the profit resulting from a \$1 bet on a single number is, considering all 38 possible outcomes: $( -1 \times \frac{37}{38} ) + ( 35 \times \frac{1}{38} )$, which is about -0.0526. Therefore one expects, on average, to lose over 5 cents for every dollar bet.

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## 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 $\langle X \rangle$ or $\mathbb{E}X$) is defined as

$\mathrm{E}(X) = \int_\Omega X\, dP$

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 x1, x2, ... and corresponding probabilities p1, p2, ... which add up to 1, then E(X) can be computed as the sum or series

 E(X) = ∑ pixi 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

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

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.

## Properties

• The expected value operator (or expectation operator) E is linear in the sense that
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.
• In general, the expectation operator and functions of random variables do not commute; that is
$\mathrm{E}(g(X)) = \int_{\Omega} g(X)\, dP \neq g(\operatorname{E}X),$
except as noted above.
• 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.

The difference, in the general case, gives rise to the covariance and correlation.

## 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.

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 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. The point at which the rod balances (its center of gravity) is E(X).

## Expectation of matrices

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

$\mathrm{E}[X] = \mathrm{E} \begin{bmatrix} x_{1,1} & x_{1,2} & \cdots & x_{1,n} \\ x_{2,1} & x_{2,2} & \cdots & x_{2,n} \\ \vdots \\ x_{m,1} & x_{m,2} & \cdots & x_{m,n} \end{bmatrix} = \begin{bmatrix} \mathrm{E}(x_{1,1}) & \mathrm{E}(x_{1,2}) & \cdots & \mathrm{E}(x_{1,n}) \\ \mathrm{E}(x_{2,1}) & \mathrm{E}(x_{2,2}) & \cdots & \mathrm{E}(x_{2,n}) \\ \vdots \\ \mathrm{E}(x_{m,1}) & \mathrm{E}(x_{m,2}) & \cdots & \mathrm{E}(x_{m,n}) \end{bmatrix}$

This property is utilized in covariance matrices.