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.

A common application of expected value is in gambling. For example, an American roulette wheel has 38 equally possible 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:

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

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."

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

$\operatorname {E} (X)=\int _{\Omega }X\,\operatorname {d} P$

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.

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

$\operatorname {E} (X)=\sum _{i}p_{i}x_{i}\,$

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

$\operatorname {E} (X)=\int _{-\infty }^{\infty }xf(x)\,\operatorname {d} x.$

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 $b$.

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

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

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

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 is always less than Y, then Expectation of X will also be less than Expectation of Y.

Linearity

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

$\operatorname {E} (aX+b)=a\operatorname {E} (X)+bor:<math>\operatorname {E} (aX+bY)=a\operatorname {E} (X)+b\operatorname {E} (Y)\,$

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:

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

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.

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:

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

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 $\alpha$:

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

Non-multiplicativity

In general, the expected value operator is not multiplicative, i.e. $\operatorname {E} (XY)$ 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.

Functional non-invariance

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

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

except as noted above.

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.

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.

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 $\operatorname {E} (X)$.

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.

See also

• 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

• "Expectation". PlanetMath.