Expected_value

"Expectation value" redirects here. For the expectation value in quantum mechanics, see expectation value (quantum mechanics).

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). 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.

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.

$$

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

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:

$$

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

(Your net is −$1 when you lose and $35 when you win.) Therefore one expects, on average, to lose over five cents for every dollar bet, and the expected value of a one dollar bet is $0.9473. In gambling or betting, a game or situation in which the expected value of the profit for the player is zero (no net gain nor loss) is commonly called a "fair game."

Contents 1 Mathematical definition 2 Conventional terminology 3 Properties 3.1 Constants 3.2 Monotonicity 3.3 Linearity 3.4 Iterated expectation 3.4.1 Iterated expectation for discrete random variables 3.4.2 Iterated expectation for continuous random variables 3.5 Inequality 3.6 Representation 3.7 Non-multiplicativity 3.8 Functional non-invariance 4 Uses and applications of the expected value 5 Expectation of matrices 6 Computation 7 Formula for non-negative integral values 8 See also 9 External links

Mathematical definition

In general, if $X\backslash ,$ is a random variable defined on a probability space $(\backslash Omega,\; \backslash Sigma,\; F(x))\backslash ,$ (where $\backslash Omega\; \backslash ,$ is the sample space, and $F(x)=P\backslash ,$ is the Cumulative Distribution Function of probability), then the expected value of $X\backslash ,$ (denoted $\backslash operatorname\{E\}(X)\backslash ,$ or sometimes $\backslash langle\; X\; \backslash rangle$ or $\backslash mathbb\{E\}(X)$) is defined as

$\backslash operatorname\{E\}(X)\; =\; \backslash int\_\backslash Omega\; X\backslash ,\; \backslash operatorname\{d\}F(x)\backslash ,$

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.

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

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

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

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:

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

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.^{[citation needed]}

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 $X\; \backslash le\; Y$ almost surely, then $\backslash operatorname\{E\}(X)\; \backslash le\; \backslash operatorname\{E\}(Y)$.

Linearity

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

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

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

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

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

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

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

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

Iterated expectation for discrete random variables

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

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

which means that $\backslash operatorname\{E\}(X|Y)$ is a function on $y$.

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) \,

$=\backslash sum\backslash limits\_y\; \backslash left(\; \backslash sum\backslash limits\_x\; x\; \backslash cdot\; \backslash operatorname\{P\}(X=x|Y=y)\; \backslash right)\; \backslash cdot\; \backslash operatorname\{P\}(Y=y)\backslash ,$

$=\backslash sum\backslash limits\_y\; \backslash sum\backslash limits\_x\; x\; \backslash cdot\; \backslash operatorname\{P\}(X=x|Y=y)\; \backslash cdot\; \backslash operatorname\{P\}(Y=y)\backslash ,$

$=\backslash sum\backslash limits\_y\; \backslash sum\backslash limits\_x\; x\; \backslash cdot\; \backslash operatorname\{P\}(Y=y|X=x)\; \backslash cdot\; \backslash operatorname\{P\}(X=x)\; \backslash ,$

$=\backslash sum\backslash limits\_x\; x\; \backslash cdot\; \backslash operatorname\{P\}(X=x)\; \backslash cdot\; \backslash left(\; \backslash sum\backslash limits\_y\; \backslash operatorname\{P\}(Y=y|X=x)\; \backslash right)\; \backslash ,$

$=\backslash sum\backslash limits\_x\; x\; \backslash cdot\; \backslash operatorname\{P\}(X=x)\; \backslash ,$

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

Hence, the following equation holds:

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

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.

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:

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

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 $X\; \backslash leq\; Y$, then $\backslash operatorname\{E\}(X)\; \backslash leq\; \backslash operatorname\{E\}(Y)$.

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:

$|\backslash operatorname\{E\}(X)|\; \backslash leq\; \backslash operatorname\{E\}(|X|)$

Representation

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

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

Non-multiplicativity

In general, the expected value operator is not multiplicative, i.e. $\backslash operatorname\{E\}(X\; Y)$ is not necessarily equal to $\backslash operatorname\{E\}(X)\; \backslash operatorname\{E\}(Y)$. If multiplicativity occurs, the $X$ and $Y$ variables are said to be uncorrelated (independent variables are a notable case of uncorrelated variables). The 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

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

A notable inequality concerning this topic is Jensen\'s inequality, involving expected values of convex (or concave) functions.

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

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

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

A very important application of the expectation value is in the field of quantum mechanics. The expectation value of a quantum mechanical operator $\backslash hat\{A\}$ operating on a quantum state vector $|\backslash psi\backslash rangle$ is written as $\backslash langle\backslash hat\{A\}\backslash rangle\; =\; \backslash langle\backslash psi|\backslash hat\{A\}|\backslash psi\backslash rangle$. The uncertainty in $\backslash hat\{A\}$ can be calculated using the formula $(\backslash Delta\; A)^2\; =\; \backslash langle\backslash hat\{A\}^2\backslash rangle\; -\; \backslash langle\backslash hat\{A\}\backslash rangle^2$.

Expectation of matrices

If $X$ is an $m\; \backslash times\; n$ 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.

Computation

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

$$

\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 $\backslash \{0,1,2,3,...\backslash \}$ 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 $\backslash sum\_\{i=\; 1\}^\backslash infty\; (1-p)^i=(1-p)/p$. 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. We used the formula for Geometric progression: $\backslash sum\_\{k=1\}^\backslash infty\; r^k=r/(1-r)$.

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
• Pascal\'s Wager
• Moment (mathematics)
• Expectation value (quantum mechanics)
• St. Petersburg Paradox

External links

• Expectation on PlanetMath

This article is licensed under the GNU Free Documentation License. It uses material from Wikipedia