Expected value

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.

Failed to parse (Missing texvc executable; please see math/README to configure.): \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:

Failed to parse (Missing texvc executable; please see math/README to configure.): \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."

Mathematical definition

In general, if Failed to parse (Missing texvc executable; please see math/README to configure.): X\,

is a random variable defined on a probability space Failed to parse (Missing texvc executable; please see math/README to configure.): (\Omega, \Sigma, F(x))\,
(where Failed to parse (Missing texvc executable; please see math/README to configure.): \Omega \,
is the sample space, and Failed to parse (Missing texvc executable; please see math/README to configure.): F(x)=P\,
is the Cumulative Distribution Function of probability), then the expected value of Failed to parse (Missing texvc executable; please see math/README to configure.): X\,
(denoted Failed to parse (Missing texvc executable; please see math/README to configure.): \operatorname{E}(X)\,
or sometimes Failed to parse (Missing texvc executable; please see math/README to configure.): \langle X \rangle
or Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{E}(X)

) is defined as

Failed to parse (Missing texvc executable; please see math/README to configure.): \operatorname{E}(X) = \int_\Omega X\, \operatorname{d}F(x)\,

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.

Failed to parse (Missing texvc executable; please see math/README to configure.): \operatorname{E}(X) = \sum_i p_i x_i\,

as in the gambling example mentioned above.

If the probability distribution of Failed to parse (Missing texvc executable; please see math/README to configure.): X

admits a probability density function Failed to parse (Missing texvc executable; please see math/README to configure.): f(x)

, then the expected value can be computed as

Failed to parse (Missing texvc executable; please see math/README to configure.): \operatorname{E}(X) = \int_{-\infty}^\infty x f(x)\, \operatorname{d}x .

It follows directly from the discrete case definition that if Failed to parse (Missing texvc executable; please see math/README to configure.): X

is a constant random variable, i.e. Failed to parse (Missing texvc executable; please see math/README to configure.): X = b
for some fixed real number Failed to parse (Missing texvc executable; please see math/README to configure.): b

, then the expected value of Failed to parse (Missing texvc executable; please see math/README to configure.): X

is also Failed to parse (Missing texvc executable; please see math/README to configure.): b

.

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

Failed to parse (Missing texvc executable; please see math/README to configure.): \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.^{[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 Failed to parse (Missing texvc executable; please see math/README to configure.): X \le Y

almost surely, then  Failed to parse (Missing texvc executable; please see math/README to configure.):  \operatorname{E}(X) \le \operatorname{E}(Y)

.

Linearity

The expected value operator (or expectation operator) Failed to parse (Missing texvc executable; please see math/README to configure.): \operatorname{E}

is linear in the sense that

Failed to parse (Missing texvc executable; please see math/README to configure.): \operatorname{E}(X + c)= \operatorname{E}(X) + c\,

Failed to parse (Missing texvc executable; please see math/README to configure.): \operatorname{E}(X + Y)= \operatorname{E}(X) + \operatorname{E}(Y)\,

Failed to parse (Missing texvc executable; please see math/README to configure.): \operatorname{E}(aX)= a \operatorname{E}(X)\,

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

Failed to parse (Missing texvc executable; please see math/README to configure.): \operatorname{E}(aX + b)= a \operatorname{E}(X) + b\,

Failed to parse (Missing texvc executable; please see math/README to configure.): \operatorname{E}(a X + b Y) = a \operatorname{E}(X) + b \operatorname{E}(Y)\,

for any two random variables Failed to parse (Missing texvc executable; please see math/README to configure.): X

and Failed to parse (Missing texvc executable; please see math/README to configure.): Y
(which need to be defined on the same probability space) and any real numbers Failed to parse (Missing texvc executable; please see math/README to configure.): a
and Failed to parse (Missing texvc executable; please see math/README to configure.): b

.

Iterated expectation

Iterated expectation for discrete random variables

For any two discrete random variables Failed to parse (Missing texvc executable; please see math/README to configure.): X,Y

one may define the conditional expectation:

Failed to parse (Missing texvc executable; please see math/README to configure.): \operatorname{E}(X|Y)(y) = \operatorname{E}(X|Y=y) = \sum\limits_x x \cdot \operatorname{P}(X=x|Y=y).

which means that Failed to parse (Missing texvc executable; please see math/README to configure.): \operatorname{E}(X|Y)

is a function on Failed to parse (Missing texvc executable; please see math/README to configure.): y

.

Then the expectation of Failed to parse (Missing texvc executable; please see math/README to configure.): X

satisfies

Failed to parse (Missing texvc executable; please see math/README to configure.): \operatorname{E} \left( \operatorname{E}(X|Y) \right)= \sum\limits_y \operatorname{E}(X|Y=y) \cdot \operatorname{P}(Y=y) \,

Failed to parse (Missing texvc executable; please see math/README to configure.): =\sum\limits_y \left( \sum\limits_x x \cdot \operatorname{P}(X=x|Y=y) \right) \cdot \operatorname{P}(Y=y)\,

Failed to parse (Missing texvc executable; please see math/README to configure.): =\sum\limits_y \sum\limits_x x \cdot \operatorname{P}(X=x|Y=y) \cdot \operatorname{P}(Y=y)\,

Failed to parse (Missing texvc executable; please see math/README to configure.): =\sum\limits_y \sum\limits_x x \cdot \operatorname{P}(Y=y|X=x) \cdot \operatorname{P}(X=x) \,

Failed to parse (Missing texvc executable; please see math/README to configure.): =\sum\limits_x x \cdot \operatorname{P}(X=x) \cdot \left( \sum\limits_y \operatorname{P}(Y=y|X=x) \right) \,

Failed to parse (Missing texvc executable; please see math/README to configure.): =\sum\limits_x x \cdot \operatorname{P}(X=x) \,

Failed to parse (Missing texvc executable; please see math/README to configure.): =\operatorname{E}(X).\,

Hence, the following equation holds:

Failed to parse (Missing texvc executable; please see math/README to configure.): \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:

Failed to parse (Missing texvc executable; please see math/README to configure.): \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 Failed to parse (Missing texvc executable; please see math/README to configure.): X \leq Y , then Failed to parse (Missing texvc executable; please see math/README to configure.): \operatorname{E}(X) \leq \operatorname{E}(Y) .

In particular, since Failed to parse (Missing texvc executable; please see math/README to configure.): X \leq |X|

and Failed to parse (Missing texvc executable; please see math/README to configure.):  -X \leq |X| 

, the absolute value of expectation of a random variable is less or equal to the expectation of its absolute value:

Failed to parse (Missing texvc executable; please see math/README to configure.): |\operatorname{E}(X)| \leq \operatorname{E}(|X|)

Representation

The following formula holds for any nonnegative real-valued random variable Failed to parse (Missing texvc executable; please see math/README to configure.): X

(such that Failed to parse (Missing texvc executable; please see math/README to configure.):  \operatorname{E}(X) < \infty 

), and positive real number Failed to parse (Missing texvc executable; please see math/README to configure.): \alpha

Failed to parse (Missing texvc executable; please see math/README to configure.): \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. Failed to parse (Missing texvc executable; please see math/README to configure.): \operatorname{E}(X Y)

is not necessarily equal to Failed to parse (Missing texvc executable; please see math/README to configure.): \operatorname{E}(X) \operatorname{E}(Y)

. If multiplicativity occurs, the Failed to parse (Missing texvc executable; please see math/README to configure.): X

and Failed to parse (Missing texvc executable; please see math/README to configure.): 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

Failed to parse (Missing texvc executable; please see math/README to configure.): \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.

Uses and applications of the expected value

The expected values of the powers of Failed to parse (Missing texvc executable; please see math/README to configure.): X

are called the moments of Failed to parse (Missing texvc executable; please see math/README to configure.): X

the moments about the mean of Failed to parse (Missing texvc executable; please see math/README to configure.): X

are expected values of powers of Failed to parse (Missing texvc executable; please see math/README to configure.): 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. 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 Failed to parse (Missing texvc executable; please see math/README to configure.): X

is a discrete random variable with values Failed to parse (Missing texvc executable; please see math/README to configure.): x_i
and corresponding probabilities Failed to parse (Missing texvc executable; please see math/README to configure.): p_i

. Now consider a weightless rod on which are placed weights, at locations Failed to parse (Missing texvc executable; please see math/README to configure.): x_i

along the rod and having masses Failed to parse (Missing texvc executable; please see math/README to configure.): p_i
(whose sum is one). The point at which the rod balances is Failed to parse (Missing texvc executable; please see math/README to configure.): \operatorname{E}(X)

.

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

Failed to parse (Missing texvc executable; please see math/README to configure.): \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. The expectation value of a quantum mechanical operator Failed to parse (Missing texvc executable; please see math/README to configure.): \hat{A}

operating on a quantum state vector Failed to parse (Missing texvc executable; please see math/README to configure.): |\psi\rangle
is written as Failed to parse (Missing texvc executable; please see math/README to configure.): \langle\hat{A}\rangle = \langle\psi|\hat{A}|\psi\rangle

. The uncertainty in Failed to parse (Missing texvc executable; please see math/README to configure.): \hat{A}

can be calculated using the formula 

Failed to parse (Missing texvc executable; please see math/README to configure.): (\Delta A)^2 = \langle\hat{A}^2\rangle - \langle\hat{A}\rangle^2 .

Expectation of matrices

If Failed to parse (Missing texvc executable; please see math/README to configure.): X

is an Failed to parse (Missing texvc executable; please see math/README to configure.): m \times n
matrix, then the expected value of the matrix is defined as the matrix of expected values:

Failed to parse (Missing texvc executable; please see math/README to configure.): \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 Failed to parse (Missing texvc executable; please see math/README to configure.): new\_value

is the Failed to parse (Missing texvc executable; please see math/README to configure.): count

-th value, and we use the previous estimate Failed to parse (Missing texvc executable; please see math/README to configure.): \operatorname{E}_\mathrm{prev}(X)

to compute Failed to parse (Missing texvc executable; please see math/README to configure.): \operatorname{E}_\mathrm{new}(X)

Failed to parse (Missing texvc executable; please see math/README to configure.): \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 Failed to parse (Missing texvc executable; please see math/README to configure.): \{0,1,2,3,...\}

we can use the following formula

for computing its expectation:

Failed to parse (Missing texvc executable; please see math/README to configure.): \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 Failed to parse (Missing texvc executable; please see math/README to configure.): p . How many tosses can we expect until the first heads? Let Failed to parse (Missing texvc executable; please see math/README to configure.): 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 Failed to parse (Missing texvc executable; please see math/README to configure.): X=0

. The expectation of Failed to parse (Missing texvc executable; please see math/README to configure.): X

may be computed by 

Failed to parse (Missing texvc executable; please see math/README to configure.): \sum_{i= 1}^\infty (1-p)^i=(1-p)/p . This is because the number of tosses is at least Failed to parse (Missing texvc executable; please see math/README to configure.): i

exactly when the first Failed to parse (Missing texvc executable; please see math/README to configure.): i
tosses yielded tails.  This matches the 

expectation of a random variable with an Exponential distribution. We used the formula for Geometric progression: Failed to parse (Missing texvc executable; please see math/README to configure.): \sum_{k=1}^\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 PlanetMathar:قيمة متوقعة

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