Expected value

The expectation value (rare and ambiguous mean) is a fundamental concept of stochastics. The expected value of a random variable describing the number that takes the random variable in the middle. It arises, for example, with unlimited repetition of the underlying experiment as an average of the results. The law of large numbers describes the form in which precisely aim the averages of the results with increasing number of experiments against the expected value, that is, as the sample mean values ​​with increasing sample size converge to the expected value.

It determines the location (position ) of the distribution of the random variables and is similar to the empirical arithmetic mean of a frequency distribution in descriptive statistics. It is calculated as a weighted according to probability average of the values ​​assumed by the random variable. He himself, however, must not be one of these values ​​. In particular, the expected value can take the values ​​.

Because the expected value depends only on the probability distribution, one also speaks of the expected value of a distribution, without reference to a random variable.

The expected value of a random variable can be regarded as a focus of the probability mass and is therefore also referred to as their first moment.

• 3.1 cubes
• 3.2 St. Petersburg game
• 3.3 random variable with density
• 3.4 General definition

• 4.1 expected value of the sum of n random variables
• 4.2 Linear Transformation
• 4.3 Expectation value of the product of n stochastically independent random variables
• 4.4 probabilities as expectation values

Motivation

The definition of the expected value is analogous to the weighted average of empirically observed numbers. Has, for example, a series of ten tests, the results cube 4, 2, 1, 3, 6, 3, 3, 1, 4, 5 supplied to the corresponding mean value

Alternatively be calculated by summarizing first the same values ​​and weighted according to their relative frequency:

In general terms the average of the numbers in eyes n throws like

Write wherein the relative frequency of eyes numeral.

The figures in the eyes dice roll can be regarded as different manifestations of a random variable. Because the relative frequencies to approximate the n probabilities for each eye numbers according to the law of large numbers with increasing sample size, the mean value against the expected value of must aspire to his calculation, the possible values ​​are weighted by their theoretical probability:

As the results of dice rolls is also the mean value depends on luck. In contrast, the expected value is a fixed measure of the distribution of the random variables.

Definitions

If a random variable discrete or has a density, then there exist the following formulas for the expected value:

Expected value of a discrete random real variables

In the real discrete case, the expected value is calculated as the sum of the products of the probabilities of each potential outcome of the experiment and the "values" of these results.

Is a real discrete random variable that takes the values ​​with the respective probabilities ( with a countable index set ), then the expected value is calculated in the case of existence with:

, So if and only has a finite expected value if the convergence condition

Expectation value of a real random variable with density function

Has a real random variable is a probability density function, ie, the size, this density with respect to the Lebesgue measure, then the expected value in the case of existence calculated as

In many applications, there is (generally improper) Riemann integrability and one has:

Is equivalent to this equation, if the distribution function of is:

(2) and (3) are under the common condition ( f density function and F is the distribution function of f ) equivalent to what can be proved with a school according agents.

General definition

The expected value is defined as corresponding to the integral with respect to the probability measure: a p - or P- integrated integrated quasi random variable from a probability space after, the Borel σ algebra is so defined is

The random variable has an expected value if and only if it is quasi- integrable, so the integrals

Are not both infinite, denote where and are the positive and the negative part of. In this case, or apply.

The expected value is exactly then finally, when is integrable, so the above integrals over and both are finite. This is equivalent to

In this case, write many authors, the expected value exists, or is a random variable with an existing expectation value, which will close the case on or off.

Expectation value of two random variables with joint density function

Have the integrable random variables and a joint probability density function, then the expected value of a function to and from the Fubini's theorem to calculate

The expected value of is then finally, when the integral

Is finite.

In particular:

From the edge density, the expected value is calculated as univariate distributions:

The edge density is given by

Examples

Dice

The experiment was a roll of the dice. As a random variable, we consider the number rolled, each of which is of the numbers 1 to 6 with a probability of 1/ 6 will be rolled.

If you roll example, 1000 times, ie, the random experiment is repeated 1000 times, along one of the numbers thrown eyes and divided by 1000 is achieved with high probability, a value near 3.5. However, it is impossible to obtain this value with a single die roll.

St. Petersburg game

The so-called St. Petersburg game is a game in which the random profit has an infinite expected value. You toss a coin. Shows you head, you get 2 € and the game is over, it shows number, one must throw again. Taking now head, you get 4 euros and the game is over, you throw back number, we may throw a third time. The expected value of the gain is infinite:

Random variable with density

Where is the real random variable with density function

Where is the Euler constant.

The expected value of is calculated as

General definition

Consider the probability space, the power set of all generations. The expected value of random variables with and is

As a discrete random variable, and with the expected value may alternatively be calculated as

Calculation rules

The expected value is linear for all the random variables can be integrated, since the integral is a linear operator. This results in the following two very useful rules:

Expected value of the sum of n random variables

The expected value of the sum of integrable random variables can be calculated as the sum of the individual expected values ​​:

This also applies to discrete random variable and even if they are not stochastically independent.

Linear transformation

Let and be two integrable random variable, then for the linear transformation:

Thus in particular

And

Expectation value of the product of n stochastically independent random variables

If the random variables are stochastically independent and integrable, the following applies:

Especially

Probabilities as expectation values

Probabilities of events can also be expressed through the expectation value. For each event

Where the indicator function of being.

This relationship is often useful, such as proof of Chebyshev 's inequality.

Expectation values ​​of functions of random variables

If back is a random variable, so you can the expected level of, rather than by means of the definition, by means of the formula

Calculate. Also in this case, the expected value exists only when

Converges.

In a discrete random variable using a Total:

If the sum is not finite, then the series must converge absolutely, so that the expected value exists.

Concept and notation

The concept of the expected value goes back to Christiaan Huygens. In a treatise on gambling of 1656, " Van rekeningh Geluck in spelen van " refers to Huygens the expected profit of a game as " het is my soo veel Weerdt ". Frans van Schooten used in his translation of Huygens ' text into Latin the term expectatio. Bernoulli took over in his Ars conjectandi introduced by van pods term in the form of valor expectationis.

In the western area is used for the operator, especially in anglophone literature. In the Russian-language literature there are descriptions. The term emphasizes the property as not dependent on chance first moment. In physics, the Bra- Ket notation is used. In particular, one size is written instead of the expected value.

Quantum mechanical expectation value

Is the wave function of a particle is in a particular state and an operator, then

The quantum mechanical expectation value of in the state. here is the real space in which the particle moves, is the dimension of, and a superscript asterisk stands for complex conjugation.

Can be as formal power series write ( and this is often so ), you use the formula

The index of the expectation value brackets will not only abbreviated as here, but sometimes omitted entirely.

The expected value of the place of stay in position representation

The expected value of the residence in the momentum representation

Where we have identified the probability density function of quantum mechanics in the spatial domain. In physics, one writes instead of (rho ).

Expected value of matrices

Is a matrix, then the expected value of the matrix is defined as: