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5.7. Moments. Dispersion. Standard deviation

Lecture



In addition to the characteristics of the position - the average, typical values ​​of a random variable - a number of characteristics are used, each of which describes a particular property of the distribution. As such characteristics, so-called moments are most often used.

The concept of moment is widely used in mechanics to describe the mass distribution (static moments, moments of inertia, etc.). The very same methods are used in probability theory to describe the basic properties of the distribution of a random variable. Most often used in practice moments of two types: primary and central.

The initial moment of the s-th order of a discontinuous random variable 5.7.  Moments.  Dispersion.  Standard deviation called the sum of the form:

5.7.  Moments.  Dispersion.  Standard deviation . (5.7.1)

Obviously, this definition coincides with the definition of the initial moment of order s in mechanics, if on the abscissa at the points 5.7.  Moments.  Dispersion.  Standard deviation concentrated masses 5.7.  Moments.  Dispersion.  Standard deviation .

For a continuous random variable X, the initial moment of the sth order is the integral

5.7.  Moments.  Dispersion.  Standard deviation . (5.7.2)

It is easy to verify that the basic characteristic of the position introduced in the preceding n ° — the mathematical expectation — is nothing but the first initial moment of the random variable. 5.7.  Moments.  Dispersion.  Standard deviation .

Using the expectation sign, you can combine the two formulas (5.7.1) and (5.7.2) into one. Indeed, formulas (5.7.1) and (5.7.2) are completely similar in structure to formulas (5.6.1) and (5.6.2), with the difference that instead of 5.7.  Moments.  Dispersion.  Standard deviation and 5.7.  Moments.  Dispersion.  Standard deviation cost accordingly 5.7.  Moments.  Dispersion.  Standard deviation and 5.7.  Moments.  Dispersion.  Standard deviation . Therefore, we can write a general definition of the initial moment 5.7.  Moments.  Dispersion.  Standard deviation th order, valid for both discontinuous and continuous values:

5.7.  Moments.  Dispersion.  Standard deviation , (5.7.3)

those. initial moment 5.7.  Moments.  Dispersion.  Standard deviation -th order of random variable 5.7.  Moments.  Dispersion.  Standard deviation called expectation 5.7.  Moments.  Dispersion.  Standard deviation -th degree of this random variable.

Before we define the central moment, we introduce a new concept of a “centered random variable”.

Let there be a random variable 5.7.  Moments.  Dispersion.  Standard deviation with mathematical expectation 5.7.  Moments.  Dispersion.  Standard deviation . Centered random variable corresponding to magnitude 5.7.  Moments.  Dispersion.  Standard deviation called the deviation of a random variable 5.7.  Moments.  Dispersion.  Standard deviation from her expectation:

5.7.  Moments.  Dispersion.  Standard deviation . (5.7.4)

We agree in the following to denote everywhere a centered random variable corresponding to a given random variable with the same letter with the symbol 5.7.  Moments.  Dispersion.  Standard deviation at the top.

It is easy to verify that the expectation of a centered random variable is zero. Indeed, for a discontinuous value

5.7.  Moments.  Dispersion.  Standard deviation ; (5.7.5)

similarly for continuous value.

Centering a random variable is obviously tantamount to transferring the origin to a middle, “central” point, the abscissa of which is equal to the mathematical expectation.

The moments of a centered random variable are called central moments. They are similar to the moments relative to the center of gravity in mechanics.

Thus, the central moment of the order s of a random variable 5.7.  Moments.  Dispersion.  Standard deviation called expectation 5.7.  Moments.  Dispersion.  Standard deviation -th degree of the corresponding centered random variable:

5.7.  Moments.  Dispersion.  Standard deviation , (5.7.6)

and for continuous - integral

5.7.  Moments.  Dispersion.  Standard deviation . (5.7.8)

In the future, in cases where there is no doubt about the random value of a given moment, for brevity, instead of 5.7.  Moments.  Dispersion.  Standard deviation and 5.7.  Moments.  Dispersion.  Standard deviation just write 5.7.  Moments.  Dispersion.  Standard deviation and 5.7.  Moments.  Dispersion.  Standard deviation .

Obviously, for any random variable, the central moment of the first order is zero:

5.7.  Moments.  Dispersion.  Standard deviation , (5.7.9)

since the expectation of a centered random variable is always zero.

We derive the relations connecting the central and initial moments of different orders. We will only draw a conclusion for discontinuous values; it is easy to see that exactly the same relations hold for continuous quantities if the final sums are replaced by integrals, and probabilities by probability elements.

Consider the second central point:

5.7.  Moments.  Dispersion.  Standard deviation

Similarly, for the third central moment we get:

5.7.  Moments.  Dispersion.  Standard deviation

Expressions for 5.7.  Moments.  Dispersion.  Standard deviation etc. can be obtained in a similar way.

Thus, for the central moments of any random variable 5.7.  Moments.  Dispersion.  Standard deviation fair formulas are:

5.7.  Moments.  Dispersion.  Standard deviation (5.7.10)

Generally speaking, moments can be considered not only relative to the origin (initial moments) or mathematical expectation (central moments), but also relative to an arbitrary point. 5.7.  Moments.  Dispersion.  Standard deviation :

5.7.  Moments.  Dispersion.  Standard deviation . (5.7.11)

However, the central moments have an advantage over all others: the first central moment, as we have seen, is always zero, and the second central moment following it has a minimum value with this frame. Let's prove it. For a discontinuous random variable 5.7.  Moments.  Dispersion.  Standard deviation at 5.7.  Moments.  Dispersion.  Standard deviation the formula (5.7.11) has the form:

5.7.  Moments.  Dispersion.  Standard deviation . (5.7.12)

Convert this expression:

5.7.  Moments.  Dispersion.  Standard deviation

Obviously, this value reaches its minimum when 5.7.  Moments.  Dispersion.  Standard deviation i.e. when the moment is taken relative to the point 5.7.  Moments.  Dispersion.  Standard deviation .

Of all the moments, the first initial moment (mathematical expectation) is most often used as the characteristics of a random variable. 5.7.  Moments.  Dispersion.  Standard deviation and the second central point 5.7.  Moments.  Dispersion.  Standard deviation .

The second central moment is called the variance of the random variable. In view of the extreme importance of this characteristic, among other things, we introduce a special designation for it. 5.7.  Moments.  Dispersion.  Standard deviation :

5.7.  Moments.  Dispersion.  Standard deviation .

According to the definition of the central moment

5.7.  Moments.  Dispersion.  Standard deviation , (5.7.13)

those. the variance of the random variable X is the mathematical expectation of the square of the corresponding centered value.

Replacing the value in expression (5.7.13) 5.7.  Moments.  Dispersion.  Standard deviation by its expression, we also have:

5.7.  Moments.  Dispersion.  Standard deviation . (5.7.14)

For the direct calculation of the variance are the formulas:

5.7.  Moments.  Dispersion.  Standard deviation , (5.7.15)

5.7.  Moments.  Dispersion.  Standard deviation (5.7.16)

- respectively for discontinuous and continuous values.

Variance of a random variable is a characteristic of dispersion, the dispersion of values ​​of a random variable around its expectation. The word "dispersion" itself means "dispersion."

If we turn to the mechanical interpretation of the distribution, then the variance is nothing but the moment of inertia of a given mass distribution relative to the center of gravity (mathematical expectation).

The variance of the random variable has the dimension of the square of the random variable; for visual dispersion characteristics, it is more convenient to use a quantity whose dimension coincides with the dimension of a random variable. For this, the square root is extracted from the dispersion. The resulting value is called the standard deviation (otherwise - "standard") of a random variable. 5.7.  Moments.  Dispersion.  Standard deviation . Standard deviation will be denoted by 5.7.  Moments.  Dispersion.  Standard deviation :

5.7.  Moments.  Dispersion.  Standard deviation , (5.7.17)

To simplify the records, we will often use the abbreviated notation for the standard deviation and variance: 5.7.  Moments.  Dispersion.  Standard deviation and 5.7.  Moments.  Dispersion.  Standard deviation . In the case when there is no doubt what random value these characteristics belong to, we will sometimes omit the xy icon 5.7.  Moments.  Dispersion.  Standard deviation and 5.7.  Moments.  Dispersion.  Standard deviation and just write 5.7.  Moments.  Dispersion.  Standard deviation and 5.7.  Moments.  Dispersion.  Standard deviation . The words "standard deviation" will sometimes be abbreviated to replace the letters with.

In practice, a formula is often used expressing the variance of a random variable through its second initial moment (the second of formulas (5.7.10)). In the new notation, it will look like:

5.7.  Moments.  Dispersion.  Standard deviation . (5.7.18)

Expected value 5.7.  Moments.  Dispersion.  Standard deviation and variance 5.7.  Moments.  Dispersion.  Standard deviation (or mean square deviation 5.7.  Moments.  Dispersion.  Standard deviation ) - the most frequently used characteristics of a random variable. They characterize the most important features of the distribution: its position and degree of dispersion. For a more detailed description of the distribution, higher order moments are used.

The third central moment serves to characterize the asymmetry (or "skewness") of the distribution. If the distribution is symmetrical with respect to the expectation (or, in a mechanical interpretation, the mass is distributed symmetrically with respect to the center of gravity), then all moments of an odd order (if they exist) are equal to zero. Indeed, in sum

5.7.  Moments.  Dispersion.  Standard deviation

with relatively symmetrical 5.7.  Moments.  Dispersion.  Standard deviation distribution law and odd 5.7.  Moments.  Dispersion.  Standard deviation each positive term corresponds to a negative term equal to it in absolute value, so that the whole sum is zero. The same is obviously true for the integral

5.7.  Moments.  Dispersion.  Standard deviation ,

which is zero, as an integral in the symmetric range of an odd function.

It is therefore natural to choose one of the odd moments as a characteristic of the asymmetry of the distribution. The simplest of them is the third central moment. It has the dimension of a cube of a random variable: in order to obtain a dimensionless characteristic, the third moment 5.7.  Moments.  Dispersion.  Standard deviation divided by the cube of the standard deviation. The value obtained is called the “asymmetry coefficient” or simply “asymmetry”; we denote it 5.7.  Moments.  Dispersion.  Standard deviation :

5.7.  Moments.  Dispersion.  Standard deviation . (5.7.19)

In fig. 5.7.1 shows two asymmetric distributions; one of them (curve I) has a positive asymmetry ( 5.7.  Moments.  Dispersion.  Standard deviation ); other (curve II) - negative ( 5.7.  Moments.  Dispersion.  Standard deviation ).

5.7.  Moments.  Dispersion.  Standard deviation

Fig. 5.7.1

The fourth central moment serves to characterize the so-called "coolness", i.e. sharpness or flatness distribution. These properties of the distribution are described by the so-called kurtosis. Excesses of random variables 5.7.  Moments.  Dispersion.  Standard deviation called magnitude

5.7.  Moments.  Dispersion.  Standard deviation . (5.7.20)

The number 3 is subtracted from the relationship 5.7.  Moments.  Dispersion.  Standard deviation because for the very important and widespread in nature normal distribution law (with which we will get acquainted in detail later) 5.7.  Moments.  Dispersion.  Standard deviation . Thus, for a normal distribution, kurtosis is zero; curves, more pointed than normal, have positive kurtosis; curves more flat-topped - negative kurtosis.

In fig. 5.7.2 are presented: normal distribution (curve I), distribution with positive kurtosis (curve II) and distribution with negative kurtosis (curve III).

5.7.  Moments.  Dispersion.  Standard deviation

Fig. 5.7.2

In addition to the above initial and central moments, in practice sometimes used the so-called absolute moments (initial and central), defined by the formulas

5.7.  Moments.  Dispersion.  Standard deviation ; 5.7.  Moments.  Dispersion.  Standard deviation .

Obviously, the absolute moments of even orders coincide with the usual moments.

Of the absolute moments, the first absolute central moment is most often used.

5.7.  Moments.  Dispersion.  Standard deviation , (5.7.21)

called average arithmetic deviation. Along with variance and standard deviation, the arithmetic average deviation is sometimes used as a scattering characteristic.

Mathematical expectation, mode, median, initial and central moments and, in particular, variance, standard deviation, asymmetry and kurtosis are the most commonly used numerical characteristics of random variables. In many problems of practice, a complete characteristic of a random variable — the law of distribution — is either not needed or cannot be obtained. In these cases, limited to an approximate description of a random variable with help. Numerical characteristics, each of which expresses some characteristic property of the distribution.

Very often, numerical characteristics are used to approximate the replacement of one distribution by another, and they usually strive to make this replacement so that several important points remain unchanged.

Example 1. One experience is performed, as a result of which an event may or may not appear. 5.7.  Moments.  Dispersion.  Standard deviation whose probability is equal 5.7.  Moments.  Dispersion.  Standard deviation . Considered a random variable 5.7.  Moments.  Dispersion.  Standard deviation - the number of occurrences 5.7.  Moments.  Dispersion.  Standard deviation (characteristic random variable of event 5.7.  Moments.  Dispersion.  Standard deviation ). Determine its characteristics: expectation, variance, standard deviation.

Decision. The series of distribution of the value is:

5.7.  Moments.  Dispersion.  Standard deviation

Where 5.7.  Moments.  Dispersion.  Standard deviation - probability of non-occurrence 5.7.  Moments.  Dispersion.  Standard deviation .

By the formula (5.6.1) we find the expected value of 5.7.  Moments.  Dispersion.  Standard deviation :

5.7.  Moments.  Dispersion.  Standard deviation .

Variance of magnitude 5.7.  Moments.  Dispersion.  Standard deviation determined by the formula (5.7.15):

5.7.  Moments.  Dispersion.  Standard deviation ,

from where

5.7.  Moments.  Dispersion.  Standard deviation .

(We offer the reader to get the same result, expressing the variance through the second initial moment).

Example 2. Three independent target shots are made; the probability of hitting each shot is 0.4. random value 5.7.  Moments.  Dispersion.  Standard deviation - number of hits. Determine the magnitude characteristics 5.7.  Moments.  Dispersion.  Standard deviation - expectation, variance, ko, asymmetry.

Decision. Distribution range 5.7.  Moments.  Dispersion.  Standard deviation has the form:

5.7.  Moments.  Dispersion.  Standard deviation

We calculate the numerical characteristics of the value 5.7.  Moments.  Dispersion.  Standard deviation :

5.7.  Moments.  Dispersion.  Standard deviation

Note that the same characteristics could be computed much simpler with the help of theorems on the numerical characteristics of functions (see Chapter 10).

Example 3. A series of independent experiments are made before the first occurrence of an event. 5.7.  Moments.  Dispersion.  Standard deviation (see example 3 n ° 5.1). Event probability 5.7.  Moments.  Dispersion.  Standard deviation in each experience is equal 5.7.  Moments.  Dispersion.  Standard deviation . Find the expectation, variance and sc. the number of experiments that will be made.

Decision. Distribution range 5.7.  Moments.  Dispersion.  Standard deviation has the form:

5.7.  Moments.  Dispersion.  Standard deviation

Expectation value 5.7.  Moments.  Dispersion.  Standard deviation expressed by the sum of a row

5.7.  Moments.  Dispersion.  Standard deviation .

Нетрудно видеть, что ряд, стоящий в скобках, представляет собой результат дифференцирования геометрической прогрессии:

5.7.  Moments.  Dispersion.  Standard deviation

Consequently,

5.7.  Moments.  Dispersion.  Standard deviation

from where

5.7.  Moments.  Dispersion.  Standard deviation .

Для определения дисперсии величины Х вычислим сначала её второй начальный момент:

5.7.  Moments.  Dispersion.  Standard deviation .

Для вычисления ряда, стоящего в скобках, умножим на q ряд:

5.7.  Moments.  Dispersion.  Standard deviation

We get:

5.7.  Moments.  Dispersion.  Standard deviation

Дифференцируя этот ряд по 5.7.  Moments.  Dispersion.  Standard deviation , we have:

5.7.  Moments.  Dispersion.  Standard deviation

Умножая на 5.7.  Moments.  Dispersion.  Standard deviation , we get:

5.7.  Moments.  Dispersion.  Standard deviation

По формуле (5.7.18) выразим дисперсию:

5.7.  Moments.  Dispersion.  Standard deviation

from where

5.7.  Moments.  Dispersion.  Standard deviation

Пример 4. Непрерывная случайная величина 5.7.  Moments.  Dispersion.  Standard deviation subject to the distribution law with a density of:

5.7.  Moments.  Dispersion.  Standard deviation

(рис. 5.7.3).

Найти коэффициент 5.7.  Moments.  Dispersion.  Standard deviation . Определить м.о., дисперсию, с.к.о., асимметрию, эксцесс величины 5.7.  Moments.  Dispersion.  Standard deviation .

5.7.  Moments.  Dispersion.  Standard deviation

Fig. 5.7.3.

Decision. For determining 5.7.  Moments.  Dispersion.  Standard deviation воспользуемся свойством плотности распределения:

5.7.  Moments.  Dispersion.  Standard deviation

from here 5.7.  Moments.  Dispersion.  Standard deviation .

Так как функция 5.7.  Moments.  Dispersion.  Standard deviation нечетная, то м.о. величины 5.7.  Moments.  Dispersion.  Standard deviation равно нулю:

5.7.  Moments.  Dispersion.  Standard deviation .

Дисперсия и с.к.о. равны, соответственно:

5.7.  Moments.  Dispersion.  Standard deviation .

Так как распределение симметрично, то 5.7.  Moments.  Dispersion.  Standard deviation .

Для вычисления эксцесса находим 5.7.  Moments.  Dispersion.  Standard deviation :

5.7.  Moments.  Dispersion.  Standard deviation

from where

5.7.  Moments.  Dispersion.  Standard deviation .

Пример 5. Случайная величина 5.7.  Moments.  Dispersion.  Standard deviation подчинена закону распределения, плотность которого задана графически на рис. 5.7.4.

Write an expression for the density of distribution. Найти м.о., дисперсию, с.к.о. и асимметрию распределения.

5.7.  Moments.  Dispersion.  Standard deviation

Fig. 5.7.4.

Decision. Выражение плотности распределения имеет вид:

5.7.  Moments.  Dispersion.  Standard deviation

Пользуясь свойством плотности распределения, находим 5.7.  Moments.  Dispersion.  Standard deviation .

Математическое ожидание величины 5.7.  Moments.  Dispersion.  Standard deviation :

5.7.  Moments.  Dispersion.  Standard deviation

Дисперсию найдем через второй начальный момент:

5.7.  Moments.  Dispersion.  Standard deviation

from here

5.7.  Moments.  Dispersion.  Standard deviation

Третий начальный момент равен

5.7.  Moments.  Dispersion.  Standard deviation

Пользуясь третьей из формул (5.7.10), выражающей 5.7.  Moments.  Dispersion.  Standard deviation через начальные моменты, имеем:

5.7.  Moments.  Dispersion.  Standard deviation

from where

5.7.  Moments.  Dispersion.  Standard deviation


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Probability theory. Mathematical Statistics and Stochastic Analysis

Terms: Probability theory. Mathematical Statistics and Stochastic Analysis