mathematical expectation discrete random variable is called the sum of the products of all its possible values ​​​​and their probabilities

Comment. It follows from the definition that the mathematical expectation of a discrete random variable is a non-random (constant) variable.

Expected value continuous random variable can be calculated by the formula

M(X) =
.

The mathematical expectation is approximately equal to(the more accurate the greater the number of trials) the arithmetic mean of the observed values ​​of the random variable.

Properties of mathematical expectation.

Property 1. The mathematical expectation of a constant value is equal to the constant itself:

Property 2. The constant factor can be taken out of the expectation sign:

Property 3. The mathematical expectation of the product of two independent random variables is equal to the product of their mathematical expectations:

M(XY) =M(X) *M(Y).

Property 4. Mathematical expectation of the sum of two random variables is equal to the sum of the mathematical expectations of the terms:

M(X+Y) =M(X) +M(Y).

12.1. Dispersion of a random variable and its properties.

In practice, it is often required to find out the dispersion of a random variable around its mean value. For example, in artillery it is important to know how closely the shells will fall close to the target that should be hit.

At first glance, it may seem that the easiest way to estimate the scattering is to calculate all possible values ​​of the deviation of a random variable and then find their average value. However, this path will not give anything, since the average value of the deviation, i.e. M, for any random variable is equal to zero.

Therefore, most often they go the other way - they use the variance to calculate.

dispersion(scattering) of a random variable is the mathematical expectation of the squared deviation of a random variable from its mathematical expectation:

D(X) = M 2 .

To calculate the variance, it is often convenient to use the following theorem.

Theorem. The dispersion is equal to the difference between the mathematical expectation of the square of the random variable X and the square of its mathematical expectation.

D (X) \u003d M (X 2) - 2.

Dispersion properties.

Property 1. Dispersion constantCequals zero:

Property 2. A constant factor can be raised beyond the sign of the variance by squaring it:

D(CX)=C 2 D(X).

Property 3. The variance of the sum of two independent random variables is equal to the sum of the variances of these variables:

D(X+Y) =D(X) +D(Y).

Property 4. The variance of the difference of two independent random variables is equal to the sum of their variances:

D(X-Y) =D(X) +D(Y).

13.1. Normalized random variables.

has a variance of 1 and an expectation of 0.

Normalized random variable V is the ratio of a given random variable X to its standard deviation σ

Standard deviation is the square root of the variance

The mathematical expectation and variance of the normalized random variable V are expressed in terms of the characteristics of X as follows:

where v is the coefficient of variation of the original random variable X.

For the distribution function F V (x) and the distribution density f V (x) we have:

F V (x) =F(σx), f V (x) =σf(σx),

where F(x) is the distribution function of the original random variable X, a f(x) is its probability density.

SCATTERING CHARACTERISTICS

From the characteristics of the position - mathematical expectation, median, mode - let's move on to the characteristics of the spread of a random variable x. dispersion D(X)= a 2 , the standard deviation a and the coefficient of variation v. The definition and properties of the variance for discrete random variables were considered in the previous chapter. For continuous random variables

The standard deviation is a non-negative value square root from dispersion:

The coefficient of variation is the ratio of the standard deviation to the mathematical expectation:

Coefficient of variation - applied when M(X)> O - measures the spread in relative units, while the standard deviation - in absolute.

Example 6. For a uniformly distributed random variable X find the variance, standard deviation and coefficient of variation. The dispersion is:

Variable substitution makes it possible to write:

where With = f - aU2.

Therefore, the standard deviation is and the coefficient of variation is:

TRANSFORMATIONS OF RANDOM VALUES

For every random variable X define three more quantities - centered Y, normalized V and given U. Centered random variable Y is the difference between the given random variable X and its mathematical expectation M(X), those. Y=X - M(X). Mathematical expectation of a centered random variable Y is equal to 0, and the variance is the variance of the given random variable:

distribution function Fy(x) centered random variable Y related to the distribution function F(x) of the original random variable X ratio:

For the densities of these random variables, the equality

Normalized random variable V is the ratio of the given random variable X to its standard deviation a, i.e. V = XIo. Mathematical expectation and variance of a normalized random variable V expressed through characteristics X So:

where v is the coefficient of variation of the original random variable x. For the distribution function Fv(x) and density fv(x) normalized random variable V we have:

where F(x)- distribution function of the original random variable x; fix) is its probability density.

Reduced random variable U is a centered and normalized random variable:

For a reduced random variable

Normalized, centered and reduced random variables are constantly used both in theoretical research and in algorithms, software products, regulatory and technical and instructive and methodological documentation. In particular, because the equalities M(U) = 0, D(lf) = 1 make it possible to simplify the substantiation of methods, formulations of theorems, and calculation formulas.

Transformations of random variables and more are used general plan. So, if U = aX + b, where a and b are some numbers, then

Example 7. If a= 1/G, b = -M(X)/G, then Y is a reduced random variable, and formulas (8) are transformed into formulas (7).

With every random variable X it is possible to connect the set of random variables Y given by the formula Y = Oh + b at various a > 0 and b. This set is called scale-shear family, generated by a random variable x. Distribution functions Fy(x) constitute a scale-shift family of distributions generated by the distribution function F(x). Instead of Y= aX + b frequently used notation

Number With is called the shift parameter, and the number d- scale parameter. Formula (9) shows that X- the result of measuring a certain value - goes to K - the result of measuring the same value, if the beginning of the measurement is moved to a point With, and then use the new unit of measure, in d times greater than the old one.

For the scale-shift family (9), the distribution X called standard. In probabilistic-statistical decision-making methods and other applied research, the standard normal distribution, the standard Weibull-Gnedenko distribution, the standard gamma distribution are used.

distribution, etc. (see below).

Other transformations of random variables are also used. For example, for a positive random variable X consider Y = IgX, where IgX- decimal logarithm of a number x. Chain of equalities

relates distribution functions X and Y.

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

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

The initial moment of the sth order of a discontinuous random variable is the sum of the form:

. (5.7.1)

Obviously, this definition coincides with the definition of the initial moment of order s in mechanics, if the masses are concentrated at the points on the x-axis.

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

. (5.7.2)

It is easy to see that the main characteristic of the position introduced in the previous n ° - the mathematical expectation - is nothing more than the first initial moment of the random variable.

Using the expectation sign, we can combine 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 and there are, respectively, and . Therefore, we can write a general definition of the initial moment of the -th order, which is valid for both discontinuous and continuous quantities:

, (5.7.3)

those. the initial moment of the th order of a random variable is the mathematical expectation of the th degree of this random variable.

Before giving the definition of the central moment, we introduce a new concept of "centered random variable".

Let there be a random variable with mathematical expectation . The centered random variable corresponding to the value is the deviation of the random variable from its mathematical expectation:

In what follows, we will agree everywhere to designate the centered random variable corresponding to the given random variable by the same letter with the icon at the top.

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

similarly for a continuous quantity.

Centering a random variable, obviously, is tantamount to moving the origin to the 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 analogous to moments about the center of gravity in mechanics.

Thus, the central moment of order s of a random variable is the mathematical expectation of the th power of the corresponding centered random variable:

, (5.7.6)

and for continuous - integral

. (5.7.8)

In what follows, in cases where there is no doubt about which random variable a given moment belongs to, for brevity we will write simply and instead of and .

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

, (5.7.9)

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

Let us derive relations connecting the central and initial moments of different orders. We will carry out the derivation only for discontinuous quantities; it is easy to verify that exactly the same relations are valid for continuous quantities, if we replace the finite sums with integrals, and the probabilities with elements of probability.

Consider the second central point:

Similarly, for the third central moment we get:

Expressions for etc. can be obtained in a similar way.

Thus, for the central moments of any random variable, the formulas are valid:

(5.7.10)

Generally speaking, the moments can be considered not only with respect to the origin (initial moments) or the mathematical expectation (central moments), but also with respect to an arbitrary point:

. (5.7.11)

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

. (5.7.12)

Let's transform this expression:

Obviously, this value reaches its minimum when , i.e. when the moment is taken with respect to the point .

Of all the moments, the first initial moment (expectation) and the second central moment are most often used as characteristics of a random variable.

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

According to the definition of the central moment

, (5.7.13)

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

Replacing in the expression (5.7.13) the value of its expression, we also have:

. (5.7.14)

To directly calculate the variance, the following formulas are used:

, (5.7.15)

(5.7.16)

Respectively for discontinuous and continuous quantities.

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

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

The variance of a random variable has the dimension of the square of the random variable; For a visual characterization of scattering, it is more convenient to use a quantity whose dimension coincides with that of a random variable. To do this, take the square root of the dispersion. The resulting value is called the standard deviation (otherwise - the "standard") of a random variable. The mean square deviation will be denoted by:

, (5.7.17)

To simplify the records, we will often use the abbreviated notation for standard deviation and variance: and . In the case when there is no doubt to which random variable these characteristics refer, we will sometimes omit the sign x y and and write simply and . The words "standard deviation" will sometimes be abbreviated by the letters s.c.o.

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

Mathematical expectation and variance (or standard deviation) are the most commonly 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 mathematical expectation (or, in the mechanical interpretation, the mass is distributed symmetrically with respect to the center of gravity), then all moments of odd order (if they exist) are equal to zero. Indeed, in total

with a distribution that is symmetric with respect to the distribution law and odd, each positive term corresponds to a negative term equal to it in absolute value, so that the whole sum is equal to zero. The same is obviously true for the integral

,

which is equal to zero as an integral in symmetric limits of an odd function.

Therefore, it is natural to choose any of the odd moments as a characteristic of the distribution asymmetry. The simplest of these is the third central moment. It has the dimension of a cube of a random variable: to obtain a dimensionless characteristic, the third moment is divided by the cube of the standard deviation. The resulting value is called the "asymmetry coefficient" or simply "asymmetry"; we will label it:

On fig. 5.7.1 shows two skewed distributions; one of them (curve I) has a positive asymmetry (); the other (curve II) is negative ().

The fourth central moment serves to characterize the so-called "coolness", i.e. peaked or flat-topped distribution. These distribution properties are described using the so-called kurtosis. The kurtosis of a random variable is the quantity

The number 3 is subtracted from the ratio because for a very important and widespread in nature normal distribution law (which we will get to know in detail later). Thus, for a normal distribution, kurtosis is zero; curves that are more pointed than normal curves have a positive kurtosis; the curves are more flat-topped - by negative kurtosis.

On fig. 5.7.2 shows: normal distribution (curve I), distribution with positive kurtosis (curve II) and distribution with negative kurtosis (curve III).

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

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

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

, (5.7.21)

called the arithmetic mean deviation. Along with dispersion and standard deviation, the arithmetic mean deviation is sometimes used as a dispersion characteristic.

Mathematical expectation, mode, median, initial and central moments and, in particular, variance, standard deviation, skewness and kurtosis are the most commonly used numerical characteristics of random variables. In many practical tasks complete characteristic random variable - the distribution law - is either not needed, or cannot be obtained. In these cases, they are 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 usually they try to make this replacement so that several important points remain unchanged.

Example 1. One experiment is performed, as a result of which an event may or may not appear, the probability of which is equal to . A random variable is considered - the number of occurrences of an event (characteristic random variable of an event ). Determine its characteristics: mathematical expectation, variance, standard deviation.

Solution. The quantity distribution series has the form:

where is the probability that the event will not occur.

According to the formula (5.6.1) we find the mathematical expectation of the value:

The dispersion of the value is determined by the formula (5.7.15):

(We invite the reader to obtain the same result by expressing the variance in terms of the second initial moment).

Example 2. Three independent shots are fired at the target; the probability of hitting each shot is 0.4. the random variable is the number of hits. Determine the characteristics of the quantity - mathematical expectation, dispersion, s.c.o., asymmetry.

Solution. The quantity distribution series has the form:

We calculate the numerical characteristics of the quantity:

Note that the same characteristics could be calculated much more simply using theorems on the numerical characteristics of functions (see Chapter 10).

Mat. Expectation Mode Median

The most important characteristic expected value , which shows the average value of a random variable.

Expected value the value of X is denoted by M[X], or m x .

For discrete random variables expected value :

The sum of the values ​​of the corresponding value by the probability of random variables.

Fashion (Mod) of a random variable X is called its most probable value.

For a discrete random variable. For a continuous random variable.

Unimodal distribution

Multi modal distribution

In general, Mod and expected value not

match.

median (Med) of a random variable X is such a value for which the probability that P(X Med). Any Med distribution can only have one.

Med divides the area under the curve into 2 equal parts. In case of unimodal and symmetrical distribution

Moments.

Most often, two types of moments are used in practice: initial and central.

Starting moment. th order of a discrete random variable X is a sum of the form:

For a continuous random variable X, the initial moment of order is the integral , it is obvious that the mathematical expectation of a random variable is the first initial moment.

Using the sign (operator) M, the initial moment of the -th order can be represented as a mat. expectation of the th power of some random variable.

Centered the random variable of the corresponding random variable X is the deviation of the random variable X from its mathematical expectation:

The mathematical expectation of a centered random variable is 0.

For discrete random variables we have:

The moments of a centered random variable are called Central moments

Central moment of order random variable X is called the mathematical expectation of the th power of the corresponding centered random variable.

For discrete random variables:

For continuous random variables:

Relation between central and initial moments of various orders

Of all the moments, the first moment (math. expectation) and the second central moment are most often used as a characteristic of a random variable.

The second central moment is called dispersion random variable. It has the designation:

By definition

For a discrete random variable:

For a continuous random variable:

The dispersion of a random variable is a characteristic of dispersion (scattering) of random variables X around its mathematical expectation.

Dispersion means scattering. The variance has the dimension of the square of a random variable.

For a visual characterization of dispersion, it is more convenient to use the value m y the same as the dimension of the random variable. For this purpose, a root is taken from the dispersion and a value is obtained, called - standard deviation (RMS) random variable X, while introducing the designation:

The standard deviation is sometimes called the "standard" of the random variable X.

Transformations of random variables

For every random variable X determine three more quantities - centered Y, normalized V and given U. Centered random variable Y is the difference between the given random variable X and its mathematical expectation M(X), those. Y = X - M(X). Mathematical expectation of a centered random variable Y is equal to 0, and the variance is the variance of the given random variable: M(Y) = 0, D(Y) = D(X). distribution function F Y(x) centered random variable Y related to the distribution function F(x) initial random variable X ratio:

F Y(x) = F(x + M(X)).

For the densities of these random variables, the equality

f Y(x) = f(x + M(X)).

Normalized random variable V is the ratio of this random variable X to its standard deviation , i.e. . Mathematical expectation and variance of a normalized random variable V expressed through characteristics X So:

,

where v is the coefficient of variation of the original random variable X. For the distribution function F V(x) and density f V(x) normalized random variable V we have:

where F(x) is the distribution function of the original random variable X, a f(x) is its probability density.

Reduced random variable U is a centered and normalized random variable:

.

For a reduced random variable

Normalized, centered and reduced random variables are constantly used both in theoretical research and in algorithms, software products, regulatory and technical and instructive and methodological documentation. In particular, because the equalities make it possible to simplify the substantiation of methods, formulations of theorems, and calculation formulas.

Transformations of random variables and more general plan are used. So if Y = aX + b, where a and b are some numbers, then

Example 7 If then Y is the reduced random variable, and formulas (8) are transformed into formulas (7).

With every random variable X you can connect a lot of random variables Y given by the formula Y = aX + b at various a> 0 and b. This set is called scale-shift family, generated by a random variable X. Distribution functions F Y(x) constitute a scale-shift family of distributions generated by the distribution function F(x). Instead of Y = aX + b frequently used notation

Number With is called the shift parameter, and the number d- scale parameter. Formula (9) shows that X- the result of measuring a certain quantity - goes into At- the result of the measurement of the same value, if the beginning of the measurement is moved to the point With, and then use the new unit of measure, in d times greater than the old one.

For the scale-shift family (9), the distribution X is called standard. In probabilistic-statistical decision-making methods and other applied research, the standard normal distribution, the standard Weibull-Gnedenko distribution, the standard gamma distribution, etc. are used (see below).

Other transformations of random variables are also used. For example, for a positive random variable X consider Y= log X, where lg X is the decimal logarithm of the number X. Chain of equalities

F Y (x) = P( lg X< x) = P(X < 10x) = F( 10x)

relates distribution functions X and Y.