The considered property of discrete distributions creates significant difficulties when tabulating and using such distributions, since it is impossible to accurately maintain typical numerical values ​​of the distribution characteristics. In particular, this is true for critical values ​​and significance levels of nonparametric statistical tests (see below), since the distributions of statistics of these tests are discrete.

Quantile order is of great importance in statistics R= S. It is called the median (random variable X or its distribution functions F(x)) and is designated Me(X). In geometry there is the concept of “median” - a straight line passing through the vertex of a triangle and dividing its opposite side in half. In mathematical statistics, the median divides in half not the side of the triangle, but the distribution of a random variable: equality F(x 0.5)= 0.5 means that the probability of getting to the left x 0.5 and the probability of getting to the right x 0.5(or directly to x 0.5) are equal to each other and equal to S, i.e.

P(X < x 0,5) = P(X > x 0.5) = S.

The median indicates the "center" of the distribution. From the point of view of one of the modern concepts - the theory of stable statistical procedures - the median is a better characteristic of a random variable than the mathematical expectation. When processing measurement results on an ordinal scale (see the chapter on measurement theory), the median can be used, but the mathematical expectation cannot.

A characteristic of a random variable such as mode has a clear meaning - the value (or values) of a random variable corresponding to the local maximum of the probability density for a continuous random variable or the local maximum of the probability for a discrete random variable.

If x 0– mode of a random variable with density f(x), then, as is known from differential calculus, .

A random variable can have many modes. So, for uniform distribution (1) each point X such that a< x < b , is fashion. However, this is an exception. Most random variables used in probabilistic statistical methods of decision making and other applied research have one mode. Random variables, densities, distributions that have one mode are called unimodal.

The mathematical expectation for discrete random variables with a finite number of values ​​is discussed in the chapter “Events and Probabilities”. For a continuous random variable X expected value M(X) satisfies the equality

which is an analogue of formula (5) from statement 2 of chapter “Events and Probabilities”.

Example 5. Expectation for a uniformly distributed random variable X equals

For the random variables considered in this chapter, all those properties of mathematical expectations and variances that were considered earlier for discrete random variables with a finite number of values ​​are true. However, we do not provide proof of these properties, since they require deepening into mathematical subtleties, which is not necessary for understanding and qualified application of probabilistic-statistical methods of decision-making.

Comment. This textbook consciously avoids mathematical subtleties associated, in particular, with the concepts of measurable sets and measurable functions, algebra of events, etc. Those wishing to master these concepts should turn to specialized literature, in particular, the encyclopedia.

Each of the three characteristics – mathematical expectation, median, mode – describes the “center” of the probability distribution. The concept of "center" can be defined in different ways - hence three different characteristics. However, for an important class of distributions—symmetric unimodal—all three characteristics coincide.

Distribution density f(x)– density of symmetric distribution, if there is a number x 0 such that

. (3)

Equality (3) means that the graph of the function y = f(x) symmetrical about a vertical line passing through the center of symmetry X = X 0 . From (3) it follows that the symmetric distribution function satisfies the relation

(4)

For a symmetric distribution with one mode, the mathematical expectation, median and mode coincide and are equal x 0.

The most important case is symmetry about 0, i.e. x 0= 0. Then (3) and (4) become equalities

(6)

respectively. The above relations show that there is no need to tabulate symmetric distributions for all X, it is enough to have tables at x > x 0.

Let us note one more property of symmetric distributions, which is constantly used in probabilistic-statistical methods of decision-making and other applied research. For a continuous distribution function

P(|X| < a) = P(-a < X < a) = F(a) – F(-a),

Where F– distribution function of a random variable X. If the distribution function F is symmetrical about 0, i.e. formula (6) is valid for it, then

P(|X| < a) = 2F(a) – 1.

Another formulation of the statement in question is often used: if

.

If and are quantiles of order and, respectively (see (2)) of a distribution function symmetric about 0, then from (6) it follows that

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

The standard deviation is the non-negative value of the square root of the variance:

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

The coefficient of variation is applied when M(X)> 0. It measures the spread in relative units, while the standard deviation is in absolute units.

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

Changing the variable makes it possible to write:

Where c = (b – a)/ 2. Therefore, the standard deviation is equal to and the coefficient of variation is:

For each random variable X determine three more quantities - centered Y, normalized V and given U. Centered random variable Y is the difference between a given random variable X and its mathematical expectation M(X), those. Y = X – M(X). Expectation of a centered random variable Y equals 0, and the variance is the variance of a 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) original random variable X ratio:

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

The densities of these random variables satisfy the equality

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

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

,

Where v– 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) – distribution function of the original random variable X, A f(x) – its probability density.

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

.

For the given random variable

Normalized, centered and reduced random variables are constantly used both in theoretical studies and in algorithms, software products, regulatory, technical and instructional documentation. In particular, because the equalities make it possible to simplify the justification of methods, the formulation of theorems and calculation formulas.

Transformations of random variables and more general ones are used. So, if Y = aX + b, Where a And b– some numbers, then

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

With each random variable X you can associate many random variables Y, given by the formula Y = aX + b at different a> 0 and b. This set is called scale-shift family, generated by the 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 often use recording

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 U– the result of measuring the same quantity if the beginning of the measurement is moved to the point With, and then use the new unit of measurement, in d times larger than the old one.

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

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

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

connects distribution functions X And Y.

When processing data, the following characteristics of a random variable are used X as moments of order q, i.e. mathematical expectations of a random variable Xq, q= 1, 2, ... Thus, the mathematical expectation itself is a moment of order 1. For a discrete random variable, the moment of order q can be calculated as

For a continuous random variable

Moments of order q also called initial moments of order q, in contrast to related characteristics - central moments of order q, given by the formula

So, dispersion is a central moment of order 2.

Normal distribution and the central limit theorem. In probabilistic-statistical methods of decision-making we often talk about normal distribution. Sometimes they try to use it to model the distribution of initial data (these attempts are not always justified - see below). More importantly, many data processing methods are based on the fact that the calculated values ​​have distributions close to normal.

Let X 1 , X 2 ,…, Xn M(X i) = m and variances D(X i) = , i= 1, 2,…, n,... As follows from the results of the previous chapter,

Consider the reduced random variable U n for the amount , namely,

As follows from formulas (7), M(U n) = 0, D(U n) = 1.

(for identically distributed terms). Let X 1 , X 2 ,…, Xn, … – independent identically distributed random variables with mathematical expectations M(X i) = m and variances D(X i) = , i= 1, 2,…, n,... Then for any x there is a limit

Where F(x)– function of standard normal distribution.

More about the feature F(x) – below (read “fi from x”, because F- Greek capital letter "phi").

The central limit theorem (CLT) gets its name because it is the central, most commonly used mathematical result of probability theory and mathematical statistics. The history of the CLT takes about 200 years - from 1730, when the English mathematician A. Moivre (1667-1754) published the first result related to the CLT (see below about the Moivre-Laplace theorem), until the twenties and thirties of the twentieth century, when Finn J.W. Lindeberg, Frenchman Paul Levy (1886-1971), Yugoslav V. Feller (1906-1970), Russian A.Ya. Khinchin (1894-1959) and other scientists obtained necessary and sufficient conditions for the validity of the classical central limit theorem.

The development of the topic under consideration did not stop there - they studied random variables that do not have dispersion, i.e. those for whom

(academician B.V. Gnedenko and others), a situation when random variables (more precisely, random elements) of a more complex nature than numbers are summed up (academicians Yu.V. Prokhorov, A.A. Borovkov and their associates), etc. .d.

Distribution function F(x) is given by the equality

,

where is the density of the standard normal distribution, which has a rather complex expression:

.

Here =3.1415925… is a number known in geometry, equal to the ratio of the circumference to the diameter, e= 2.718281828... - the base of natural logarithms (to remember this number, please note that 1828 is the year of birth of the writer L.N. Tolstoy). As is known from mathematical analysis,

When processing observation results, the normal distribution function is not calculated using the given formulas, but is found using special tables or computer programs. The best “Tables of mathematical statistics” in Russian were compiled by corresponding members of the USSR Academy of Sciences L.N. Bolshev and N.V. Smirnov.

The form of the density of the standard normal distribution follows from mathematical theory, which we cannot consider here, as well as the proof of the CLT.

For illustration, we provide small tables of the distribution function F(x)(Table 2) and its quantiles (Table 3). Function F(x) symmetrical about 0, which is reflected in Table 2-3.

Table 2.

Standard normal distribution function.

If the random variable X has a distribution function F(x), That M(X) = 0, D(X) = 1. This statement is proven in probability theory based on the type of probability density. It is consistent with a similar statement for the characteristics of the reduced random variable U n, which is quite natural, since the CLT states that with an unlimited increase in the number of terms, the distribution function U n tends to the standard normal distribution function F(x), and for any X.

Table 3.

Quantiles of the standard normal distribution.

Let us introduce the concept of a family of normal distributions. By definition, a normal distribution is the distribution of a random variable X, for which the distribution of the reduced random variable is F(x). As follows from the general properties of scale-shift families of distributions (see above), a normal distribution is a distribution of a random variable

Where X– random variable with distribution F(X), and m = M(Y), = D(Y). Normal distribution with shift parameters m and scale is usually indicated N(m, ) (sometimes the notation is used N(m, ) ).

As follows from (8), the probability density of the normal distribution N(m, ) There is

Normal distributions form a scale-shift family. In this case, the scale parameter is d= 1/ , and the shift parameter c = - m/ .

For the central moments of the third and fourth order of the normal distribution, the following equalities are valid:

These equalities form the basis of classical methods for verifying that observations follow a normal distribution. Nowadays it is usually recommended to test normality using the criterion W Shapiro - Wilka. The problem of normality testing is discussed below.

If random variables X 1 And X 2 have distribution functions N(m 1 , 1 ) And N(m 2 , 2 ) accordingly, then X 1+ X 2 has a distribution Therefore, if random variables X 1 , X 2 ,…, Xn N(m, ) , then their arithmetic mean

has a distribution N(m, ) . These properties of the normal distribution are constantly used in various probabilistic and statistical methods of decision-making, in particular, in the statistical regulation of technological processes and in statistical acceptance control based on quantitative criteria.

Using the normal distribution, three distributions are defined that are now often used in statistical data processing.

Distribution (chi - square) – distribution of a random variable

where are the random variables X 1 , X 2 ,…, Xn independent and have the same distribution N(0,1). In this case, the number of terms, i.e. n, is called the “number of degrees of freedom” of the chi-square distribution.

Distribution t Student's t is the distribution of a random variable

where are the random variables U And X independent, U has a standard normal distribution N(0.1), and X– chi distribution – square c n degrees of freedom. Wherein n is called the “number of degrees of freedom” of the Student distribution. This distribution was introduced in 1908 by the English statistician W. Gosset, who worked at a beer factory. Probabilistic and statistical methods were used to make economic and technical decisions at this factory, so its management forbade V. Gosset to publish scientific articles under his own name. In this way, trade secrets and “know-how” in the form of probabilistic and statistical methods developed by V. Gosset were protected. However, he had the opportunity to publish under the pseudonym "Student". The history of Gosset-Student shows that for another hundred years, managers in Great Britain were aware of the greater economic efficiency of probabilistic-statistical methods of decision-making.

The Fisher distribution is the distribution of a random variable

where are the random variables X 1 And X 2 are independent and have chi-square distributions with the number of degrees of freedom k 1 And k 2 respectively. At the same time, the couple (k 1 , k 2 ) – a pair of “degrees of freedom” of the Fisher distribution, namely, k 1 is the number of degrees of freedom of the numerator, and k 2 – number of degrees of freedom of the denominator. The distribution of the random variable F is named after the great English statistician R. Fisher (1890-1962), who actively used it in his works.

Expressions for the chi-square, Student and Fisher distribution functions, their densities and characteristics, as well as tables can be found in the specialized literature (see, for example,).

As already noted, normal distributions are now often used in probabilistic models in various applied areas. What is the reason for this two-parameter family of distributions being so widespread? It is clarified by the following theorem.

Central limit theorem(for differently distributed terms). Let X 1 , X 2 ,…, Xn,… - independent random variables with mathematical expectations M(X 1 ), M(X 2 ),…, M(X n), ... and variances D(X 1 ), D(X 2 ),…, D(X n), ... respectively. Let

Then, if certain conditions are true that ensure the small contribution of any of the terms in U n,

for anyone X.

We will not formulate the conditions in question here. They can be found in specialized literature (see, for example,). “The clarification of the conditions under which the CPT operates is the merit of the outstanding Russian scientists A.A. Markov (1857-1922) and, in particular, A.M. Lyapunov (1857-1918).”

The central limit theorem shows that in the case when the result of a measurement (observation) is formed under the influence of many causes, each of them making only a small contribution, and the total result is determined additively, i.e. by addition, then the distribution of the measurement (observation) result is close to normal.

It is sometimes believed that for the distribution to be normal, it is sufficient that the result of the measurement (observation) X is formed under the influence of many reasons, each of which has a small impact. This is wrong. What matters is how these causes operate. If additive, then X has an approximately normal distribution. If multiplicatively(i.e. the actions of individual causes are multiplied and not added), then the distribution X close not to normal, but to the so-called. logarithmically normal, i.e. Not X, and log X has an approximately normal distribution. If there is no reason to believe that one of these two mechanisms for the formation of the final result is at work (or some other well-defined mechanism), then about distribution X nothing definite can be said.

It follows from the above that in a specific applied problem, the normality of measurement results (observations), as a rule, cannot be established from general considerations; it should be checked using statistical criteria. Or use nonparametric statistical methods that are not based on assumptions about the membership of the distribution functions of measurement results (observations) to one or another parametric family.

Continuous distributions used in probabilistic and statistical methods of decision making. In addition to the scale-shift family of normal distributions, a number of other families of distributions are widely used - lognormal, exponential, Weibull-Gnedenko, gamma distributions. Let's look at these families.

Random value X has a lognormal distribution if the random variable Y= log X has a normal distribution. Then Z= log X = 2,3026…Y also has a normal distribution N(a 1 ,σ 1), where ln X- natural logarithm X. The density of the lognormal distribution is:

From the central limit theorem it follows that the product X = X 1 X 2 … Xn independent positive random variables X i, i = 1, 2,…, n, at large n can be approximated by a lognormal distribution. In particular, the multiplicative model of the formation of wages or income leads to the recommendation to approximate the distributions of wages and income by logarithmically normal laws. For Russia, this recommendation turned out to be justified - statistical data confirms it.

There are other probabilistic models that lead to the lognormal law. A classic example of such a model was given by A.N. Kolmogorov, who, from a physically based system of postulates, came to the conclusion that the particle sizes when crushing pieces of ore, coal, etc. in ball mills have a lognormal distribution.

Let's move on to another family of distributions, widely used in various probabilistic-statistical methods of decision-making and other applied research - the family of exponential distributions. Let's start with a probabilistic model that leads to such distributions. To do this, consider the “flow of events”, i.e. a sequence of events occurring one after another at certain points in time. Examples include: call flow at a telephone exchange; flow of equipment failures in the technological chain; flow of product failures during product testing; flow of customer requests to the bank branch; flow of buyers applying for goods and services, etc. In the theory of event flows, a theorem similar to the central limit theorem is valid, but it is not about the summation of random variables, but about the summation of event flows. We consider a total flow composed of a large number of independent flows, none of which has a predominant influence on the total flow. For example, a call flow entering a telephone exchange is composed of a large number of independent call flows originating from individual subscribers. It has been proven that in the case when the characteristics of flows do not depend on time, the total flow is completely described by one number - the intensity of the flow. For the total flow, consider the random variable X- the length of the time interval between successive events. Its distribution function has the form

(10)

This distribution is called exponential distribution because formula (10) involves the exponential function e -λ x. The value 1/λ is a scale parameter. Sometimes a shift parameter is also introduced With, the distribution of a random variable is called exponential X + s, where the distribution X is given by formula (10).

Exponential distributions are a special case of the so-called. Weibull - Gnedenko distributions. They are named after the names of the engineer V. Weibull, who introduced these distributions into the practice of analyzing the results of fatigue tests, and the mathematician B.V. Gnedenko (1912-1995), who received such distributions as limits when studying the maximum of the test results. Let X- a random variable characterizing the duration of operation of a product, complex system, element (i.e. resource, operating time to a limiting state, etc.), duration of operation of an enterprise or the life of a living being, etc. Failure intensity plays an important role

(11)

Where F(x) And f(x) - distribution function and density of a random variable X.

Let us describe the typical behavior of the failure rate. The entire time interval can be divided into three periods. On the first of them the function λ(x) has high values ​​and a clear tendency to decrease (most often it decreases monotonically). This can be explained by the presence in the batch of product units in question with obvious and hidden defects, which lead to a relatively rapid failure of these product units. The first period is called the “break-in period” (or “break-in”). This is what the warranty period usually covers.

Then comes a period of normal operation, characterized by an approximately constant and relatively low failure rate. The nature of failures during this period is sudden (accidents, errors of operating personnel, etc.) and does not depend on the duration of operation of the product unit.

Finally, the last period of operation is the period of aging and wear. The nature of failures during this period is in irreversible physical, mechanical and chemical changes in materials, leading to a progressive deterioration in the quality of a product unit and its final failure.

Each period has its own type of function λ(x). Let us consider the class of power dependences

λ(x) = λ 0bx b -1 , (12)

Where λ 0 > 0 and b> 0 - some numeric parameters. Values b < 1, b= 0 and b> 1 correspond to the type of failure rate during the periods of running-in, normal operation and aging, respectively.

Relationship (11) at a given failure rate λ(x)- differential equation for a function F(x). From the theory of differential equations it follows that

(13)

Substituting (12) into (13), we obtain that

(14)

The distribution given by formula (14) is called the Weibull - Gnedenko distribution. Because the

then from formula (14) it follows that the quantity A, given by formula (15), is a scale parameter. Sometimes a shift parameter is also introduced, i.e. Weibull-Gnedenko distribution functions are called F(x - c), Where F(x) is given by formula (14) for some λ 0 and b.

The Weibull-Gnedenko distribution density has the form

(16)

Where a> 0 - scale parameter, b> 0 - form parameter, With- shift parameter. In this case, the parameter A from formula (16) is associated with the parameter λ 0 from formula (14) by the relationship specified in formula (15).

The exponential distribution is a very special case of the Weibull-Gnedenko distribution, corresponding to the value of the shape parameter b = 1.

The Weibull-Gnedenko distribution is also used in constructing probabilistic models of situations in which the behavior of an object is determined by the “weakest link”. There is an analogy with a chain, the safety of which is determined by the link that has the least strength. In other words, let X 1 , X 2 ,…, Xn- independent identically distributed random variables,

X(1)=min( X 1, X 2,…, X n), X(n)=max( X 1, X 2,…, X n).

In a number of applied problems, they play an important role X(1) And X(n) , in particular, when studying the maximum possible values ​​("records") of certain values, for example, insurance payments or losses due to commercial risks, when studying the elasticity and endurance limits of steel, a number of reliability characteristics, etc. It is shown that for large n the distributions X(1) And X(n) , as a rule, are well described by Weibull-Gnedenko distributions. Fundamental contribution to the study of distributions X(1) And X(n) contributed by the Soviet mathematician B.V. Gnedenko. The works of V. Weibull, E. Gumbel, V.B. are devoted to the use of the results obtained in economics, management, technology and other fields. Nevzorova, E.M. Kudlaev and many other specialists.

Let's move on to the family of gamma distributions. They are widely used in economics and management, theory and practice of reliability and testing, in various fields of technology, meteorology, etc. In particular, in many situations, the gamma distribution is subject to such quantities as the total service life of the product, the length of the chain of conductive dust particles, the time the product reaches the limiting state during corrosion, the operating time to k-th refusal, k= 1, 2, …, etc. The life expectancy of patients with chronic diseases and the time to achieve a certain effect during treatment in some cases have a gamma distribution. This distribution is most adequate for describing demand in economic and mathematical models of inventory management (logistics).

The gamma distribution density has the form

(17)

The probability density in formula (17) is determined by three parameters a, b, c, Where a>0, b>0. Wherein a is a form parameter, b- scale parameter and With- shift parameter. Factor 1/Γ(а) is normalizing, it was introduced to

Here Γ(a)- one of the special functions used in mathematics, the so-called “gamma function”, after which the distribution given by formula (17) is named,

At fixed A formula (17) specifies a scale-shift family of distributions generated by a distribution with density

(18)

A distribution of the form (18) is called the standard gamma distribution. It is obtained from formula (17) at b= 1 and With= 0.

A special case of gamma distributions for A= 1 are exponential distributions (with λ = 1/b). With natural A And With=0 gamma distributions are called Erlang distributions. From the works of the Danish scientist K.A. Erlang (1878-1929), an employee of the Copenhagen Telephone Company, who studied in 1908-1922. the functioning of telephone networks, the development of queuing theory began. This theory deals with probabilistic and statistical modeling of systems in which a flow of requests is serviced in order to make optimal decisions. Erlang distributions are used in the same application areas in which exponential distributions are used. This is based on the following mathematical fact: the sum of k independent random variables exponentially distributed with the same parameters λ and With, has a gamma distribution with a shape parameter a =k, scale parameter b= 1/λ and shift parameter kc. At With= 0 we obtain the Erlang distribution.

If the random variable X has a gamma distribution with a shape parameter A such that d = 2 a- integer, b= 1 and With= 0, then 2 X has a chi-square distribution with d degrees of freedom.

A random variable X with a gvmma distribution has the following characteristics:

Expected value M(X) =ab + c,

Variance D(X) = σ 2 = ab 2 ,

The coefficient of variation

Asymmetry

Excess

The normal distribution is an extreme case of the gamma distribution. More precisely, let Z be a random variable having a standard gamma distribution given by formula (18). Then

for any real number X, Where F(x)- standard normal distribution function N(0,1).

In applied research, other parametric families of distributions are also used, of which the most famous are the system of Pearson curves, the Edgeworth and Charlier series. They are not considered here.

Discrete distributions used in probabilistic and statistical methods of decision making. The most commonly used are three families of discrete distributions - binomial, hypergeometric and Poisson, as well as some other families - geometric, negative binomial, multinomial, negative hypergeometric, etc.

As already mentioned, the binomial distribution occurs in independent trials, in each of which with probability R event appears A. If the total number of trials n given, then the number of tests Y, in which the event appeared A, has a binomial distribution. For a binomial distribution, the probability of being accepted as a random variable is Y values y is determined by the formula

Number of combinations of n elements by y, known from combinatorics. For all y, except 0, 1, 2, …, n, we have P(Y= y)= 0. Binomial distribution with fixed sample size n is specified by the parameter p, i.e. binomial distributions form a one-parameter family. They are used in the analysis of data from sample studies, in particular, in the study of consumer preferences, selective control of product quality according to single-stage control plans, when testing populations of individuals in demography, sociology, medicine, biology, etc.

If Y 1 And Y 2 - independent binomial random variables with the same parameter p 0 , determined from samples with volumes n 1 And n 2 accordingly, then Y 1 + Y 2 - binomial random variable having distribution (19) with R = p 0 And n= n 1 + n 2 . This remark extends the applicability of the binomial distribution by allowing the results of several groups of tests to be combined when there is reason to believe that the same parameter corresponds to all these groups.

The characteristics of the binomial distribution were calculated earlier:

M(Y) = n.p., D(Y) = n.p.( 1- p).

In the section "Events and Probabilities" the law of large numbers is proven for a binomial random variable:

for anyone . Using the central limit theorem, the law of large numbers can be refined by indicating how much Y/ n differs from R.

De Moivre-Laplace theorem. For any numbers a and b, a< b, we have

Where F(X) is a function of standard normal distribution with mathematical expectation 0 and variance 1.

To prove it, it suffices to use the representation Y in the form of a sum of independent random variables corresponding to the outcomes of individual tests, formulas for M(Y) And D(Y) and the central limit theorem.

This theorem is for the case R= ½ was proven by the English mathematician A. Moivre (1667-1754) in 1730. In the above formulation, it was proven in 1810 by the French mathematician Pierre Simon Laplace (1749 - 1827).

Hypergeometric distribution occurs during selective control of a finite set of objects of volume N according to an alternative criterion. Each controlled object is classified either as having the attribute A, or as not having this characteristic. The hypergeometric distribution has a random variable Y, equal to the number of objects that have the attribute A in a random sample of volume n, Where n< N. For example, number Y defective units of product in a random sample of volume n from batch volume N has a hypergeometric distribution if n< N. Another example is the lottery. Let the sign A ticket is a sign of “being a winner”. Let the total number of tickets N, and some person acquired n of them. Then the number of winning tickets for this person has a hypergeometric distribution.

For a hypergeometric distribution, the probability of a random variable Y accepting the value y has the form

(20)

Where D– the number of objects that have the attribute A, in the considered set of volume N. Wherein y takes values ​​from max(0, n - (N - D)) to min( n, D), other things y the probability in formula (20) is equal to 0. Thus, the hypergeometric distribution is determined by three parameters - the volume of the population N, number of objects D in it, possessing the characteristic in question A, and sample size n.

Simple random volume sampling n from the total volume N is a sample obtained as a result of random selection in which any of the sets of n objects have the same probability of being selected. Methods for randomly selecting samples of respondents (interviewees) or units of piece goods are discussed in the instructional, methodological, and regulatory documents. One of the selection methods is this: objects are selected one from another, and at each step, each of the remaining objects in the set has the same chance of being selected. In the literature, the terms “random sample” and “random sample without return” are also used for the type of samples under consideration.

Since the volumes of the population (batch) N and samples n are usually known, then the parameter of the hypergeometric distribution to be estimated is D. In statistical methods of product quality management D– usually the number of defective units in a batch. The distribution characteristic is also of interest D/ N– level of defects.

For hypergeometric distribution

The last factor in the expression for variance is close to 1 if N>10 n. If you make a replacement p = D/ N, then the expressions for the mathematical expectation and variance of the hypergeometric distribution will turn into expressions for the mathematical expectation and variance of the binomial distribution. This is no coincidence. It can be shown that

at N>10 n, Where p = D/ N. The limiting ratio is valid

and this limiting relation can be used when N>10 n.

The third widely used discrete distribution is the Poisson distribution. The random variable Y has a Poisson distribution if

,

where λ is the Poisson distribution parameter, and P(Y= y)= 0 for all others y(for y=0 it is designated 0! =1). For Poisson distribution

M(Y) = λ, D(Y) = λ.

This distribution is named after the French mathematician S. D. Poisson (1781-1840), who first obtained it in 1837. The Poisson distribution is the limiting case of the binomial distribution, when the probability R the implementation of the event is small, but the number of tests n great, and n.p.= λ. More precisely, the limit relation is valid

Therefore, the Poisson distribution (in the old terminology “distribution law”) is often also called the “law of rare events.”

The Poisson distribution originates in event flow theory (see above). It has been proven that for the simplest flow with constant intensity Λ, the number of events (calls) that occurred during the time t, has a Poisson distribution with parameter λ = Λ t. Therefore, the probability that during the time t no event will occur, equal to e - Λ t, i.e. the distribution function of the length of the interval between events is exponential.

The Poisson distribution is used in analyzing the results of sample marketing surveys of consumers, calculating the operational characteristics of statistical acceptance control plans in the case of small values ​​of the acceptance level of defects, to describe the number of breakdowns of a statistically controlled technological process per unit of time, the number of “service requirements” received per unit of time in queuing system, statistical patterns of accidents and rare diseases, etc.

Descriptions of other parametric families of discrete distributions and the possibilities of their practical use are considered in the literature.

In some cases, for example, when studying prices, output volumes or total time between failures in reliability problems, the distribution functions are constant over certain intervals in which the values ​​of the studied random variables cannot fall.

In probability theory, one has to deal with random variables, all of whose values ​​cannot be enumerated. For example, it is impossible to take and “iterate” all the values ​​of the random variable $X$ - the service time of the clock, since time can be measured in hours, minutes, seconds, milliseconds, etc. You can only specify a certain interval within which the values ​​of the random variable lie.

Continuous random variable is a random variable whose values ​​completely fill a certain interval.

Distribution function of a continuous random variable

Since it is not possible to enumerate all the values ​​of a continuous random variable, it can be specified using the distribution function.

Distribution function random variable $X$ is called a function $F\left(x\right)$, which determines the probability that the random variable $X$ will take a value less than some fixed value $x$, that is, $F\left(x\right )=P\left(X< x\right)$.

Properties of the distribution function:

1 . $0\le F\left(x\right)\le 1$.

2 . The probability that the random variable $X$ will take values ​​from the interval $\left(\alpha ;\ \beta \right)$ is equal to the difference between the values ​​of the distribution function at the ends of this interval: $P\left(\alpha< X < \beta \right)=F\left(\beta \right)-F\left(\alpha \right)$.

3 . $F\left(x\right)$ - non-decreasing.

4 . $(\mathop(lim)_(x\to -\infty ) F\left(x\right)=0\ ),\ (\mathop(lim)_(x\to +\infty ) F\left(x \right)=1\ )$.

Example 1
0,\x\le 0\\
x,\ 0< x\le 1\\
1,\ x>1
\end(matrix)\right.$. The probability of a random variable $X$ falling into the interval $\left(0.3;0.7\right)$ can be found as the difference between the values ​​of the distribution function $F\left(x\right)$ at the ends of this interval, that is:

$$P\left(0.3< X < 0,7\right)=F\left(0,7\right)-F\left(0,3\right)=0,7-0,3=0,4.$$

Probability distribution density

The function $f\left(x\right)=(F)"(x)$ is called the probability distribution density, that is, it is the first-order derivative taken from the distribution function $F\left(x\right)$ itself.

Properties of the function $f\left(x\right)$.

1 . $f\left(x\right)\ge 0$.

2 . $\int^x_(-\infty )(f\left(t\right)dt)=F\left(x\right)$.

3 . The probability that the random variable $X$ will take values ​​from the interval $\left(\alpha ;\ \beta \right)$ is $P\left(\alpha< X < \beta \right)=\int^{\beta }_{\alpha }{f\left(x\right)dx}$. Геометрически это означает, что вероятность попадания случайной величины $X$ в интервал $\left(\alpha ;\ \beta \right)$ равна площади криволинейной трапеции, которая будет ограничена графиком функции $f\left(x\right)$, прямыми $x=\alpha ,\ x=\beta $ и осью $Ox$.

4 . $\int^(+\infty )_(-\infty )(f\left(x\right))=1$.

Example 2 . A continuous random variable $X$ is defined by the following distribution function $F(x)=\left\(\begin(matrix)
0,\x\le 0\\
x,\ 0< x\le 1\\
1,\ x>1
\end(matrix)\right.$. Then the density function $f\left(x\right)=(F)"(x)=\left\(\begin(matrix)
0,\x\le 0\\
1,\ 0 < x\le 1\\
0.\x>1
\end(matrix)\right.$

Expectation of a continuous random variable

The mathematical expectation of a continuous random variable $X$ is calculated using the formula

$$M\left(X\right)=\int^(+\infty )_(-\infty )(xf\left(x\right)dx).$$

Example 3 . Let's find $M\left(X\right)$ for the random variable $X$ from example $2$.

$$M\left(X\right)=\int^(+\infty )_(-\infty )(xf\left(x\right)\ dx)=\int^1_0(x\ dx)=(( x^2)\over (2))\bigg|_0^1=((1)\over (2)).$$

Variance of a continuous random variable

The variance of a continuous random variable $X$ is calculated by the formula

$$D\left(X\right)=\int^(+\infty )_(-\infty )(x^2f\left(x\right)\ dx)-(\left)^2.$$

Example 4 . Let us find $D\left(X\right)$ for the random variable $X$ from example $2$.

$$D\left(X\right)=\int^(+\infty )_(-\infty )(x^2f\left(x\right)\ dx)-(\left)^2=\int^ 1_0(x^2\ dx)-(\left(((1)\over (2))\right))^2=((x^3)\over (3))\bigg|_0^1-( (1)\over (4))=((1)\over (3))-((1)\over (4))=((1)\over(12)).$$

Dispersion continuous random variable X, the possible values ​​of which belong to the entire Ox axis, is determined by the equality:

Purpose of the service. The online calculator is designed to solve problems in which either distribution density f(x) or distribution function F(x) (see example). Usually in such tasks you need to find mathematical expectation, standard deviation, plot functions f(x) and F(x).

Instructions. Select the type of source data: distribution density f(x) or distribution function F(x).

The distribution density f(x) is given:

The distribution function F(x) is given:

A continuous random variable is specified by a probability density
(Rayleigh distribution law - used in radio engineering). Find M(x) , D(x) .

The random variable X is called continuous , if its distribution function F(X)=P(X< x) непрерывна и имеет производную.
The distribution function of a continuous random variable is used to calculate the probability of a random variable falling into a given interval:
P(α< X < β)=F(β) - F(α)
Moreover, for a continuous random variable, it does not matter whether its boundaries are included in this interval or not:
P(α< X < β) = P(α ≤ X < β) = P(α ≤ X ≤ β)
Distribution density a continuous random variable is called a function
f(x)=F’(x) , derivative of the distribution function.

Properties of distribution density

d) the sum of the probabilities of opposite events is equal to one, therefore P(X≥5)+P(X<5)=1. Отсюда, используя условие, в силу которого при х>4 function F(x)=1, we get P(X≥5) = 1-P(X<5) = 1-F(5) = 1-1 = 0.

The random variable X is specified by the distribution function

Find the probability that, as a result of four independent trials, the value of X will take a value belonging to the interval (0.25, 0.75) exactly three times.

The probability that X will take a value contained in the interval (a, b) is equal to the increment of the distribution function on this interval: P(a

P(0.25< X <0.75) = F(0.75)-F(0.25) = 0.5

Therefore, , or From here, or.

The random variable X is specified on the entire Ox axis by the distribution function. Find a possible value that satisfies the condition: with probability, random X as a result of the test will take a value greater

Solution. Events and are opposite, therefore . Hence, . Since, then.

By definition of the distribution function, .

Therefore, , or . From here, or.

Discrete random variable X is given by the distribution law

So, the required distribution function has the form

Discrete random variable X is given by the distribution law

Find the distribution function and draw its graph.

Given the distribution function of a continuous random variable X

Find the distribution density f(x).

The distribution density is equal to the first derivative of the distribution function:

At x=0 the derivative does not exist.

A continuous random variable X is specified by the distribution density in the interval ; outside this interval. Find the probability that X will take a value belonging to the interval .

Let's use the formula. By condition, and. Therefore, the required probability

A continuous random variable X is given by the distribution density in the interval ; outside this interval. Find the probability that X will take a value belonging to the interval .

Let's use the formula. By condition, and . Therefore, the required probability

The distribution density of a continuous random variable X in the interval (-π/2, π/2) is equal to f(x)=(2/π)*cos2x ; outside this interval f(x)=0. Find the probability that in three independent trials X will take exactly twice the value contained in the interval (0, π/4).

Let's use the formula P(a

P(0

Answer: π+24π.

fx=0, at x≤0cosx, at 0

We use the formula

If x ≤0, then f(x)=0, therefore,

F(x)=-∞00dx=0.

If 0

F(x)=-∞00dx+0xcosxdx=sinx.

If x≥ π2, then

F(x)=-∞00dx+0π2cosxdx+π2x0dx=sinx|0π2=1.

So, the required distribution function

Fx=0, at x≤0sinx, at 0 π2.

The distribution density of a continuous random variable X is given:

Fx=0, at x≤0sinx, at 0 π2.

Find the distribution function F(x).

We use the formula

The distribution density of a continuous random variable X is specified on the entire Ox axis by the equality . Find the constant parameter C.

.

. (*)

.

Thus,

The distribution density of a continuous random variable is specified on the entire axis by the equality Find the constant parameter C.

Solution. The distribution density must satisfy the condition. We require that this condition be satisfied for the given function:

.

. (*)

Let's first find the indefinite integral:

.

Then we calculate the improper integral:

Thus,

Substituting (**) into (*), we finally get .

The distribution density of a continuous random variable X in the interval is equal to ; outside this interval f(x) = 0. Find the constant parameter C.

.

. (*)

Let's first find the indefinite integral:

Then we calculate the improper integral:

(**)

Substituting (**) into (*), we finally get .

The distribution density of a continuous random variable X is specified in the interval by the equality ; outside this interval f(x) = 0. Find the constant parameter C.

Solution. The distribution density must satisfy the condition, but since f(x) outside the interval is equal to 0, it is enough for it to satisfy: We require that this condition be satisfied for the given function:

.

. (*)

Let's first find the indefinite integral:

Then we calculate the improper integral:

(**)

Substituting (**) into (*), we finally get .

The random variable X is specified by the distribution density ƒ(x) = 2x in the interval (0,1); outside this interval ƒ(x) = 0. Find the mathematical expectation of the value X.

R decision. We use the formula

Substituting a = 0, b = 1, ƒ(x) = 2x, we get

Answer: 2/3.

The random variable X is specified by the distribution density ƒ(x) = (1/2)x in the interval (0;2); outside this interval ƒ(x) = 0. Find the mathematical expectation of the value X.

R decision. We use the formula

Substituting a = 0, b = 2, ƒ(x) = (1/2)x, we get

M(X) = = 4/3

Answer: 4/3.

The random variable X in the interval (–s, s) is specified by the distribution density

ƒ (x) = ; outside this interval ƒ(x) = 0. Find the mathematical expectation of the value X.

R decision. We use the formula

Substituting a = –с, b = c, ƒ(x) = , we get

Considering that the integrand is odd and the limits of integration are symmetrical about the origin, we conclude that the integral is equal to zero. Therefore, M(X) = 0.

This result can be obtained immediately if we take into account that the distribution curve is symmetrical with respect to the straight line x = 0.

The random variable X in the interval (2, 4) is specified by the distribution density f(x)=

. From this it can be seen that at x = 3 the distribution density reaches a maximum; hence, . The distribution curve is symmetrical about the straight line x=3, therefore .

The random variable X in the interval (3, 5) is specified by the distribution density f(x)= ; outside this interval f(x)=0. Find the mode, mathematical expectation and median of X.

Solution. Let us represent the distribution density in the form . From this it can be seen that at x = 3 the distribution density reaches a maximum; hence, . The distribution curve is symmetrical about the straight line x=4, therefore .

The random variable X in the interval (-1, 1) is specified by the distribution density ; outside this interval f(x)=0. Find: a) fashion; b) median X.

Distribution density probabilities X call the function f(x)– the first derivative of the distribution function F(x):

The concept of probability distribution density of a random variable X not applicable for discrete quantities.

Probability distribution density f(x)– called the differential distribution function:

Property 1. Distribution density is a non-negative quantity:

Property 2. The improper integral of the distribution density in the range from to is equal to unity:

Example 1.25. Given the distribution function of a continuous random variable X:

f(x).

Solution: The distribution density is equal to the first derivative of the distribution function:

1. Given the distribution function of a continuous random variable X:

Find the distribution density.

2. The distribution function of a continuous random variable is given X:

Find the distribution density f(x).

1.3. Numerical characteristics of continuous random

quantities

Expected value continuous random variable X, the possible values ​​of which belong to the entire axis Oh, is determined by the equality:

It is assumed that the integral converges absolutely.

a,b), That:

f(x)– distribution density of a random variable.

Dispersion continuous random variable X, the possible values ​​of which belong to the entire axis, is determined by the equality:

A special case. If the values ​​of a random variable belong to the interval ( a,b), That:

The probability that X will take values ​​belonging to the interval ( a,b), is determined by the equality:

.

Example 1.26. Continuous random variable X

Find the mathematical expectation, variance and probability of hitting a random variable X in the interval (0;0.7).

Solution: The random variable is distributed over the interval (0,1). Let us determine the distribution density of a continuous random variable X:

a) Mathematical expectation :

b) Variance

V)

Tasks for independent work:

1. Random variable X given by the distribution function:

M(x);

b) variance D(x);

X into the interval (2,3).

2. Random variable X

Find: a) mathematical expectation M(x);

b) variance D(x);

c) determine the probability of a random variable hitting X into the interval (1;1.5).

3. Random variable X is given by the cumulative distribution function:

Find: a) mathematical expectation M(x);

b) variance D(x);

c) determine the probability of a random variable hitting X in the interval

1.4. Laws of distribution of a continuous random variable

1.4.1. Uniform distribution

Continuous random variable X has a uniform distribution on the segment [ a,b], if on this segment the probability distribution density of the random variable is constant, and outside it it is equal to zero, i.e.:

Rice. 4.

; ; .

Example 1.27. A bus on a certain route moves uniformly at intervals of 5 minutes. Find the probability that a uniformly distributed random variable X– the waiting time for the bus will be less than 3 minutes.

Solution: Random value X– uniformly distributed over the interval .

Probability density: .

In order for the waiting time not to exceed 3 minutes, the passenger must appear at the stop within 2 to 5 minutes after the previous bus leaves, i.e. random value X must fall into the interval (2;5). That. required probability:

Tasks for independent work:

1. a) find the mathematical expectation of a random variable X distributed uniformly in the interval (2;8);

b) find the variance and standard deviation of the random variable X, distributed uniformly in the interval (2;8).

2. The minute hand of an electric clock moves abruptly at the end of every minute. Find the probability that at a given moment the clock will show a time that differs from the true time by no more than 20 seconds.

1.4.2. Exponential distribution

Continuous random variable X is distributed according to the exponential law if its probability density has the form:

where is the parameter of the exponential distribution.

Thus

Rice. 5.

Numerical characteristics:

Example 1.28. Random value X– operating time of a light bulb - has an exponential distribution. Determine the probability that the operating time of the light bulb will be at least 600 hours if the average operating time is 400 hours.

Solution: According to the conditions of the problem, the mathematical expectation of a random variable X equals 400 hours, therefore:

;

The required probability, where

Finally:

Tasks for independent work:

1. Write the density and distribution function of the exponential law if the parameter .

2. Random variable X

Find the mathematical expectation and variance of a quantity X.

3. Random variable X is given by the probability distribution function:

Find the mathematical expectation and standard deviation of a random variable.

1.4.3. Normal distribution

Normal is called the probability distribution of a continuous random variable X, whose density has the form:

Where A– mathematical expectation, – standard deviation X.

The probability that X will take a value belonging to the interval:

, Where

– Laplace function.

A distribution for which ; , i.e. with probability density called standard.

Rice. 6.

Probability that the absolute value is rejected less than a positive number:

.

In particular, when a= 0 the equality is true:

Example 1.29. Random value X normally distributed. Standard deviation. Find the probability that the deviation of a random variable from its mathematical expectation in absolute value will be less than 0.3.

Solution: .

Tasks for independent work:

1. Write the probability density of the normal distribution of the random variable X, knowing that M(x)= 3, D(x)= 16.

2. Expectation and standard deviation of a normally distributed random variable X respectively equal to 20 and 5. Find the probability that as a result of the test X will take the value contained in the interval (15;20).

3. Random measurement errors are subject to the normal law with standard deviation mm and mathematical expectation a= 0. Find the probability that out of 3 independent measurements the error of at least one will not exceed 4 mm in absolute value.

4. A certain substance is weighed without systematic errors. Random weighing errors are subject to the normal law with a standard deviation r. Find the probability that weighing will be carried out with an error not exceeding 10 g in absolute value.