Individual distribution, its properties and methods of analysis. Pareto distribution law. Pareto distribution An excerpt characterizing the Pareto Distribution

Where x m , k > 0 (\displaystyle x_(m),k>0). Then they say that X (\displaystyle X) has a Pareto distribution with parameters x m (\displaystyle x_(m)) And k (\displaystyle k). , . His 20 to 80 rule (which states: 20% of the population owns 80% of the wealth) however depends on the specific value k, and it is argued that in fact there are significant quantitative deviations, for example, Pareto’s own data for Britain in Cours d'économie politique it is said that there approximately 30% of the population owns 70% of the total income.

The Pareto distribution is not only found in economics. The following examples can be given.

V. Pareto discovered the effect of concentrated tension. IN in this case meaning that focusing on vital activities has the greatest impact on achieving the desired results. This leads to the 20/80 rule: concentrating 20% ​​of the time on the most important problems can lead to 80% of the results. The remaining 80% of the time provides only the remaining 20% ​​of the results (Fig. 1).

In 1897, the Italian economist V. Pareto invented a formula showing that all benefits are distributed unevenly. In most cases, the largest share of income or wealth belongs to a small number of people. M.S. Lorenz (an American economist) illustrated this theory with a diagram. Dr. D.M. Juran used a diagram to classify quality problems into a few essential and many non-essential and called this method Pareto analysis. He pointed out that in most cases the vast majority of defects and associated losses arise from a relatively small number of causes.

Application of Pareto's law in marketing analysis

Consider the application of the Pareto principle, also called Pareto's law, in the marketing analysis of buyers or suppliers.

An Italian scientist involved in sociology and economics, Vilfredo Pareto (1848-1923), formulated a universal law in 1897, which was based on large quantities statistical measurements carried out by the author in many fields of activity, including finance, sales, economics, sociology, and manufacturing. The main postulate of this law says: 20% of the effort produced gives 80% of the result, and the remaining 80% of the effort adds only 20% to the result. Pareto applied the same ratio regarding the time spent on certain work.

The main provisions of the Pareto law can be summarized as follows:

• In any process, there are a very large number of factors or causes influencing it, of which only a few have a decisive impact. This rule “works” in many other areas. For example, it most often manifests itself in production processes when conducting quality analysis and identifying the causes of manufacturing defects. Is the Pareto rule applicable in geopolitics? Rather, one can give a positive answer, since out of almost 200 countries in the world, only 20-30 are the most influential. And in sports? Of the 400 rated professional tennis players in the world, only the first few dozen mainly share prize funds and compete in matches.
• Most of the efforts put into business are spent ineffectively, without the necessary return, and only a small part of them leads to a positive result.
• It is difficult to understand a large number of individual events and accept correct solution based on their analysis, since these events can be very different in nature and essence.

From a marketing perspective, this refers to the analysis of customers and suppliers, of which some businesses have hundreds or thousands.

IN practical application Pareto's rule of 20:80 ratio is not strictly followed. This is a conditionally generalized relationship. In reality, the ratio can range from 5:95 to 30:70 depending on many factors.

A typical form of distribution of the number of buyers depending on the share of income generated is shown in Fig. 3.9.

Rice. 3.9. The relationship between purchase volumes and customer profitability

In Fig. 3.10 specified areas A-D, which distinguish groups of clients with different levels of profitability. Area A denotes a group of unprofitable clients, area B - break-even clients bringing zero or almost zero profit, area C - a group of buyers bringing profit, but not high, and area D - a group of very profitable clients.

Rice. 3.10. Distribution of buyers by income

A significantly smaller portion of shoppers (about 20%) accounts for the majority of purchases (about 80%), while a limited portion of a retail store's assortment (about 20%) generates the bulk of revenue (about 80%). For example, CJSC Klinsky Beer Plant had three groups of wholesalers. In table Figure 3.18 shows sales volumes as a percentage of total revenues for these three groups.

Table 3.18. Distribution of income from buyers

Within the specified framework, the application of the Pareto rule can simplify marketing analysis, facilitate decision-making on organizing work with suppliers and customers, and help in developing principles for relationships with them.

Let's consider the procedure for constructing a Pareto diagram and the principle of identifying customer groups. Let us assume that, based on the results of the work of the enterprise, let’s call it “Finishing”, over a certain period of time, services were provided in the quantities and for consumers indicated in the table. 3.19. Table form 3.19 is quite common in practice, since it is a standard table for operational reporting.

Table 3.19. Summary reporting on performance results

In table 3.19 reflects the consumers of the enterprise’s services, the number of services purchased, the cost of a unit of the purchased service and the amount of payment received. In Fig. 3.11 shows the distribution of shares of buyers in total volume sales

Rice. 3.11. Distribution of consumption shares

However, based on the form of presentation of the calculated shares of consumption, it is impossible to identify trends and typical consumer groups that are necessary for the analysis. Therefore, the next stage of constructing the Pareto distribution is to rank consumers, for example, according to the criterion of gross income received. In practice, other criteria may include maximum profit, number of one-time purchases, multiplicity of purchases, etc. In Table. Figure 3.20 shows the distribution of consumers in descending order of purchase amount.

Table 3.20. Ranking of consumer hemlines in purchases

The corresponding distribution of ranked consumers is shown in Fig. 3.12. This form of presentation is very clear, since groups of consumers with relatively high and low shares are clearly visible.

Let us carry out an additional construction that will make it possible to obtain the cumulative (total) distribution of consumers’ shares in total income. To do this, starting from the second, we will add the sum of the shares of all previous consumers to the current one (see the “Total share” column of Table 3.21). The cumulative distribution of consumers is plotted in Fig. 3.13.

Rice. 3.12. Ranking of consumers by share of income generated

Rice. 3.13. Cumulative distribution of consumers in total sales

All consumers in total contribute 100% of the income. From the distribution, one can obtain formal signs of classifying a consumer into one or another group: primary or secondary importance, taking into account the ratio of 20: 80.

Let's analyze the distribution shown in Fig. 3.14. Let us approximate it to the corresponding curve shape. The standard Pareto distribution is described by the density function (for a positive parameter c) by the following formula:

f(x) = c / x c +1

c is the distribution (shape) parameter; c > 0, x≥ 1.

Depending on the number of consumers (from 1 to infinity) and the volumes of consumption of each of them (from equal shares to very different ones), the shape of the Pareto distribution can take the form shown in Fig. 3.14.

Rice. 3.14. Possible forms of Pareto distribution

The ratio of 80:20 is considered to be a normal and fairly stable situation in the business of a particular enterprise. If the analysis shows that the ratio is, for example, 90: 10, then this is a signal that the enterprise becomes financially dependent on a small part of its partners or customers, who at a certain point in time may stop working with this enterprise and disrupt its entire system. A situation where the ratio is, for example, 50:50, may mean that the company is not paying attention to defining its target segments and is scattering its efforts without good results. The above does not apply to cases where the enterprise is a subsidiary of a large company and provides certain parts of the operation or is unitary.

Pareto distribution

Let us move from the expression for the Pareto curve (1.2) to the Pareto distribution of the random variable X(in the above examples, this is the amount of income) in terms of probability theory and mathematical statistics.

First, let's move on to the probabilistic interpretation of the size of persons with income X not lower than this X, represented by (1.2), dividing this expression by the total number Y population with an income of at least X.

Considering that according to the Pareto law, as stated earlier, income (or other random variable) begins to be distributed starting from a certain value X 0 , it is necessary to introduce this variable into (1.7), despite the fact that we previously got rid of it for convenience. This can be done by normalizing X on X 0 :

Let's replace:

Then: . (1.9)

But in probability theory it is customary to consider not the probability expressed by (1.9), but the so-called distribution function of a random variable, which is the complement of (1.9) to unity. Distribution function F(x), determining the probability that a random variable X will take a value less than this X, for the Pareto distribution has the form:

Corresponding probability density p(x) is found as the derivative of the distribution function and determines the probability that the random variable will take a value equal to X. For the Pareto distribution, the probability density is given by:

Distributions similar to the Pareto distribution in that they are limited on one side by the values ​​that a random variable can take are called truncated distributions. They are usually used in studies when the dynamics of the behavior of not the entire population of objects under study is important, but only some part of it or even the tail of the distribution, or if part of the population is distributed according to one law, and part according to another.

Let's consider important characteristic Pareto distribution, which determines the areas of its application in research. To do this, we find the mathematical expectation of this distribution:

Thus, we can see that the mathematical expectation of the Pareto distribution can be finite or infinite depending on the parameter. As stated earlier, in economic studies of income distribution the condition is met, thus it is possible to find the mathematical expectation ( average level income distributed according to the Pareto law). The second case of the Pareto distribution at is a heavy-tailed distribution (the concept is discussed below) and has found application in catastrophe theory as a distribution from which the probability of the occurrence of rare, but significant in scale, events is determined.

Let's consider another interesting characteristic that determines the sum of accumulated values X random variable, let us denote it (in the previously discussed examples this is the total amount of income of all persons falling within a given income interval) between the values X 1 And X 2 . This value can be determined as follows:

Moreover, it will reflect reality more accurately, the greater the distance between X 1 And X 2 . It is clear that at the behavior of this function will depend on the parameter in the same way as the mathematical expectation found above.

When using this function to calculate, for example, the total income of persons who receive income from a certain value X 1 up to the maximum income received in the country by one person, X max, it is more expedient to take as X 2 this is the meaning X max which can be expressed like this:

where are the values ​​that the random variable takes, in the example under consideration - income, in each specific case.

Expression (1.14) can be used if the necessary information about the maximum value is available X max. In this case, the total effect (1.13) will be final for any value of the parameter and expression (1.13) can be used to predict the total effects of a random variable X, distributed according to the Pareto law, even if this distribution has a heavy tail. Let us describe how expression (1.13) can be made even more effective when analyzing these random variables. Let us assume (and we can say this with a high degree of confidence) that the quantity X max depends on the number of events that occurred or objects observed P. And this, in turn, of course, depends on time t, thus we get:

It would also be logical to assume that both the parameter and A(this is certainly true for economic and social phenomena, and perhaps also for natural ones):

Now we can rewrite (1.13) for X max And x 0 as:

Having sufficient quantity statistical data, you can calculate the type and parameters (1.15) and (1.16). Thus, we will obtain a dynamic model that describes the accumulated total effect of a random variable distributed according to the Pareto law

$kurtosis\; =$\frac(6(k^3+k^2-6k-2))(k(k-3)(k-4)) $k>4$}} Pareto distribution|

Definition

entropy = $\backslash ln\backslash left(\backslash frac(k)(x\_\backslash mathrm(m))\backslash right)\; -\; \backslash frac(1)(k)\; -\; 1$ is such that its distribution is given by the equality:

Where $mgf\; =undefined|$. Then they say that $\backslash ln\backslash left(\backslash frac(k)(x\_\backslash mathrm(m))\backslash right)\; -\; \backslash frac(1)(k)\; -\; 1$ has a Pareto distribution with parameters $char\; =$ And $k\backslash left(\backslash Gamma(-k)(x\_\backslash mathrm(m)^k(-it)^k-(-ix\_\backslash mathrm(m)t)^k)+\backslash right.$\left.+E_\mathrm(k+1)(-ix_\mathrm(m)t)\right)

$|notation\; =$

P(k, x_m)< x_m \end{matrix} \right..

in probability theory, a two-parameter family of absolutely continuous distributions that are power laws. Called by the name of Vilfredo Pareto. It is found in the study of various phenomena, in particular social, economic, physical and others. Outside the field of economics, sometimes also called the Bradford distribution.

Let the random variable

$X$,

F_X(x)=P(X

$x\_m,k>0$, $x\_m$.

k

Vilfredo Pareto originally used this distribution to describe the distribution of wealth as well as the distribution of income. His 20 to 80 rule (which states that 20% of the population owns 80% of the wealth) however depends on the specific value k, and it is argued that in fact there are significant quantitative deviations, for example, Pareto’s own data for Britain in Cours d'économie politique it is said that there approximately 30% of the population owns 70% of the total income.

The Pareto distribution is not only found in economics. The following examples can be given:

see also

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Notes

An excerpt characterizing the Pareto Distribution

In November 1805, Prince Vasily was supposed to go to an audit in four provinces. He arranged this appointment for himself in order to visit his ruined estates at the same time, and taking with him (at the location of his regiment) his son Anatoly, he and he would go to Prince Nikolai Andreevich Bolkonsky in order to marry his son to the daughter of this rich man old man. But before leaving and these new affairs, Prince Vasily needed to resolve matters with Pierre, who, however, Lately spent whole days at home, that is, with Prince Vasily, with whom he lived, he was funny, excited and stupid (as a lover should be) in the presence of Helen, but still did not propose.
“Tout ca est bel et bon, mais il faut que ca finisse,” [All this is good, but we must end it] - Prince Vasily said to himself one morning with a sigh of sadness, realizing that Pierre, who owed him so much (well, yes Christ be with him!), is not doing very well in this matter. “Youth... frivolity... well, God bless him,” thought Prince Vasily, feeling his kindness with pleasure: “mais il faut, que ca finisse.” After Lelya’s name day tomorrow, I will call someone, and if he does not understand what he must do, then it will be my business. Yes, it's my business. I am the father!