MATHEMATICS / 4. Applied mathematic

Candidate of Physical and Mathematical Science Iskakova A.

L.N. Gumilev Eurasian National University, Astana, Kazakhstan

The probability model for the dynamics of delinquent behavior of minors in the EU countries

The problem of social urbanization in the EU countries is very acute. For example, according to the Committee on Legal Statistics and Special Accounts of the Prosecutor General's Office of the Republic of Kazakhstan, about eight thousand minors who commit crimes are identified each year, almost half of them are subject to criminal liability.

One of the indicators characterizing the social health of society is the smallest deviation from social norms. Obviously, the following factors influence the dynamics of crime among adolescents: economic (price growth, low incomes of the bulk of the population, demographic structure of the population), social (sharp deterioration of the psychological climate in the families of the unemployed, alienation of parents from the responsibility for raising children, forced search by minors Their own sources of income, devaluation of family values, the institution of marriage as the basis for the normal life of people in society) and legal factors (changes in criminal law that expand or narrow the sphere of criminal and punishable, change the classification and qualification of crimes, and the detection of crimes).

The probabilistic study of all the quantitative indicators of crimes is based on the probability of the influence of the relevant factors. From the course of probability theory it is obvious that these factors can be considered as polynomially distributed. However, the probability distribution of the sum of polynomially distributed random variables and its application in social studies in the scientific literature is available in [1-3].

However, if we consider situations in which unknown phenomena were imposed on explored events, in other words, implicit assumptions, then many unresolved problems remain.

Any crime committed by minors is a consequence of the influence of a group of factors. Suppose that the crime is influenced by d factors with some degree of effect. We define each factor by one of the possible numbers l₁, l₂, ..., l_d with the corresponding probability values p₁, ..., p_d, and

Let us be interested in the number of crimes for a certain period. Suppose the number of crimes n can be affected by d factors with possible repetitions. Moreover, the factor l₁ influenced the crime n u r₁ times, the factor l₂ affected the crime n r₂ times, and so on, the factor crime n rd times. Moreover, ld influenced the for each i = 1, ..., d, r_i takes the value either 0 or 1. Obviously,

. (1)

Theorem 1. The probability that the sum of numbers on k influencing factors with repetitions per crime is n is determined by the formula

. (2)

Evidence. Of course, if there are subdivisions of n into l₁, l₂, ..., l_d, then (1) has one or more solutions. The probability of each partition of n by l₁, l₂, ..., l_d is determined by the polynomial distribution. Thus, we have arrived at the proof of the theorem. The theorem is proved.

Ex. 1. When reviewing the analysis of the dynamics of juvenile delinquency in the city Leon (France), we have the data presented in Table 1.

Table 1.

Dynamics of juvenile delinquency in the city Leon (France)

Year	2007	2008	2009	2010	2011	2012	2013	2014	2015
Number of crimes	60	21	17	21	1	23	22	10	9

Assume that the economic factor can affect the state of crime among adolescents with a probability of 0.7, the 2nd factor is 0.2, the third factor is 0.1. Presumptive variants of the breakdown of factors influencing the dynamics of juvenile delinquency in the city Leon (France) are presented in Table 2.

Table 2.

Presumptive variants of the breakdown of factors influencing the dynamics of juvenile delinquency in the city Leon (France) are presented in Table 2.

Year		2007	2008	2009	2010	2011	2012	2013	2014	2015
Number of crimes		60	21	17	21	1	23	22	10	9
Variant 1	F1	30	20	15	15	1	10	20	5	5
	F2	20	1	1	6	0	10	1	3	3
	F3	10	0	1	5	0	3	1	2	1
Variant 2	F1	-	-	16	-	0	-	-	-	5
	F2	-	-	1	-	1	-	-	-	4
	F3	-	-	0	-	0	-	-	-	0

Let us assume that only two variants of the factor splitting are possible, which are presented in Table 2. So for the first for 2015, we have from

Reference

1. Ayman I. Statistical Research for Probabilistic Model of Distortions of Remote Sensing //Journal of Physics: Conference Series. – IOP Publishing, 2016. – T. 738. - №.1. – C. 012004.

2. Ayman I. Construction of the most suitable unbiased estimate distortions of radiation processes from remote sensing data //Journal of Physics: Conference Series. – IOP Publishing, 2014. – Т. 490. – №. 1. – С. 012113.

3. Iskakova A.S. Determination of the most suitable unbiased estimate for a weather forecast being correc //SibirskiiZhurnalIndustrial’noiMatematiki. – 2002. –T.5. - №.1. – C.79-84.