Mathematic/ 5. Mathematical modeling

A. Iskakova, D. Salipov

L.N. Cumilyov Eurasian National University, Kazakhstan

About approximation of a exponential function for lottery system

In an official message among the challenges put forward by the President of the Republic of Kazakhstan in Epistle 16 (10 July 2012), it was sounded challenge of develop of a new model of an effective system of national lottery system.

However, for decades the issue about lottery systems simply not recognized as one of the important processes in the formation of social and economic development. Until recently it was approved untenable philosophical approach that the lottery system (as well as other socio-economic processes) must develop independently.

Therefore there is a need for new and effective models of the lottery system. However, the models that are used at present, has the general theoretical nature. At the same time, modern society in the face of government and business more elegantly refers to a lottery system to an urgent requirement to develop predictive function in accordance with the capabilities of high information technologies. Adjustable urbanization directly affects the construction of the successful model of the lottery system. Social processes are among the difficult to formalize objects.

The relevance of effective lottery system increases, given not only the acceleration of social processes, and their close connection with transient transformation of modern Kazakhstan in the context of globalization.

Let's we have the results of a statistical information specific lottery game of k units of statistical periods. Present this data as follows: 1,…, k - statistical periods, х₁,…, х_k are statistical dates. We are interested in the functional relationship between x_i and y_i, where i accepts any positive finite values.

Let’s y is function of one variable with two parameters a and b. As a selection set of functions, of which we have an empirical relationship, consider:

1) the linear function

;

2) the exponential function

;

3) the rational function

;

4) the logarithmic function

;

5) the power function

;

6) the hyperbolic function

;

7) the rational function

.

For the best choice of the form of the analytic dependence y=f(x,a,b) make the following intermediate calculations. On a given interval independent variable chosen point sufficiently secure and, if possible, far removed from each other. We assume that this x₁ and x_k.

We calculate the the arithmetic mean

,

the geometric mean

and the harmonic mean

.

According to the calculated values of the independent variables are the corresponding values of the variable we find , , .

We calculate the the arithmetic mean for extreme values

,

the geometric mean

and the harmonic mean

.

After the performed calculations we define estimation following errors:

, , , ,

, , .

The following theorem allows us to define the approach to functional dependence statistics of lottery games.

Theorem. Let’s .

1. If e=e₁ then an analytical dependence for this chart is a good approximation of a linear function

.

2. If e=e₂ then an analytical dependence for this chart is a good approximation of a exponential function

.

3. If e=e₃ then an analytical dependence for this chart is a good approximation of a rational function

;

4. If e=e₄ then an analytical dependence for this chart is a good approximation of a logarithmic function

.

5. If e=e₅ then an analytical dependence for this chart is a good approximation of a power function

.

6. If e=e₆ then an analytical dependence for this chart is a good approximation of a hyperbolic function

.

7. If e=e₇ then an analytical dependence for this chart is a good approximation of a rational function

.

The proof of Theorem can be found in different textbook on such directions as "Numerical Methods", "Mathematical Analysis", "Theory of Probability and Mathematical Statistics."

Thus, from the proposed theorem we can determine the type empirical function f(x,a,b). The coefficients a and b of empirical function f(x,a,b) we can determine by several methods, the optimal of which is the method of least squares.

Literature

1. M. G. Kendall. "The advanced theory of statistics (vol. I). Distribution theory (2nd edition)". Charles Griffin & Company Limited, 1945.

2. Papoulis А. Probability, random variables, and stochastic processes (3rd edition). McGrow-Hill Inc., 1991.

3. J. F. Kenney and E. S. Keeping. Mathematics of Statistics. Part I & II. D. Van Nostrand Company, Inc., 1961, 1959.

4. Blagouchine А. V. and E. Moreau: "Unbiased Adaptive Estimations of the Fourth-Order Cumulant for Real Random Zero-Mean Signal", IEEE Transactions on Signal Processing, vol. 57, no. 9, pp. 3330–3346, September 2009.