Stochastic Derivation of an Integral Equation for Probability Generating Functions

Transcription

1 Journal of Informatics and Mathematical Sciences Volume 5 (2013), Number 3, RGN Publications htt:// Stochastic Derivation of an Integral Equation for Probability Generating Functions Panagiotis T. Artikis and Constantinos T. Artikis Abstract Functional, integral and differential equations of transformed robability generating functions are generally recognized as owerful analytical tools for establishing characterizations of discrete robability distributions. The resent aer establishes a characterization of the distribution of an imortant integral art model by incororating an integral equation based on three fundamental transformed robability generating functions. Interretations of such a characterization in analyzing and imlementing information risk frequency reduction oerations are also established. 1. Introduction Stochastic multilicative models incororating two nonnegative random variables are generally recognized as very strong analytical tools of several fundamental areas of robability theory [3]. Moreover, such stochastic models have very useful alications in economics, management, oerational research, risk theory, insurance, reliability theory, logistics, engineering, informatics, and other fundamental ractical discilines. If the two nonnegative random variables of a stochastic multilicative model are indeendent, then the distribution function of this model belongs to the class of scale mixtures of distribution functions [6]. This class has substantially contributed to the investigation of unimodality, infinite divisibility, selfdecomosability, stability and other significant theoretical roerties with articular ractical imortance of many fundamental classes of distribution functions [8]. The literature concentrating on the investigation of the theoretical roerties and ractical alications of stochastic multilicative models incororating two nonnegative and indeendent random variables is very broad 2010 Mathematics Subject Classification. 91B30; 65R20; 91B70. Key words and hrases. Integral equation; Stochastic models; Risk theory. Coyright 2013 Panagiotis T. Artikis et al. This is an oen access article distributed under the Creative Commons Attribution License, which ermits unrestricted use, distribution, and reroduction in any medium, rovided the original work is roerly cited.

2 158 Panagiotis T. Artikis and Constantinos T. Artikis [9]. Proerties and ractical alications in information risk management of a stochastic multilicative model incororating two nonnegative and indeendent random variables, one of which follows the standard uniform distribution and the other is the sum of two nonnegative and indeendent random variables, have been established by Artikis and Artikis [1]. The resent aer is devoted to the establishment of roerties and alications in information risk management of a discrete analogue of such a stochastic multilicative model by making use of the stochastic model M = [ΠB], where Π, B are indeendent random variables with Π taking values in the set N = {1, 2, 3,... }, B following the standard uniform distribution and [ΠB] denoting the integral art of ΠB. If P(Π = π) = π, π = 1, 2,... is the robability function of Π and P Π (z) is the robability generating function of Π, then P(M = m) = π=m+1 is the robability function of M and P M (z) = 1 1 P Π (β) dβ 1 z β z π, m = 0, 1, 2,... π is the robability generating function of M [7]. The resent aer concentrates on the imlementation of two uroses. The first urose is the establishment of a characterization of the distribution of the integral art model, based on the roduct of two indeendent random variables one of which follows the geometric distribution and the other follows the standard uniform distribution, by making use of an integral equation incororating three fundamental transformed robability generating functions. The second urose is the interretation of such a characterization in the area of stochastic modelling of information risk frequency reduction oerations. 2. Certain Transformations of Probability Generating Functions The resent section of the aer mainly concentrates on the consideration of certain known transformations of robability generating functions corresonding to discrete random variables with values in the set N 0 = {0, 1, 2,... } and finite mean.

3 Stochastic Derivation of an Integral Equation for Probability Generating Functions 159 Let X be a discrete random variable with values in the set N 0, robability generating function P X (z) and finite mean µ, then 1 P U (z) = µ(z 1) log P X (z), (2.1) is a robability generating function of a discrete random variable U with values in the set N 0 and robability function having a unique mode at the oint 0 if and only if, X is infinitely divisible [2]. The formula (2.1) is a transformation which mas the robability generating function P X (z) into the robability generating function P U (z). An integral equation incororating the robability generating function P U (z) and the robability generating function of an integral art model, based on the roduct of two indeendent random variables one of which is distributed as the random variable X +1 and the other follows the standard uniform distribution, has been investigated by Artikis et al [2]. Moreover, the formula P X (z) = ex[µ(z 1)P U (z)] (2.2) is a transformation which mas the robability generating function P U (z) into the robability generating function P X (z). Let V be a discrete random variable with values in the set N 0, robability generating function P V (z) and finite mean θ, then P J (z) = 1 P V (z) θ(1 z) (2.3) is the robability generating function of the renewal distribution corresonding to the distribution of the random variable V [4]. Formula (2.3) is a transformation which mas the robability generating function P V (z) into the renewal robability generating function P J (z). The resent aer makes use of the transformation in (2.2) and the transformation in (2.3) to establish a characterization of a discrete distribution. 3. Characterizing the Distribution of an Integral Part Model The resent section of the aer establishes a characterization of the integral art model based on the roduct of two indeendent random variables, one of which is geometrically distributed and the other follows the standard uniform distribution [2]. Theorem. Let W, T and L be indeendent random variables with W following the standard uniform distribution and T, L distributed as the random variable S with robability generating function P S (z) = ex[κ(z 1)P Y (z)], κ > 0, where P Y (z) is the robability generating function of a discrete random variable Y with values in the set N 0, finite mean and robability function having a unique mode

4 160 Panagiotis T. Artikis and Constantinos T. Artikis at the oint 0. The robability generating function of the random variable Y has the form P Y (z) = q(z 1) log, 1 qz where = 1 κ + 1, q = 1 if and only if the stochastic model C = [(L + T + 1)W ], denoting the integral art of (L + T + 1)W, is equally distributed with the random variable R following the renewal distribution corresonding to the distribution of the random variable S. Proof. Only the sufficient condition will be roved since the necessity condition can be roved by reversing the argument. It is readily shown that the indeendence of the random variables W, T, L imlies the indeendence of the random variables L + T + 1, W. Hence, it is also readily shown that the robability generating function of the integral art model is given by C = [(L + T + 1)W ] P C (z) = 1 1 z 1 z ex[2κ(w 1)P Y (w)] dw. (3.1) Moreover, the robability generating function of the random variable R is given by P R (z) = 1 ex[κ(z 1)P Y (z)]. (3.2) κ(1 z) From (3.1), (3.2) and the assumtion that the random variables R, C = [(L + T + 1)W ] are equal in distribution, we get the integral equation 1 ex[κ(z 1)P Y (z)] κ(1 z) = 1 1 z 1 z ex[2κ(w 1)P Y (w)] dw. (3.3)

5 Stochastic Derivation of an Integral Equation for Probability Generating Functions 161 If we multily both sides of the integral equation in (3.3) by κ(1 z) and then differentiate, we get the differential equation ex[κ(z 1)P Y (z)] d dz [κ(z 1)P Y (z)] = κ ex[2κ(z 1)P Y (z)] (3.4) which satisfies the boundary condition P Y (1) = 1. Since the robability generating function P S (z) = ex[(κ(z 1)P Y (z)] has no real roots, then the differential equation in (3.4) can be written in the form ex[ κ(z 1)P Y (z)] d dz [ κ(z 1)P Y (z)] = κ (3.5) Integrating the differential equation in (3.5) with due regard to the above boundary condition we get that with and ex[ κ(z 1)P Y (z)] = κ + 1 κz (3.6) From (3.6) it follows that P Y (z) = q(z 1) log 1 qz = 1 κ + 1 q = κ κ + 1. (3.7) It is of some theoretical imortance to mention that the above robability generating function is derived from the transformation (2.1) if the random variable X follows the geometric distribution with robability generating function P X (z) = 1 qz. It is easily demonstrated that the random variable with robability generating function P Y (z) in (3.7) can be written in the form of the integral art model Y = [DH], where D, H are indeendent random variables with D following the geometric distribution with robability generating function P D (z) = z 1 qz and H following the uniform distribution with robability density function f H (h) = 1.

6 162 Panagiotis T. Artikis and Constantinos T. Artikis An interretation of the integral art model C = [(L + T + 1)W ] incororated by the resent aer for characterizing the distribution of the integral art model Y = [DH], in the area of risk frequency reduction oerations is the following. We suose that the random variable L denotes the frequency of a risk and the random variable T + 1 denotes the frequency of another risk. We also suose that the risks are of the same tye. Since the random variable W follows the standard uniform distribution, then the integral art model C = [(L + T + 1)W ] can be interreted as the total risk frequency after alying a risk frequency reduction oeration to these risks. The general recognition of risk frequency reduction oerations as extremely imortant, for roactive treatment of risks threatening information systems of modern comlex organizations, substantially suorts the alicability of the roosed interretation of the integral art model incororated by this aer in analyzing and imlementing such information risk management oerations [5]. 4. Conclusions The establishment of a characterization for the distribution of an integral art model, based on the roduct of two indeendent random variables, one of which follows the geometric distribution and the other follows the standard uniform distribution, is the theoretical contribution of the aer. The ractical contribution of the aer consists of interreting such a characterization in the area of stochastic modelling of information risk frequency reduction oerations. References [1] P. Artikis and C. Artikis, Stochastic derivation of an integral equation for characteristic functions, Journal of Informatics and Mathematical Sciences 1 (2009), [2] T. Artikis, S. Loukas and D. Jerwood, A transformed geometric distribution in stochastic modelling, Mathematical and Comuter Modelling 27 (1998), [3] C. Goldie, A class of infinitely divisible distributions, Proc. Cambridge Phil. Soc. 63 (1967), [4] R. Guta, On the characterization of survival distributions in reliability by roerties of their renewal densities, Communications in Statistics Theory and Methods 8 (1979), [5] Y. Haimes, Risk Modelling, Assessment and Management, John Wiley & Sons, Inc., Hoboken, New Jersey, [6] J. Keilson and F. Steutel, Mixtures of distributions, moment inequalities and measures of exonentiality and normality, Ann. Probab. 2 (1974),

7 Stochastic Derivation of an Integral Equation for Probability Generating Functions 163 [7] N. Krishnaji, A characteristic roerty of the Yule distribution, Sankhya A 32 (1970), [8] P. Medgyessy, On a new class of unimodal infinitely divisible distribution functions and related toics, Studia Scientiarum Mathematicarum Hungarica 2 (1967), [9] F. Steutel and K. van Harn, Infinite Divisibility of Probability Distributions on the Real Line, Marcel Dekker, New York, Panagiotis T. Artikis, Section of Informatics and Mathematics, Pedagogical Deartment, University of Athens, 20 Hiocratous Str., Athens, Greece. tartikis@gmail.com Constantinos T. Artikis, Ethniki Asfalistiki Insurance Comany, Syngrou Avenue, Athens, Greece. ctartikis@gmail.com Received January 2, 2010 Acceted November 19, 2012