The Equilibrium Distribution of Counting Random Variables

Author(s)
Shuanming Li

ABSTRACT

We study the high order equilibrium distributions of a counting random variable. Properties such as moments, the probability generating function, the stop--loss transform and the mean residual lifetime, are derived. Expressions are obtained for higher order equilibrium distribution functions under mixtures and convolutions of a counting distribution. Recursive formulas for higher order equilibrium distribution functions of the (*a*,*b*,0) -family of distributions are given.

We study the high order equilibrium distributions of a counting random variable. Properties such as moments, the probability generating function, the stop--loss transform and the mean residual lifetime, are derived. Expressions are obtained for higher order equilibrium distribution functions under mixtures and convolutions of a counting distribution. Recursive formulas for higher order equilibrium distribution functions of the (

KEYWORDS

Counting Random Variable, Equilibrium Distribution, Stop-Loss Transform, Mean Residual Life, (*a*,
*b*,
0) Family,
Recursive Formulas,
Probability Generating Function

Counting Random Variable, Equilibrium Distribution, Stop-Loss Transform, Mean Residual Life, (

Cite this paper

nullS. Li, "The Equilibrium Distribution of Counting Random Variables,"*Open Journal of Discrete Mathematics*, Vol. 1 No. 3, 2011, pp. 127-135. doi: 10.4236/ojdm.2011.13016.

nullS. Li, "The Equilibrium Distribution of Counting Random Variables,"

References

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[2] E. Fagiuoli and F. Pellerey, “Preservation of Certain Classes of Life Distributions under Poisson Shock Models,” Journal of Applied Probability, Vol. 31, No. 2, 1994, pp. 458-465. doi:10.2307/3215038

[3] A.K. Nanda, H. Jain and H. Singh, “On Closure of Some Partial Orderings under Mixture,” Journal of Applied Probability, Vol. 33, No. 3, 1996, pp. 698-706. doi:10.2307/3215351

[4] O. Hesselager, S. Wang and G.E. Willmot, “Exponential and Scale Mixture and Equilibrium Distributions,” Scandinavian Actuarial Journal, Vol. 20, No. 2, 1994, pp. 125-142.

[5] M. Bowers, H. Gerber, J. Kickman, D. Jones and C. Nesbitt, “Actuarial Mathematics,” 2nd Edition, Society of Actuaries, Schaumburg, 1997.

[6] X. Lin and G. E. Willmot, “Analysis of a Defective Renewal Equation Arising in Ruin Theory,” Insurance: Mathematics and Economics, Vol. 25, No. 1, 1999, pp. 63-84. doi:10.1016/S0167-6687(99)00026-8

[7] X. Lin and G. E. Willmot, “The Moments of the Time of Ruin, the Surplus before Ruin, and the Deficit at Ruin,” Insurance: Mathematics and Economics, Vol. 27, No. 1, 2000, pp. 19-44. doi:10.1016/S0167-6687(00)00038-X

[8] G. E. Willmot, “Bounds for Compound Distributions Based on Mean Residual Lifetimes and Equilibrium Distributions,” Insurance: Mathematics and Economics, Vol. 21, No. 1, 1997, pp. 25-42. doi:10.1016/S0167-6687(97)00016-4

[9] G. E. Willmot and J. Cai, “Aging and other Distributional Properties of Discrete Compound Geometric Distributions,” Insurance: Mathematics and Economics, Vol. 28, No. 3, 2001, pp. 361-379. doi:10.1016/S0167-6687(01)00062-2

[10] R. W. Hamming, “Numerical methods for scientists and Engineers,” 2nd Edition, Dover, New-York, 1973.

[11] H. H. Panjer, “Recursive Evaluation of a Family of Compound Distributions,” ASTIN Bulletin, Vol. 12, No. 1, 1981, pp. 22-26.

[12] S. A. Klugman, H. H. Panjer and G. E. Willmot, “Loss Models,” Wiley, New-York, 1998.

[13] S. Li and J. Garrido, “On the Time Value Ruin of Discrete Time Risk Process,” Working Paper 02-18, Universidad Carlos III of Madrid, Madrid, 2002.

[1] E. Fagiuoli and F. Pellerey, “New Partial Orderings and Applications,” Naval Research Logistics, Vol. 40, No. 6, 1993, pp. 829-842. doi:10.1002/1520-6750(199310)40:6<829::AID-NAV3220400607>3.0.CO;2-D

[2] E. Fagiuoli and F. Pellerey, “Preservation of Certain Classes of Life Distributions under Poisson Shock Models,” Journal of Applied Probability, Vol. 31, No. 2, 1994, pp. 458-465. doi:10.2307/3215038

[3] A.K. Nanda, H. Jain and H. Singh, “On Closure of Some Partial Orderings under Mixture,” Journal of Applied Probability, Vol. 33, No. 3, 1996, pp. 698-706. doi:10.2307/3215351

[4] O. Hesselager, S. Wang and G.E. Willmot, “Exponential and Scale Mixture and Equilibrium Distributions,” Scandinavian Actuarial Journal, Vol. 20, No. 2, 1994, pp. 125-142.

[5] M. Bowers, H. Gerber, J. Kickman, D. Jones and C. Nesbitt, “Actuarial Mathematics,” 2nd Edition, Society of Actuaries, Schaumburg, 1997.

[6] X. Lin and G. E. Willmot, “Analysis of a Defective Renewal Equation Arising in Ruin Theory,” Insurance: Mathematics and Economics, Vol. 25, No. 1, 1999, pp. 63-84. doi:10.1016/S0167-6687(99)00026-8

[7] X. Lin and G. E. Willmot, “The Moments of the Time of Ruin, the Surplus before Ruin, and the Deficit at Ruin,” Insurance: Mathematics and Economics, Vol. 27, No. 1, 2000, pp. 19-44. doi:10.1016/S0167-6687(00)00038-X

[8] G. E. Willmot, “Bounds for Compound Distributions Based on Mean Residual Lifetimes and Equilibrium Distributions,” Insurance: Mathematics and Economics, Vol. 21, No. 1, 1997, pp. 25-42. doi:10.1016/S0167-6687(97)00016-4

[9] G. E. Willmot and J. Cai, “Aging and other Distributional Properties of Discrete Compound Geometric Distributions,” Insurance: Mathematics and Economics, Vol. 28, No. 3, 2001, pp. 361-379. doi:10.1016/S0167-6687(01)00062-2

[10] R. W. Hamming, “Numerical methods for scientists and Engineers,” 2nd Edition, Dover, New-York, 1973.

[11] H. H. Panjer, “Recursive Evaluation of a Family of Compound Distributions,” ASTIN Bulletin, Vol. 12, No. 1, 1981, pp. 22-26.

[12] S. A. Klugman, H. H. Panjer and G. E. Willmot, “Loss Models,” Wiley, New-York, 1998.

[13] S. Li and J. Garrido, “On the Time Value Ruin of Discrete Time Risk Process,” Working Paper 02-18, Universidad Carlos III of Madrid, Madrid, 2002.