# On Sum and Geometric Sum of independent New Quasi Lindley Random Variables and its Applications

## DOI:

https://doi.org/10.24297/jam.v22i.9528## Keywords:

Moments, Laplace transformation, New quasi-Lindley distribution (NQLD), Sum of random variables, Geometric sum, Random sum## Abstract

The Laplace transformation method is used to drive the distribution of the sum Sn of n-fixed random variables, which has a new quasi Lindley distribution with two parameters θ and α, NQLD (θ,α). The sum of NQLD (SUNQLD) distribution is obtained in pdf and cdf formats. It is discussed how to calculate the random sum SN of a random number of NQLD random variables. The random sum of the NQLD distribution's pdf and cdf are calculated. When *N* has a geometric distribution, the geometric sum of NQLD distribution (GSN QLD) was introduced as an example of a random number of NQLD random variables. For all cases, some statistical measures are determined. The distribution's parameters are estimated using the maximum likelihood method. To test the viability and efficiency of the proposed distributions SNQLD and GSNQLD, lifetime count data sets from acute myeloid leukaemia are fitted. The results should become accepted knowledge in the fields of probability theory and its allied sciences. In addition, the histogram, fitted probability density function (pdf), and P-P plots for the analyzed real data set are presented.

### Downloads

## References

A. Z. Afify, Z. M. Nofal, and N. S. Butt, Transmuted complementary Weibull geometric distribution, Pak. J. Stat. Oper. Res., 10 (2014), pp. 435-454. https://doi.org/10.18187/pjsor.v10i4.836

Y. M. Amer, D. H. Abdel Hady, and R. M. Shalabi On a Sum and Difference of Two Quasi Lindley Distributions: Theory and Applications, American Journal of Applied Mathematics and Statistics 9(1) (2021), pp. 12-23. https://doi.org/10.12691/ajams-9-1-3

L. Bon, and V. Kalashnikov, Some estimates of geometric sums, Statistics & Probability Letters, 55 (2001), pp. 89-97. https://doi.org/10.1016/S0167-7152(01)00136-5

C. Chesneau, L. Tomy, and J. Gillariose, On a Sum and Difference of Two Lindley Distributions: Theory and Applications, Revstat - Statistical Journal 18(5) (2020), pp. 673-695.

S. Chouia, and H. Zeghdoudi, The XLindley Distribution: Properties and Application. Journal of Statistical Theory and Applications, 20(2) (2021), pp. 318–327. https://doi.org/10.2991/jsta.d.210607.001

R. A.Ganaie, V. Rajagopalan, and A. A. Rather, Weighted new quasi Lindley distribution with Properties and Applications, Journal of Xi'an University of Architecture Technology, 12(2) (2020), pp. 561 – 575.

R. A. Ganaie, V. Rajagopalan, and A. A. Rather, Length biased Weighted New Quasi Lindley Distribution: Statistical Properties and Applications, Pak. j. stat. oper. res., 17(1) (2021), pp. 123-136. https://doi.org/10.18187/pjsor.v17i1.3084

M. E. Ghitany, B. Atieh, and S. Nadarajah, Lindley Distribution and Its Application, Mathematics and Computers in Simulation, 78 (2008), pp. 493-506. https://doi.org/10.1016/j.matcom.2007.06.007

M. Girondot, and J. Barry, Computation of the Distribution of the Sum of Independent Negative Binomial Random Variables, Math. Comput. Appl. 28(3) (2023), No. 63. https://doi.org/10.3390/mca28030063

A. S. Hassan, , and M. Abd-Allah, Power Quasi Lindley Power Series Class of Distributions: Theory and Applications, Thailand Statistician. 21(2) (2023), pp. 314-336.

N. Kolev, E. T.Kolkovska, and J. A.Lopez-Mimbela, Joint probability generating function for a vector of arbitrary indicator variables Journal of Computational and Applied Mathematics, 186 (2006), pp. 89-98. https://doi.org/10.1016/j.cam.2005.03.066

D.V. Lindley, Fiducial distributions and Bayes' theorem, Journal of the Royal Statistical Society, Series B, 20 (1958), pp. 102-107. https://doi.org/10.1111/j.2517-6161.1958.tb00278.x

N. Y. Ma, and R. P. King, The n-fold convolution of generalized exponential-sum distribution functions, Applied Mathematics and Computation, 142 (2003), pp. 23-33. https://doi.org/10.1016/S0096-3003(02)00281-3

[N. N. Midhu, I. Dewan, K. K. Sudheesh, and E. P. Sreedevi, On approximation and estimation of distribution function of sum of independent random variables, Stat Papers (2023). https://doi.org/10.1007/s00362-023-01413-4

M. M. Mohie El-Din, A. A. Teamah, A. M. Salem, and A. T. Abd El-Bar, Random sum of mid truncated Lindley distribution, Journal of Advanced Research in Statistics and Probability, 2(1) (2010), pp. 27 – 36.

S. Nadarajah, Y. Zhang, and T. K. Pog´any, On sums of independent generalized Pareto random variables with applications to insurance and and cat bonds, Probability in the Engineering and Informational Sciences, 32 (2017), pp. 296–305. https://doi.org/10.1017/S0269964817000055

R. Shanker, and A. G. Amanuel, A new quasi Lindley distribution, International Journal of Statistics and Systems, 8(2) (2013), pp. 143 – 156.

R. Shanker, and A. A. Mishra, Quasi Lindley Distribution, African Journal of Mathematics and Computer Science Research, 6(4) (2013), pp. 64 – 71.

R. Shanker, and A. A. Mishra, two-parameter Lindley distribution, Statistics in Transition-new series, 14(1) (2013), pp. 45- 56.

R. Shanker, S. Sharma, and R. Shanker, A two-parameter Lindley distribution for modeling waiting and survival times data, Applied Mathematics, 4 (2013), pp. 363 – 368. https://doi.org/10.4236/am.2013.42056

S. H. Sung, Complete convergence for weighted sums of random variables, Statistics & Probability Letters. 77 (2007), pp. 303-311. https://doi.org/10.1016/j.spl.2006.07.010

A. A. Teamah, and A. R. El-Alosey, Random sum of mixtures of exponential distributions. Journal of Applied Mathematics, 16 (2004), pp. 237-247.

G. E. Willmot, and J. Cai, Monotonicity and aging properties of random sums, Statistics & Probability Letters. 73 (2005), pp. 381-392. https://doi.org/10.1016/j.spl.2005.04.013

## Downloads

## Published

## How to Cite

*JOURNAL OF ADVANCES IN MATHEMATICS*,

*22*, 61–71. https://doi.org/10.24297/jam.v22i.9528

## Issue

## Section

## License

Copyright (c) 2023 Alaa Rafat El Alosey

This work is licensed under a Creative Commons Attribution 4.0 International License.

All articles published in *Journal of Advances in Linguistics* are licensed under a Creative Commons Attribution 4.0 International License.