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This manuscript aims to study the intervention-based probability model. Statistical and reliability properties such as the expressions for, cumulative density function (CDF), mean deviations about mean and median, rth order central and non-central moments, â€generation functions†for moments have been derived. Moreover, the expression for reliability function, hazard rate, reverse hazard rate, aging intensity, mean residual life function, stress-strength reliability, and entropy metrics due to R´enyi and Shannon are also derived. Monte Carlo simulation study performance of maximum likelihood estimates (MLEs) has been carried out, followed by calculations of Average Bias (ABias), and Mean Square Error (MSE). The applicability of the model in real-life situations has been discussed by analyzing the two real-life data sets.


Bias Entropy Intervnetion Mean Square Error Monte Carlo Simulaion

Article Details

Author Biography

Sudesh Pundir, Pondicherry University - INDIA

Assistant Professor,  Department of Statistics, Pondicherry University - INDIA.

How to Cite
BHAT, V. A., & Pundir, S. (2022). Intervened Exponential Distribution: Properties and Applications. Pakistan Journal of Statistics and Operation Research, 18(1), 71-84.


  1. Balakrishnan, K. (2019). Exponential Distribution: theory, methods and applications. CRC Press. DOI:
  2. Barlow, R. E., & Proschan, F. (1975). Statistical theory of reliability and life testing. New York: Holt Rinehart and Winston.
  3. Dhanavanthan, P. (1998). Compound intervened Poisson distribution. Biometrical Journal, 40(5), 641-664, doi:10.1002/(SICI)1521. DOI:<641::AID-BIMJ641>3.0.CO;2-F
  4. Dhanavanthan, P. (2000). Estimation of the parameters of compound intervened Poisson distribution. Biometrical Journal, 42(3), 315-320,$<$315::AID-BIMJ315$>$3.0.CO;2-E. DOI:<315::AID-BIMJ315>3.0.CO;2-E
  5. Efron, B. (1988). Logistic regression, survival analysis, and the Kaplan-Meier curve. Journal of the American Statistical Association, 83(402), 414-425, doi:10.1080/01621459.1988.10478612. DOI:
  6. Finkelstein, M. (2008). Failure rate modelling for reliability and risk. Springer Science & Business Media.
  7. Fuller, E. R., Jr, Freiman, S. W., Quinn, J. B., Quinn, G. D., & Carter, W. C. (1994). Fracture mechanics approach to the design of glass aircraft windows: a case study. In P. Klocek (Red), Window and Dome Technologies and Materials IV, International Society for Optics and Photonics, SPIE. (Vol 2286, bll 419-430, doi :10.1117/12.187363). DOI:
  8. Glaser, R. E. (1980). Bathtub and related failure rate characterizations. Journal of the American Statistical Association, 75(371), 667-672, doi: 10.1080/01621459.1980.10477530. DOI:
  9. Huang, M. L., & Fung, K. Y. (1989). Intervened truncated Poisson distribution. SankhyÄ: The Indian Journal of Statistics, Series B, 51(3), 302-310, doi:
  10. Jiang, R., Ji, P., & Xiao, X. (2003). Aging property of unimodal failure rate models. Reliability Engineering & System Safety, 79(1), 113-116, doi:10.1016/S0951-8320(02)00175-8. DOI:
  11. Kayid, M., & Izadkhah, S. (2014). Mean inactivity time function, associated orderings, and classes of life distributions. IEEE Transactions on Reliability, 63(2), 593-602, doi:10.1109/TR.2014.2315954. DOI:
  12. Marshall, A. W., & Olkin, I. (2007). Life distributions. Verlag New York: Springer.
  13. Rényi, A. (1961). On measures of entropy and information. In Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Volume 1: Contributions to the Theory of Statistics, 547-561. University of California Press.
  14. Scollnik, D. P. (1995). Bayesian analysis of an intervened Poisson distribution. Communications in Statistics - Theory and Methods, 24(3), 735-754, doi:10.1080/03610929508831519. DOI:
  15. Scollnik, D. P. (2006). On the intervened generalized Poisson distribution. Communications in Statistics - Theory and Methods, 35(6), 953-963, doi:10.1080/03610920600672278. DOI:
  16. Shaked, M., & Shanthikumar, J. G. (2007). Stochastic orders. New York: Springer. DOI:
  17. Shanmugam, R. (1985). An intervened Poisson distribution and Its medical application. Biometrics, 41(4), 1025-1029, doi:10.2307/2530973. DOI:
  18. Shanmugam, R., Bartolucci, A. A., & Singh, K. P. (2002). The analysis of neurologic studies using an extended exponential model. Mathematics and Computers in Simulation, 59(1-3), 81-85, doi:10.1016/S0378-4754(01)00395-0. DOI:
  19. Shannon, C. E. (1948). A mathematical theory of communication. The Bell System Technical Journal, 27(3), 379-423, doi:10.1002/j.1538-7305.1948.tb01338.x. DOI:
  20. Zacks, S. (1992). Introduction to reliability analysis: probability models and statistical methods. New York: Springer. DOI: