Summary
Some new type of modifications of binomial and Poisson distributions, are discussed. First, we consider Bernoulli trials of lengthn with success ratep up to time whenm times of successes occur, and then, changing the success rate to γp, we continue the remaining trial. The distribution of number of successes is called the modified binomial distribution. The Poisson limit (n tends to infinity andp tends to 0, keepingnp=λ) of the modified binomial is called the modified Poisson distribution. The probability functions of modified binomial and Poisson distributions are given (Section 1).
A new concept of (m, γ)-modification is introduced and fundamental theorem which gives the relations between the factorial moments of any probability function and the factorial moments of its (m, γ)-modification, is presented. Then some lower order moments of the modified binomial and Poisson distributions are given explicitly (Section 2).
The modified Poisson ofm=2 is fitted to the distribution of number of children for Japanese women in some age group. The fitting procedure is also presented (Section 3). Some historical sketch concerning the modification and generalization of binomial and Poisson distributions is given in Appendix.
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References
Adelson, R. M. (1966). Compound Poisson distributions,Operat. Res. Quart.,17, 73–75.
Birnbaum, Z. W. and Tingey, F. (1951). One-sided confidence contours for probability distribution functions,Ann. Math. Statist.,22, 592–596.
Chaddha, R. L. (1956). A case of contagion in binomial distribution,Classical and contagious discrete distributions Statist. Pub. Soc., Calcutta.
Consul, P. C. (1974). A simple urn model dependent upon predetermined strategy,Sankhyã,36, B, 391–399.
Consul, P. C. and Jain, G. C. (1973). A generalization of the Poisson distribution,Technometrics,15, 791–799.
Dandekar, V. M. (1955). Certain modified forms of binomial and Poisson distributions,Sankhyã,15, 237–250.
Eggenberger, F. and Pólya, G. (1923). Über die Statistik verketteter Vorgänge.Zeit. angew. Math. Mech.,1, 179–289.
Feller, W. (1943). On a general class of contagious distributionsAnn. Math. Statist.,14, 389–400.
Galliher, H. P., Morse, P. M. and Simond, M. (1959). Dynamics of two classes of continuous-review inventory systemsOperat. Res.,7, 362–384.
Greenwood, M. and Yule, G. U. (1920). An enquiry into the nature of frequency distributions representative of multiple happenings,J. R. Statist. Soc., A,83, 255–279.
Gurland, J. (1957). Some interrelations among compound and generalized distributions,Biometrika,44, 265–268.
Inst. Population Problems (1973). Summary of the 6th fertility survey in 1972,Res. Series, No. 200, 41.
Inst. Population Problems (1978). Summary of the 7th fertility survey in 1977,Res. Series, No. 219, 12.
Jain, G. C. and Consul, P. C. (1971). A generalized negative binomial distribution,SIAM J. Appl. Math.,21, 501–513.
Johnson, N. L. and Kotz, S. (1969).Distributions in Statistical Discrete Distribution, John Wiley & Sons, New York.
Neyman, J. (1939). On a new class of ‘contagious’ distributions, applicable in entomology and bacteriology,Ann. Math. Statist.,10, 35–57.
Rutherford, R. S. G. (1954). On a contagious distribution,Ann. Math. Statist.,25, 703–713.
Schelling, H. (1951). Distribution of the ordinal number of simultaneous events which last during a finite time.Ann. Statist. Math.,22, 452–455.
Snyder, D. L. (1975)Random Point Processes, John Wiley & Sons, New York.
Suzuki, G. (1976). Modified binomial distribution model—Bernoulli trials with controllable success rate,Proc. Inst. Statist. Math.,24, 41–46.
Thomas M. (1949). A generalization of Poisson's binomial limit for use in ecology,Biometrika,36, 18–25.
Woodbury, M. A. (1949). On a probability distribution,Ann. Math. Statist.,20, 311–313.
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Suzuki, G. Further modified forms of binomial and poisson distributions. Ann Inst Stat Math 32, 143–159 (1980). https://doi.org/10.1007/BF02480320
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DOI: https://doi.org/10.1007/BF02480320