Further modified forms of binomial and poisson distributions

  • Giitiro Suzuki


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.


Poisson Distribution Probability Function Probability Generate Function Factorial Moment Bernoulli Trial 


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  1. [1]
    Adelson, R. M. (1966). Compound Poisson distributions,Operat. Res. Quart.,17, 73–75.CrossRefGoogle Scholar
  2. [2]
    Birnbaum, Z. W. and Tingey, F. (1951). One-sided confidence contours for probability distribution functions,Ann. Math. Statist.,22, 592–596.MathSciNetCrossRefGoogle Scholar
  3. [3]
    Chaddha, R. L. (1956). A case of contagion in binomial distribution,Classical and contagious discrete distributions Statist. Pub. Soc., Calcutta.Google Scholar
  4. [4]
    Consul, P. C. (1974). A simple urn model dependent upon predetermined strategy,Sankhyã,36, B, 391–399.MathSciNetzbMATHGoogle Scholar
  5. [5]
    Consul, P. C. and Jain, G. C. (1973). A generalization of the Poisson distribution,Technometrics,15, 791–799.MathSciNetCrossRefGoogle Scholar
  6. [6]
    Dandekar, V. M. (1955). Certain modified forms of binomial and Poisson distributions,Sankhyã,15, 237–250.MathSciNetzbMATHGoogle Scholar
  7. [7]
    Eggenberger, F. and Pólya, G. (1923). Über die Statistik verketteter Vorgänge.Zeit. angew. Math. Mech.,1, 179–289.zbMATHGoogle Scholar
  8. [8]
    Feller, W. (1943). On a general class of contagious distributionsAnn. Math. Statist.,14, 389–400.MathSciNetCrossRefGoogle Scholar
  9. [9]
    Galliher, H. P., Morse, P. M. and Simond, M. (1959). Dynamics of two classes of continuous-review inventory systemsOperat. Res.,7, 362–384.MathSciNetCrossRefGoogle Scholar
  10. [10]
    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.CrossRefGoogle Scholar
  11. [11]
    Gurland, J. (1957). Some interrelations among compound and generalized distributions,Biometrika,44, 265–268.CrossRefGoogle Scholar
  12. [12]
    Inst. Population Problems (1973). Summary of the 6th fertility survey in 1972,Res. Series, No. 200, 41.Google Scholar
  13. [13]
    Inst. Population Problems (1978). Summary of the 7th fertility survey in 1977,Res. Series, No. 219, 12.Google Scholar
  14. [14]
    Jain, G. C. and Consul, P. C. (1971). A generalized negative binomial distribution,SIAM J. Appl. Math.,21, 501–513.MathSciNetCrossRefGoogle Scholar
  15. [15]
    Johnson, N. L. and Kotz, S. (1969).Distributions in Statistical Discrete Distribution, John Wiley & Sons, New York.zbMATHGoogle Scholar
  16. [16]
    Neyman, J. (1939). On a new class of ‘contagious’ distributions, applicable in entomology and bacteriology,Ann. Math. Statist.,10, 35–57.CrossRefGoogle Scholar
  17. [17]
    Rutherford, R. S. G. (1954). On a contagious distribution,Ann. Math. Statist.,25, 703–713.MathSciNetCrossRefGoogle Scholar
  18. [18]
    Schelling, H. (1951). Distribution of the ordinal number of simultaneous events which last during a finite time.Ann. Statist. Math.,22, 452–455.MathSciNetCrossRefGoogle Scholar
  19. [19]
    Snyder, D. L. (1975)Random Point Processes, John Wiley & Sons, New York.zbMATHGoogle Scholar
  20. [20]
    Suzuki, G. (1976). Modified binomial distribution model—Bernoulli trials with controllable success rate,Proc. Inst. Statist. Math.,24, 41–46.MathSciNetzbMATHGoogle Scholar
  21. [21]
    Thomas M. (1949). A generalization of Poisson's binomial limit for use in ecology,Biometrika,36, 18–25.MathSciNetCrossRefGoogle Scholar
  22. [22]
    Woodbury, M. A. (1949). On a probability distribution,Ann. Math. Statist.,20, 311–313.CrossRefGoogle Scholar

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© The Institute of Statistical Mathematics, Tokyo 1980

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  • Giitiro Suzuki

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