# Further modified forms of binomial and poisson distributions

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## Summary

Some new type of modifications of binomial and Poisson distributions, are discussed. First, we consider Bernoulli trials of length*n* with success rate*p* up to time when*m* times of successes occur, and then, changing the success rate to γ*p*, we continue the remaining trial. The distribution of number of successes is called the modified binomial distribution. The Poisson limit (*n* tends to infinity and*p* tends to 0, keeping*np*=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 of*m*=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.

## Keywords

Poisson Distribution Probability Function Probability Generate Function Factorial Moment Bernoulli Trial## Preview

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