Poisson Distribution

Poisson distribution is a discrete probability distribution and is very widely used in statistical work. It was originated by a French mathematician, Simcon Denis Poisson (1781 – 1840), in 1837. Poisson distribution may be expected in cases where the chance of any individual event being a success is small. The distribution is used to describe the behavior of rare events such as the number of accidents on road, number of printing mistakes in a book, etc. and has been called “the law of improbable events”. In recent years the statisticians have had a renewed interest in the occurrence of comparatively rare events, such as serious floods, accidental release of radiation from a nuclear, and the like.

How Poisson distribution is a limiting form of Binomial distribution can be proved as follows:
Proof. In case of binomial distribution the probability of r successes is given by

            P (r) =

        = n (n-1)…. (n – r +1) p^rq^n-r
           ______  r!
Put p = m/n therefore, q = 1 – p = 1 – m/n

We now get

p (r) = n (n-1)…. (n – r + 1)  (m/n)^r (1-m/n)^n-r
        r!

= (1-1/n) (1-2/n)…. (1-r-1/n)m^r    {(1-m/n)ⁿ}
     ________   r! _______           {(1-m/n)^r}

For fixed r, as n → ∞

(1-1/n)……….. (1- r-1/n). (1-m/n)^r all tend to 1 and (1- m/n)ⁿ to e^-m.

Hence in he limiting case
P (r) = e^-mm^r
    ____   r!




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