(1) Poisson distribution was developed by french mathematician , Simen Denis Poisson.

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(2) Definition of Poisson Distribution   :

A random variable x is defined to follow poisson distribution with a prameter m , it is denoted by
x ∿ P(m) .
Getting x successes in poisson distribution is given by .

P(X = x) =

e^-m . m^x/x!

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(3) If x ∿ P(x) , probability mass function of x is given by

P(X = x) =

e^-m . m^x/x!

for x = 0,1,2,3----------

= 0    otherwise .

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(4) It is discrete type probability distribution .

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(5) It is uniparametric distribution, having only one parameter m .

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(6) The mean of poisson distribution is m . variance of this distribution is also equal to m .
So in poisson distribution mean = variance = m.

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(7) Poisson distribution could be unimodal or bimodal depends on the value of the parameter m.
Mode the largest integer contained in m (unimodal) = If m is a non integer .
Mode (m) and (m - 1)(bimodal) = If m is an integer .

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(8) Poisson approximation to Binomial Distribution :

The number of independent trials of a binomial distribution say n, are very large (tends to infinite) and p the probability of success of random variable x is very small (tends to zero), so that m = np remains finite , then a binomial distribution with parameters n and p can be approximated by a poisson distribution with parameter m = np .

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(9) Additive property of poisson distribution :
Let x and y are two independent variables having parameters m₁ and m₂ respectively, then z = x + y will also follow poisson distribution with parameter (m₁ + m₂).
if x ∿ p(m₁), y ∿p(m₂)
then z = x + y ∿ p(m₁ + m ₂)

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(10) Application of poisson distribution

Poisson distribution is applicable when number of independent trials are very large and probability of success (occurrence) is very small .

(a) The distribution of the printing mistakes per 200 pages (in a book) .

(b) The distribution of number of telephone calls recieved at a particular exchange in a specific time interval .

(c) The distribution of number of defective items produced in a production process .

(d) The distribution of the number of suicides committed per day in a district and so on .

(11) Recurrence relation for probability is given by

p(x + 1) =

m/x + 1

. p(x)