POISSON DISTRIBUTION

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
POISSON
DISTRIBUTION
INTRODUCTION :
Poisson distribution is a discrete probability
distribution and it is widely used in statistical work.
This distribution was developed by French
mathematician Dr. Simon Denis Poisson in 1837 and the
distribution is named after him. The Poisson distribution
is used in those situations where the probability of the
happening of an event is very small , i.e., the event rarely
occurs. For example, the probability of defective items in
a manufacturing company is very small, the probability
of the accidents on road is very small, etc. all these are
examples of such events where the probability of
occurrence is very small.
Poisson distribution as Limiting form of
binomial distribution :-

 Poisson distribution is derived as a limiting
form of binomial distribution under certain
condition :(1) n, the number of trials is infinitely large, i.e.,
n→∞
(2) P, the probability of success is very small and
q, the probability of failure is very large, i.e., p→0,
q→1.
(3) the average number of successes (np) is equal to
positive finite quantity (m) i.e., no=m, where,
m is the parameter of the distribution.
Definition of Poisson
distribution:-

From binomial equation:
P(X=x)
e-m . mx
x!
Where P(X-x) = probability of obtaining x number of
success.
M= np = parameter of distribution
e= 2.7183 (base of natural logarithm )
Properties/Characteristics of Poisson
Distribution

 Discrete Probability Distribution: the Poisson Distribution
is a discrete probability distribution in which the number of
success are given in whole numbers such as 0, 1, 2…, etc.
 Value of p and q: the poisson distribution is used in those
situation where the probability of occurrence of an event is
very large (i.e., q-1) and the value of n is also infinitely large.
 Main Parameter: It has only one parameter m and its value
is equal to np, i.e., m=np. The entire distribution can be
known from this parameter.


Roles/Use/examples Of Poisson
distribution :

It is used in statical quality control to count the number of
defects of an item.
(1) In Biology to count the the number of bacteria.
(2) insurance problem to count the number of causalities.
(3) To count the number of typing errors per page in a
typed.
(4) To count the number of incoming telephone calls in a
town.
(5) To count the number of defective blades in a lot of
manufacture
blades in a factory.
(6) To count the number of death at a particular crossing
in a town as
result of a road accident.
(7) To count the number of suicides committed by the
lover point in a
year.
Application Of Poisson
Distribution


Application of Poisson Distribution
Formula.

Fitting of Poisson Distribution.
(A) Application of Poisson
Distribution Formula

We study the application distribution formula in two different
situation:
(1) when the of p is given :
When we are giving probability of success relating to a problem, i.e., is
given, the uses of the poisson distribution formula can be illustrated by
following examples:
Example: it is given that 2% of the screw manufactured by a company
are defective. Use poisson distribution to find the probability that a
packet of 100 screws contains:
Solution: Let p= probability of a defective screw= 2%=2/100
In the usual nation, we are given
np = 2/100, n =100
m = np = 100*2/100=2


When the value of m is
given

 When we are given the value of th parameter m
relating to a problem, the uses of Poisson
distribution formula can be illustrated by the
following examples :
Between the hours of 2 P.M. and 4 P.M., the
average number of phone calls per minute
coming into a switch board of a company is 2.5
find the probability that during one particular
minutes there will be (1) no phone calls at all (2)
exactly 3 calls, (3) at least 2 calls .




FITTING OF POISSON DISTRIBUTION:-

The following is adopted for fitting a Poisson distribution to the
observed data:
(1) Firstly, we computer mean (X) from the observed frequency
data by
using the formal: n X=fx/N
We use value this mean as the parameter of the poisson
distribution, i.e., X=m
(2) The value of e-m is obtained. If the value of e-m is not given in
the question, then the following formula is used to compete:
e-m = Reciprocal (Antilog (m*0.4343))
(3) Then, we compute the probability of 0,1,2,3 or x success by
using the Poisson Distribution formula:
P(X=x) = e-m. mx
(4) The expected or theoretical frequencies are then obtained bym
multiply each probability with N, total frequency. Thus14
Alternative Method:

The expected frequencies can be calculated in as easy way as
follows:
(1) First we calculate the ƒe(o) = N .e-m
(2) Other expected frequencies can be calculated
as follow:
ƒe(o)=N . P(0) N , e-m
ƒe(1)= m/1 . ƒe(o)
ƒe(2)=m/2 . ƒe(1)
ƒe(3)=m/3 . ƒe(2)
ƒe(4)=m/4 . ƒe(3) and so on.
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