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.