Page 205 - DMTH404_STATISTICS

Unit 14: Theoretical Probability Distributions

14.6.7 Uses of Poisson Distribution Notes

(i) This distribution is applicable to situations where the number of trials is large and the
probability of a success in a trial is very small.
(ii) It serves as a reasonably good approximation to binomial distribution when n  20 and
p  0.05.

14.7 Summary

 Binomial distribution is a theoretical probability distribution which was given by James
Bernoulli.
Let n be the total number of repeated trials, p be the probability of a success in a trial and
q be the probability of its failure so that q = 1 - p.

 Let r be a random variable which denotes the number of successes in n trials. The possible
values of r are 0, 1, 2, ...... n. We are interested in finding the probability of r successes out
of n trials, i.e., P(r).
 Binomial distribution is often used in various decision making situations in business.
Acceptance sampling plan, a technique of quality control, is based on this distribution.
With the use of sampling plan, it is possible to accept or reject a lot of items either at the
stage of its manufacture or at the stage of its purchase.

 The binomial distribution is not applicable when the probability of a success p does not
remain constant from trial to trial. In such a situation the probabilities of the various
values of r are obtained by the use of Hypergeometric distribution.
 When n, the number of trials become large, the computation of probabilities by using the
binomial probability mass function becomes a cumbersome task. Usually, when n ³ 20 and
p £ 0.05, Poisson distribution can be used as an approximation to binomial with parameter
m = np.

14.8 Keywords

Binomial distribution is a theoretical probability distribution which was given by James
Bernoulli.
Probability distribution: The purpose of fitting a distribution is to examine whether the observed
frequency distribution can be regarded as a sample from a population with a known probability
distribution.
Geometrical distribution: When r = 1, the Pascal distribution can be written as

P ( ) n = n 1 C pq n 1 = pq n 1 , where n = 1,2,3,.....
0
Geometrical distribution: Here n is a random variable which denotes the number of trials
required to get a success. This distribution is known as geometrical distribution.

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