Page 185 - DMTH404_STATISTICS

Unit 14: Theoretical Probability Distributions

Objectives Notes

After studying this unit, you will be able to:
 Discuss Binomial Distribution
 Describe Hypergeometric Distribution

 Explain Pascal Distribution
 Discuss Geometrical Distribution
 Describe Uniform Distribution (Discrete Random Variable)

 Explain Poisson Distribution
Introduction

The study of a population can be done either by constructing an observed (or empirical) frequency
distribution, often based on a sample from it, or by using a theoretical distribution. We have
already studied the construction of an observed frequency distribution and its various summary
measures. Now we shall learn a more scientific way to study a population through the use of
theoretical probability distribution of a random variable. It may be mentioned that a theoretical
probability distribution gives us a law according to which different values of the random
variable are distributed with specified probabilities. It is possible to formulate such laws either
on the basis of given conditions (a priori considerations) or on the basis of the results (a posteriori
inferences) of an experiment.
If a random variable satisfies the conditions of a theoretical probability distribution, then this
distribution can be fitted to the observed data.

14.1 Binomial Distribution

Binomial distribution is a theoretical probability distribution which was given by James
Bernoulli. This distribution is applicable to situations with the following characteristics:
1. An experiment consists of a finite number of repeated trials.

2. Each trial has only two possible, mutually exclusive, outcomes which are termed as a
'success' or a 'failure'.
3. The probability of a success, denoted by p, is known and remains constant from trial to
trial. The probability of a failure, denoted by q, is equal to 1 - p.
4. Different trials are independent, i.e., outcome of any trial or sequence of trials has no effect
on the outcome of the subsequent trials.
The sequence of trials under the above assumptions is also termed as Bernoulli Trials.

14.1.1 Probability Function or Probability Mass Function

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).

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