Page 282 - DCOM203_DMGT204_QUANTITATIVE_TECHNIQUES_I

Unit 13: Binomial Probability Distribution

Notes
Pascal Distribution
Notes

In binomial distribution, we derived the probability mass function of the number of
successes in n (fixed) Bernoulli trials. We can also derive the probability mass function of
the number of Bernoulli trials needed to get r (fixed) successes. This distribution is known
as Pascal distribution. Here r and p become parameters while n becomes a random variable.
We may note that r successes can be obtained in r or more trials i.e. possible values of the
random variable are r, (r + 1), (r + 2), ...... etc. Further, if n trials are required to get r
successes, the nth trial must be a success. Thus, we can write the probability mass function
of Pascal distribution as follows:

Probability of r 1 successes Probability of a success
P n
out of n 1 trials in nth trial
n 1 r 1 n r n 1 r n r
C r 1 p q p C r 1 p q
where n = r, (r + 1), (r + 2), ... etc.

r rq
It can be shown that the mean and variance of Pascal distribution are p and p 2 respectively.

This distribution is also known as Negative Binomial Distribution because various values
- r
r
of P(n) are given by the terms of the binomial expansion of p (1 - q) .
Self Assessment

State whether the following statements are true or false:
1. 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.
2. It is not possible to formulate various laws either on the basis of given conditions (a priori
considerations) or on the basis of the results (a posteriori inferences) of an experiment.
3. If a random variable satisfies the conditions of a theoretical probability distribution, then
this distribution can be fitted to the observed data.

4. The knowledge of the theoretical probability distribution is of no use in the understanding
and analysis of a large number of business and economic situations.
5. It is possible to test a hypothesis about a population, to take decision in the face of
uncertainty, to make forecast, etc.
6. Theoretical probability distributions can be divided into two broad categories, viz. discrete
and continuous probability distributions.
7. Binomial distribution is a theoretical probability distribution which was given by James
Bernoulli.

8. In Binomial distribution, an experiment consists of a finite number of repeated trials.
9. Each trial has only two possible, mutually exclusive, outcomes which are termed as a
‘success’ or a ‘failure’.

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