Paper 4: Fundamentals of Business Mathematics & Statistic

10.1 THEORETICAL DISTRIBUTION

Broadly speaking, the frequency distributions are of two types : Observed Frequency Distribution and
Theoretical Frequency Distribution. The distributions, which are based on actual data or experimentation
are called the observed frequency distribution. On the other hand, the distributions based on expectations
on the basis of past experience is known as Theoretical Frequency Distribution or Expected Frequency
Distribution or Probability Distributions. In short, the observed frequency distribution is based on actual
sample studies whereas the theoretical distribution is based on expectations on the basis of previous
experience or theoretical considerations. For example, we toss a coin 200 times. We may get 80 heads and
120 tails ; but our expectation is 100 heads and 100 tails, because the chance is 50% heads and 50% tails. On
the basis of this expectation we can test whether a given coin is unbiased or not. If a coin is tossed 100
times we may get 40 heads and 60 tails. This is our observation. Our expectation is 50% heads and 50% tails.
Now the question is whether this discrepancy is due to sampling fluctuation or is due to the fact that the
coin is biased. The word expected or expectation is used in the sense of an average. When a coin is tossed
for a large number of times, we will on an average get close to 50% heads and 50% tails. The following are
important distributions.

1. Binomial Distribution Discrete Probability Distribution


2. Poisson Distribution Discrete Probability Distribution


3. Normal Distribution Continuous Probability Distribution.

10.2 Binomial Distribution 10.

This distribution was discovered by a Swiss mathematician Jame Bernoulli (1654-1705) and is also known as
Bernoulli Distribution. He discovered this theory and published it in the year 1700 dealing with dichotomous
classification of events one possessing and the other not possessing. The probability of occurrence of an
event is p and its non-occurrence is q. The distribution can be used under the following conditions :

1. The number of trials is finite and fixed.


2. In every triail there are only two possible outcomes success or failure.


3. The trials are independent. The outcome of one trial does not affect the other trial.


4. p, the probability of success from trial to trial is fixed and q the probability of failure is equal to 1-p. This
   is the same in all the trials.
   For instance, a card is drawn from a pack of 52 cards. The probability of getting a king is 4/52. Before a
   second draw, the card drawn is replaced. But if the card is not replaced, we cannot have binomial
   distribution.
   Another example, a head or a tail can be had on a toss of coin ; a card drawn may be black or red ; an
   item inspected from a batch may be defective or non-defective. In each experiment the outcome can be
   classified as success or failure. Success is generally denoted by p and failure is 1 – p= q.

Study Note - 10

THEORETICAL DISTRIBUTION

FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS I 10.1

This Study Note includes

10.1 Theoretical Distribution

10.2 Binomial Distribution

10.3 Poisson Distribution 10.

10.4 Normal Distribution 10.