Page 283 - DCOM203_DMGT204_QUANTITATIVE_TECHNIQUES_I

Quantitative Techniques – I

Notes 10. Theoretical probability distribution gives us a law according to which different values of
the random variable are distributed with non-specified probabilities.

13.3 Fitting of Binomial Distribution

The fitting of a distribution to given data implies the determination of expected (or theoretical)
frequencies for different values of the random variable on the basis of this data.
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.
To fit a binomial distribution to the given data, we find its mean. Given the value of n, we can
compute the value of p and, using n and p, the probabilities of various values of the random
variable. These probabilities are multiplied by total frequency to give the required expected
frequencies. In certain cases, the value of p may be determined by the given conditions of the
experiment.

Example: The following data give the number of seeds germinating (X) out of 10 on
damp filter for 80 sets of seed. Fit a binomial distribution to the data.
X : 0 1 2 3 4 5 6 7 8 9 10
f : 6 20 28 12 8 6 0 0 0 0 0
Solution:
Here the random variable X denotes the number of seeds germinating out of a set of 10 seeds.
The total number of trials n = 10.

0 6 1 20 2 28 3 12 4 8 5 6 174
The mean of the given data X 2.175
80 80
2.175
Since mean of a binomial distribution is np, np = 2.175. Thus, we get p 0.22 (approx.)
10
Further, q = 1 - 0.22 = 0.78.
10 X 10 X
Using these values, we can compute P X C X 0.22 0.78 and then expected frequency
[= N × P(X)] for X = 0, 1, 2, ...... 10. The calculated probabilities and the respective expected
frequencies are shown in the following table:

Approximated Approximated
X P X N P X X P X N P X
Frequency Frequency
0 0.0834 6.67 6 6 0.0088 0.71 1
1 0.2351 18.81 19 7 0.0014 0.11 0
2 0.2984 23.87 24 8 0.0001 0.01 0
3 0.2244 17.96 18 9 0.0000 0.00 0
4 0.1108 8.86 9 10 0.0000 0.00 0
5 0.0375 3.00 3 Total 1.0000 80

13.3.1 Features of Binomial Distribution

1. It is a discrete probability distribution.
2. It depends upon two parameters n and p. It may be pointed out that a distribution is
known if the values of its parameters are known.

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