Fit a binomial distribution of the following data

X 0 1 2 3 4 5 6
f 5 8 15 14 10 6 2

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Solution:

Let x be the random variable following Binomial distribution with parameter n = 6

x f fx
0 5 0
1 8 8
2 15 30
3 14 42
4 10 40
5 6 30
6 2 12
N = Σf = 60 Σfx = 162

N = 60 = total frequency

Then mean of observed data (x̄) = \(\frac{\sum fx}{\sum f}\)

= \(\frac{162}{60}\) = \(\frac{27}{10}\) = 2.7

Then np = \(\frac{27}{10}\)

or, 6 × p = \(\frac{27}{10}\)

or, p = \(\frac{27}{6 \times 10}\) = \(\frac{9}{20}\)

And

q = 1 – p = 1 – \(\frac{9}{20}\) = \(\frac{11}{20}\)

Then the probability mass function becomes

P(X=x) = p(x) = \(C(6, x)\left ( \frac{9}{20} \right )^x \left ( \frac{11}{20} \right )^{6-x}\)

The expected frequency is obtained by substituting the values of x = 0, 1, 2, 3, . . . . . .

x \(p(x) = C(6, x)\left ( \frac{9}{20} \right )^x \left ( \frac{11}{20} \right )^{6-x}\) Expected frequency = N × p(x)
0 0.277 1.662 2
1 0.1359 8.154 8
2 0.2708 16.248 16
3 0.3032 18.192 18
4 0.186 11.166 11
5 0.0608 3.654 4
6 0.0083 0.492 1
1 60 60
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