# Fit a binomial distribution of the following data X 0 1 2 3 4 5 6 f 5 8 15 14 10 6 2

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