An experiment is repeated 100 times and a fitted binomial distribution is obtained. Calculate the Expected frequencies for the fitted binomial distribution.

An experiment is repeated 100 times and a fitted binomial distribution is obtained as follows:

`\P(X = x) = C_x^5 (0.4)^x (0.6)^{n - x}, x = 0,1,2,3,4,5`

Calculate the Expected frequencies for the fitted binomial distribution.

Solution:

The expected frequency for a particular value of x in a binomial distribution is given by the product of the probability of that value and the total number of trials (in this case, 100). So for x = 0, the expected frequency is:

Expected frequency for `x = 0`

`\P(X = 0) * 100 = C_0^5 (0.4)^0 (0.6)^{5 - 0} * 100 = 0.077 * 100 = 7.7`

Similarly, we can calculate the expected frequencies for the other values of x:

Expected frequency for `x = 1`

`\P(X = 1) * 100 = C_1^5 (0.4)^1 (0.6)^{5 - 1} * 100 = 0.234 * 100 = 23.4`

Expected frequency for `x = 2`

`\P(X = 2) * 100 = C_2^5 (0.4)^2 (0.6)^{5 - 2} * 100 = 0.312 * 100 = 31.2`

Expected frequency for `x = 3`

`\P(X = 3) * 100 = C_3^5 (0.4)^3 (0.6)^{5 - 3} * 100 = 0.194 * 100 = 19.4`

Expected frequency for `x = 4`

`\P(X = 4) * 100 = C_4^5 (0.4)^4 (0.6)^{5 - 4} * 100 = 0.053 * 100 = 5.3`

Expected frequency for `x = 5`

`\P(X = 5) * 100 = C_5^5 (0.4)^5 (0.6)^{5 - 5} * 100 = 0.006 * 100 = 0.6`

So the expected frequencies for the fitted binomial distribution are 7.7, 23.4, 31.2, 19.4, 5.3, and 0.6 for x = 0, 1, 2, 3, 4, and 5, respectively.