Question

If X is binomially distributed with mean 3.20 and variance 1.152, find the complete binomial probability distribution

Answer 1

Let u - mean, d - variance

Since we have binpmial distribution, then u = np, d = np(1-p), where n - number of experiments, p - probability of succes in experiment

Then we have system of two equations $\begin{Bmatrix} np = 3.2 \\ np(1-p) = 1.152 \end{Bmatrix}$ . After solving it, we receive n = 5 and p = 0.64

So, given distribution is Bin(5, 0.64)

Complete binomial probability distribution is the sum of P(x≤n) for every n from 0 to 5, where x is the number of successful experiments. Let's mark the complete binomial probability distribution as P(n) for n from 0 to 5.

$P(0) = 0$

$P(1) = 0 + P(1) = {5 \choose 1}*(0.64)^{1}*(0.36)^{4} = 0.0537$

$P(2) = 0.0537 + P(2) =0.0537 {5 \choose 2}*(0.64)^{2}*(0.36)^{3}=0.245$

$P(3) = 0.245 + P(3) = 0.245+ {5 \choose 3}*(0.64)^{3}*(0.36)^{2}=0.585$

$P(4) = 0.585 + P(4) =0.585+ {5 \choose 4}*(0.64)^{4}*(0.36)^{1} = 0.887$

$P(5) = 1$