Question

For the binomial distribution ${(p+q)}^n$ whose mean is $20$ and variance is $16$, the pair $(n,p)$ is

• A. $\left(100,\frac{1}{5}\right)$
• B. $\left(100,\frac{2}{5}\right)$
• C. $\left(50,\frac{1}{5}\right)$
• D. $\left(50,\frac{2}{5}\right)$

Answer 1

The correct option is A

$\left(100,\frac{1}{5}\right)$

Explanation for the correct answer:

Find the value of $\left(n,p\right)$

$\begin{array}{rcl}Mean& =& np\\ 20& =& np\_\_\_\_\_\left(1\right)\\ Variance& =& npq\\ 16& =& npq\_\_\_\_\_\left(2\right)\end{array}$

Divide$\left(2\right)$ by $\left(1\right)$

$\begin{array}{rcl}\frac{npq}{np}& =& \frac{16}{20}\\ q& =& \frac{4}{5}\\ p& =& 1-\frac{4}{5}\left[\mathbf{\because }\mathit{p}\mathbf{+}\mathit{q}\mathbf{=}\mathbf{1}\right]\\ & =& \frac{1}{5}\end{array}$

$\begin{array}{rcl}n\left(\frac{1}{5}\right)& =& 20\left[\mathit{S}\mathit{u}\mathit{b}\mathit{s}\mathit{t}\mathit{i}\mathit{t}\mathit{u}\mathit{t}\mathit{e}\mathbf{}\mathit{t}\mathit{h}\mathit{e}\mathbf{}\mathit{v}\mathit{a}\mathit{l}\mathit{u}\mathit{e}\mathbf{}\mathit{o}\mathit{f}\mathbf{}\mathit{p}\mathbf{}\mathit{i}\mathit{n}\mathbf{}\mathbf{\left(}\mathbf{1}\right)\right]\\ \mathit{n}& \mathbf{=}& \mathbf{100}\end{array}$

Hence, Option(A) is the correct answer.