In a binomial distribution, the sum and the product of the mean and the variance are $(25 / 3)$ and $(50 / 3)$ respectively. Find the distribution.

Home » In a binomial distribution, the sum and the product of the mean and the variance are and respectively. Find the distribution.

In a binomial distribution, the sum and the product of the mean and the variance are $(25 / 3)$ and $(50 / 3)$ respectively. Find the distribution.

$\begin{array}{l} \text { Mean }+\text { Variance }=n p+n p q=n p(1+q)=25 / 3 \\ \text { Variance }=n^{2} p^{2} q=n 2=50 / 3 \ldots(i) \\ n^{2} p^{2}(1+q)^{2}=625 / 9 \ldots \text { (ii) } \end{array}$
Dividing (i) by (ii), we get,
$\begin{array}{l} \frac{q}{(q+1)^{2}}=\frac{\frac{50}{3}}{\frac{625}{9}}=\frac{6}{25} \\ \Rightarrow 6 q 2-13 q+6=0 \\ \Rightarrow q=2 / 3 \text { or } 3 / 2 \end{array}$
$\Rightarrow$ But as q can not be greater than 1 thus, $q=2 / 3$ .
$\begin{array}{l} \Rightarrow \mathrm{p}=1 / 3 \\ \Rightarrow \mathrm{n}=15 \end{array}$
Binomial distribution,
${ }^{15} \mathrm{C}_{\mathrm{r}} \cdot\left(\frac{1}{3}\right)^{\mathrm{r}} \cdot\left(\frac{2}{3}\right)^{(15-\mathrm{r})}$