Question

Determine the binomial distribution whose mean is $5$ and S.D. is $\sqrt{2.5}$.

Answer 1

Let the parameters of the distribution be $n,P$.

Given the mean is $5\Rightarrow np=5$

$S.D$ is $\sqrt{2.5}=\Rightarrow \sqrt{npq}=\sqrt{2.5}$

$\Rightarrow npq=2.5$ where $q=1-p$.

So, $q=\frac{2.5}{5}=\frac{1}{2}$

$p=1-q=1-\frac{1}{2}=\frac{1}{2}$

$np=5\Rightarrow n=10$

$\therefore$ The binomial distribution is ,

$P(x-r)={}^{10}{C}_r{\left(\frac{1}{2}\right)}^2{\left(\frac{1}{2}\right)}^{10-r}$ for $r=0,1,......10$

$={}^{10}{C}_r{\left(\frac{1}{2}\right)}^{10}$ for $r=0,1,.....10.$