Question

Find the binomial distribution whose mean is 5 and variance $\frac{10}{3}.$

Answer 1

Mean of binomial distribution, i.e. np =5

Variance, i.e. npq =$\frac{10}{3}$
$$\begin{gathered} q=\frac{\mathrm{Variance}}{\mathrm{Mean}}=\frac{2}{3} \\ \text{and}p=1-q=\frac{1}{3} \\ np\mathit{}=5 \\ \Rightarrow n=15 \\ \therefore P(X=r)={}^{15}C_r{\left(\frac{1}{3}\right)}^r{\left(\frac{2}{3}\right)}^{15-r},\mathit{}r=0,1,2......15 \end{gathered}$$