Question

The mean and variance of a random variable $X$ having a binomial distribution are $4$ and $2$ respectively, find the value of $P(X=1)$ .

• A. $1/4$
• B. $1/16$
• C. $1/8$
• D. $1/32$

Answer 1

Answer: (D)

$1/32$

Answer 2

Given, mean, $np=4$ ?.(i) Variance, $npq=2$ ?.(ii) $\frac{npq}{np}=\frac{2}{4}$
$\Rightarrow$ $q=\frac{1}{2}$
$\therefore$ $p=\frac{1}{2}$ From Eq (i), we get $n=\frac{4}{1/2}=8$ Now, $P(X=1){{=}^{n}}{{C}_{1}}{{p}^{1}}{{q}^{n-1}}$
${{\left( \frac{1}{2} \right)}^{1}}{{\left( \frac{1}{2} \right)}^{7}}=\frac{8!}{1!7!}\times \frac{1}{{{2}^{8}}}=\frac{8}{{{2}^{8}}}=\frac{1}{32}$