Question

Let $X$ be a random variable having binomial distribution $B(7,p)$. If $P(X=3)=5P(X=4)$, then the sum of the mean and the variance of $X$ is :

• A. $\frac{105}{16}$
• B. $\frac{7}{16}$
• C. $\frac{77}{36}$
• D. $\frac{49}{16}$

Answer 1

Answer: (C)

$B(7,p)$
$n=7p=p$
given
$P(x=3)=5P(x=4)$
${}^7{C}_3\times p^3(1-p{)}^4=5\cdot {}^7{C}_4{p}^4(1-p{)}^3$
$\frac{{}^7{C}_3}{5\times {}^7{C}_4}=\frac{p}{1-p}$
$1-p=5p$
$6p=1$
$p=\frac{1}{6}\Rightarrow q=\frac{5}{6}$
$n=7$
Mean $=np=7\times \frac{1}{6}=\frac{7}{6}$
Var $=npq=7\times \frac{1}{6}\times \frac{5}{6}=\frac{35}{36}$
Sum
$=\frac{7}{6}+\frac{35}{36}$
$=\frac{42+35}{36}$
$=\frac{77}{36}$