Question

Let $X$ be a random variable having binomial distribution $B (7, p )$. If $P (X=3)=5 P (X=4)$, 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)

$B (7, p )$
$n =7\,\, p = p$
given
$P(x=3)=5 P(x=4)$
${ }^{7} C_{3} \times p^{3}(1-p)^{4}=5 \cdot{ }^{7} C_{4} p^{4}(1-p)^{3}$
$\frac{{ }^{7} C_{3}}{5 \times{ }^{7} C_{4}}=\frac{p}{1-p}$
$1- p =5 p$
$6 p =1$
$p=\frac{1}{6} \Rightarrow q=\frac{5}{6}$
$n =7$
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
$=\frac{7}{6}+\frac{35}{36}$
$=\frac{42+35}{36}$
$=\frac{77}{36}$