MCQ
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$\frac{105}{16}$
- ✓$\frac{77}{36}$
- C$\frac{7}{16}$
- D$\frac{49}{16}$
$n =7 \quad p = p$
given
$P(x=3)=5 P(x=4)$
${ }^{7} C_{3} \times p^{3}(1-p)^{4}=5^{7} C_{4} p^{4}(1-p)^{3}$
$\frac{{ }^{7} C_{3}}{5 \times{ }^{7} C_{4}}=\frac{p}{1-p}$
$1- p =5 p$
$6 p =1$
$p=\frac{1}{6} \Rightarrow q=\frac{5}{6}$
$n =7$
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
$=\frac{7}{6}+\frac{35}{36}$
$=\frac{42+35}{36}$
$=\frac{77}{36}$
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