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)

$\frac{77}{36}$

Answer 2

It is provided that,

$P(X = 3) = 5P(X = 4) \;and \;n = 7$

$⇒ ^7C_3p^3q^4 = 5⋅^7C_4p^4q^3$

$⇒ q = 5p$ and also $p + q = 1$

$⇒p=\frac{1}{6}$ and $q=\frac{5}{6}$

So, Mean =$\frac{7}{6}$ and variance =$\frac{35}{36}$

Therefore, $Mean + Variance$ = $\frac{7}{6}+\frac{35}{36}$

= $\frac{77}{36}$

Hence, the correct option is (C): $\frac{77}{36}$