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

JEE Main · 2022 · 27 Jun 2022 · Shift 1 · Probability

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 :

Answer: (C)

For binomial distribution

B (7, p)

, given

P (X = 3) = 5 P (X = 4)

(3 7) p^{3} (1 - p)^{4} = 5 (4 7) p^{4} (1 - p)^{3}

Simplifying:

\frac{( 3 7 )}{5 ( 4 7 )} = \frac{p}{1 - p} ⟹ \frac{1}{5} = \frac{p}{1 - p} ⟹ 1 - p = 5 p ⟹ p = \frac{1}{6}

Thus

q = 1 - p = \frac{5}{6}

.

Mean:

n p = 7 \times \frac{1}{6} = \frac{7}{6}

Variance:

n pq = 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}

JEE Main · 2022 · 27 Jun 2022 · Shift 1 · Single correct option