Question

In a binomial distribution $B(n, p)$, the sum and the product of the mean and the variance are 5 and 6 respectively, then $6(n+p-q)$ is equal to

• A. 50
• B. 53
• C. 52
• D. 51

Answer 1

The Correct Option is C

Using the formulas for mean and variance in a binomial distribution: \[ \text{Mean} = np, \quad \text{Variance} = npq \] Given: \[ np + npq = 5, \quad npq = 6 \] \[ np(1 + q) = 5, \quad 6(n p^2 q) = 6 \] \[ (1 + q)^2 = 25 \Rightarrow q^2 + 12q + 6 = 25q \] Solving for \( q \), we get: \[ q = \frac{2}{3} \] \[ p = \frac{1}{3} \] \[ n = 9 \] Now, computing \( 6(n + p - q) \): \[ 6(9 + \frac{1}{3} - \frac{2}{3}) = 52 \]