Question

For a Binomial distribution with mean 4 and variance 2, the value of \( n \) is:

• A. \( 2 \)
• B. \( 4 \)
• C. \( 6 \)
• D. \( 8 \)

Hint:

For a Binomial Distribution: \[ E(X) = n p, \quad V(X) = n p (1 - p). \] Use these formulas to determine \( n \) and \( p \).

Answer 1

The Correct Option is C

Step 1: Using the binomial formulas. - Mean of a binomial distribution is given by: \[ E(X) = n p. \] - Variance of a binomial distribution is: \[ V(X) = n p (1 - p). \] Step 2: Substituting given values. \[ 4 = n p, \quad 2 = n p (1 - p). \] Step 3: Expressing \( p \) in terms of \( n \). \[ p = \frac{4}{n}. \] Step 4: Solving for \( n \). \[ 2 = n \left( \frac{4}{n} \right) (1 - \frac{4}{n}). \] \[ 2 = 4(1 - \frac{4}{n}). \] \[ \frac{2}{4} = 1 - \frac{4}{n}. \] \[ \frac{1}{2} = 1 - \frac{4}{n}. \] \[ \frac{4}{n} = \frac{1}{2}. \] \[ n = 6. \] Step 5: Selecting the correct option. Since \( n = 6 \), the correct answer is (C).