Question

Let \(X\) follows a binomial \((n, p)\) distribution, the skewness of the distribution will be zero if:

• A. \(p = \frac{1}{2}\)
• B. \(p = \frac{3}{4}\)
• C. \(p = \frac{2}{3}\)
• D. \(p = 1\)

Hint:

Exam Tip:
For the binomial distribution:

• Skewness \(= \frac{1 - 2p}{\sqrt{np(1 - p)}}\).
• Symmetric when \(p = 0.5\).
• Skewed right when \(p < 0.5\).
• Skewed left when \(p > 0.5\).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Skewness measures the asymmetry of a distribution. For a binomial distribution, the skewness is given by: \[ \text{Skewness} = \frac{1 - 2p}{\sqrt{np(1 - p)}} \]

Step 2: Key Formula or Approach:
For skewness to be zero, the numerator must be zero: \[ 1 - 2p = 0 \Rightarrow p = \frac{1}{2} \]

Step 3: Detailed Explanation:
The binomial distribution is symmetric when \(p = 0.5\).
When \(p < 0.5\), the distribution is positively skewed (tail to the right).
When \(p > 0.5\), the distribution is negatively skewed (tail to the left).
So, the skewness is zero only when \(p = 0.5\).

Step 4: Final Answer:
Therefore, option (A) is correct.