Question

The skewness in a binomial distribution will be zero, if :

• A. p \(<\) 1/2
• B. p = 1/2
• C. p \(>\) 1/2
• D. p \(<\) q

Hint:

A Binomial distribution is perfectly symmetric when success and failure are equally likely (\(p = q = 0.5\)).
Symmetric distributions always have a skewness of zero.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Skewness is a measure of the asymmetry of a probability distribution about its mean.
For any probability distribution, a skewness value of zero indicates that the distribution is perfectly symmetric.
Key Formula or Approach:
For a Binomial distribution \(X \sim \text{Binomial}(n, p)\), the coefficient of skewness (\(\gamma_1\)) is given by:
\[ \gamma_1 = \frac{q - p}{\sqrt{n \cdot p \cdot q}} \]
Where \(q = 1 - p\) is the probability of failure.

Step 2: Detailed Explanation:
Let us set the skewness coefficient to zero to find the required condition:
\[ \gamma_1 = 0 \implies \frac{q - p}{\sqrt{n \cdot p \cdot q}} = 0 \]
Since the denominator \(\sqrt{n \cdot p \cdot q}\) is non-zero for any non-degenerate binomial distribution (\(n \ge 1\) and \(0 < p < 1\)), the numerator must be zero:
\[ q - p = 0 \]
\[ q = p \]
Since \(p + q = 1\), we substitute \(q = p\) into the equation:
\[ p + p = 1 \]
\[ 2p = 1 \implies p = \frac{1}{2} \]
When \(p = \frac{1}{2}\), the probability of success is equal to the probability of failure (\(q = \frac{1}{2}\)).
Under this condition, the Binomial distribution becomes perfectly symmetric, and its skewness is zero.
If \(p < \frac{1}{2}\), the distribution is positively skewed (skewed to the right).
If \(p > \frac{1}{2}\), the distribution is negatively skewed (skewed to the left).

Step 3: Final Answer:
The correct option is (B).