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 (\(p = q = 0.5\)).
Symmetrical distributions always have a skewness coefficient of zero.

Answer 1

The Correct Option is B

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

Step 2: Detailed Explanation:
Let us set the skewness coefficient of the Binomial distribution to zero:
\[ \gamma_1 = 0 \implies \frac{q-p}{\sqrt{n p q}} = 0 \]
Since the denominator \(\sqrt{n p q}\) is non-zero (assuming \(0 < p < 1\) and \(n \ge 1\)), we must have:
\[ q - p = 0 \implies q = p \]
Since \(p + q = 1\), substitute \(q = p\) into the equation:
\[ p + p = 1 \implies 2p = 1 \implies p = \frac{1}{2} \]
When \(p = 1/2\), the probability of success is equal to the probability of failure (\(q = 1/2\)).
At this value, the Binomial distribution becomes perfectly symmetrical, resulting in a skewness of zero.
If \(p < 1/2\), the distribution is positively skewed.
If \(p > 1/2\), the distribution is negatively skewed.

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