Question

The Binomial distribution with probability of success as $p$, will be symmetric if

• A. $p < 1/2$
• B. $p > 1/2$
• C. $p = 0$
• D. $p = 1/2$

Hint:

A binomial distribution is perfectly symmetric only when success and failure are equally likely ($p = q = 0.5$). For other values of $p$, it asymptotically approaches symmetry as the sample size $n$ becomes very large.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The skewness of a Binomial distribution depends entirely on its probability parameters $p$ (success) and $q = 1-p$ (failure).

Step 2: Detailed Explanation:
The skewness parameter $\gamma_1$ of a Binomial distribution is given by:
\[ \gamma_1 = \frac{q - p}{\sqrt{npq}} \]
For a distribution to be perfectly symmetric, its skewness must be equal to zero:
\[ \gamma_1 = 0 \implies q - p = 0 \implies q = p \]
Since $p + q = 1$, we can substitute $q = p$:
\[ p + p = 1 \implies 2p = 1 \implies p = \frac{1}{2} \]
- If $p < 1/2$, the distribution is positively skewed (skewed to the right).
- If $p > 1/2$, the distribution is negatively skewed (skewed to the left).
- If $p = 1/2$, the distribution is perfectly symmetric.
Therefore, the Binomial distribution is symmetric when $p = 1/2$.

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