Question

Given below are two statements
Statement I: For normal distribution, all central moments exist.
Statement II: For normal distribution Mean deviation (MD): Quartile deviation (QD): Standard Deviation (SD) is 10:12:15.
In light of the above statements, choose the correct answer from the options given below

• A. Statement I is true but Statement II is false
• B. Statement I is false but Statement II is true
• C. Both Statement I and II are false
• D. Both Statement I and II are true

Hint:

Always remember the relative order of dispersion measures for a Normal distribution: $\text{QD} < \text{MD} < \text{SD}$, which directly maps to the ascending numerical order $10 < 12 < 15$.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This question evaluates fundamental properties and parameters of the Normal (Gaussian) distribution.

Step 2: Detailed Explanation:
Let us analyze the two statements:
- Statement I: The normal distribution has a moment-generating function that exists for all real values.
This mathematically guarantees that all of its central moments ($\mu_r$) exist, are finite, and are well-defined. Thus, Statement I is true.
- Statement II: For a normal distribution, the relationships between the standard deviation ($\sigma$), mean deviation ($\text{MD}$), and quartile deviation ($\text{QD}$) are:
\[ \text{Quartile Deviation (QD)} \approx 0.6745 \sigma \approx \frac{2}{3} \sigma \]
\[ \text{Mean Deviation (MD)} \approx 0.7979 \sigma \approx \frac{4}{5} \sigma \]
Let us find the ratio:
\[ \text{QD} : \text{MD} : \text{SD} \approx \frac{2}{3} \sigma : \frac{4}{5} \sigma : 1 \sigma \]
Multiply by 15 to clear the denominators:
\[ \text{QD} : \text{MD} : \text{SD} \approx 10 : 12 : 15 \]
The correct ratio is $\text{QD} : \text{MD} : \text{SD} = 10 : 12 : 15$.
However, Statement II states that $\text{MD} : \text{QD} : \text{SD}$ is $10:12:15$.
Since $\text{MD}$ is greater than $\text{QD}$ ($\text{MD} \approx 0.8\sigma > \text{QD} \approx 0.67\sigma$), their positions cannot be swapped in this numerical ratio.
Therefore, Statement II is false.
Thus, Statement I is true but Statement II is false.

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