Question

If kurtosis is defined as \[ \left(\frac{m_4}{\sigma^4}-3\right), \]
where \(m_4\) is the \(4^{th}\) order central moment and \(\sigma\) is the standard deviation, then its value is positive for

• A. a leptokurtic distribution
• B. a mesokurtic distribution
• C. a platykurtic distribution
• D. a normal distribution

Hint:

Positive excess kurtosis indicates a sharper peak and heavier tails than the normal distribution, which is characteristic of leptokurtic distributions.

Answer 1

The Correct Option is A

Step 1: Recall the definition of kurtosis.
Kurtosis is a measure of the peakedness or flatness of a probability distribution. The excess kurtosis is defined as
\[ \beta_2-3=\frac{m_4}{\sigma^4}-3 \] where
\[ m_4=\text{fourth central moment} \] and
\[ \sigma=\text{standard deviation}. \]

Step 2: Understand the types of kurtosis.
\[ \text{Excess kurtosis} > 0 \] indicates a leptokurtic distribution.
\[ \text{Excess kurtosis} = 0 \] indicates a mesokurtic distribution such as the normal distribution.
\[ \text{Excess kurtosis} < 0 \] indicates a platykurtic distribution.

Step 3: Analyze the options.

(A) Leptokurtic distribution: Has positive excess kurtosis. Correct.

(B) Mesokurtic distribution: Has zero excess kurtosis. Incorrect.

(C) Platykurtic distribution: Has negative excess kurtosis. Incorrect.

(D) Normal distribution: Excess kurtosis is zero. Incorrect.

Step 4: Final conclusion.
Therefore, kurtosis defined as
\[ \left(\frac{m_4}{\sigma^4}-3\right) \] is positive for a leptokurtic distribution.
\[ \boxed{\text{Leptokurtic distribution}} \]