Question

For any discrete distribution, what is the value of $\beta_2$?

• A. $\beta_2 > 1$
• B. $\beta_2 \geq 1$
• C. $\beta_2 < 1$
• D. $\beta_2 \leq 1$

Hint:

Pearson's inequality states that $\beta_2 \ge \beta_1 + 1$. Since $\beta_1$ is a squared quantity (always non-negative), $\beta_2$ must be at least 1 for any valid probability distribution.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The coefficient $\beta_2$ (kurtosis) is a measure of the peakedness or flat-toppedness of a probability distribution curve.
It is mathematically defined using the moments of the distribution.
Key Formula or Approach:
The formula for $\beta_2$ is: \[ \beta_2 = \frac{\mu_4}{\mu_2^2} \] where $\mu_4$ is the fourth central moment and $\mu_2$ is the second central moment (variance).

Step 2: Detailed Explanation:
By Pearson's inequality, for any real-valued random variable, the relation between the skewness coefficient ($\beta_1$) and kurtosis ($\beta_2$) is: \[ \beta_2 \geq \beta_1 + 1 \]
Since $\beta_1 = \frac{\mu_3^2}{\mu_2^3} \geq 0$, it follows directly that: \[ \beta_2 \geq 1 \]
For any discrete distribution (and indeed any probability distribution with finite moments), the value of $\beta_2$ can never be less than 1.
In the limiting case of a discrete two-point symmetric distribution with equal probabilities, the value of $\beta_2$ reaches its absolute minimum value of 1.
Therefore, for any discrete distribution, the value of $\beta_2$ must be greater than or equal to 1.

Step 3: Final Answer:
The value of $\beta_2$ is $\beta_2 \geq 1$.