Question

Let \( X \) follow a beta distribution with parameters \( m (> 0) \) and 2. If \( P(X \leq \frac{1}{2}) = \frac{1}{2} \), then \( \text{Var}(X) \) equals

• A. \( \frac{1}{10} \)
• B. \( \frac{1}{20} \)
• C. \( \frac{1}{25} \)
• D. \( \frac{1}{40} \)

Hint:

For a beta distribution, the variance formula \( \text{Var}(X) = \frac{m \cdot 2}{(m + 2)^2 (m + 3)} \) is useful for calculating the spread of the distribution once the parameters are known.

Answer 1

The Correct Option is B

Step 1: Use the properties of the beta distribution.
For a beta distribution with parameters \( m \) and 2, the mean is \( \mu = \frac{m}{m + 2} \) and the variance is: \[ \text{Var}(X) = \frac{m \cdot 2}{(m + 2)^2 (m + 3)} \] We are given that \( P(X \leq \frac{1}{2}) = \frac{1}{2} \), which helps us solve for \( m \). Step 2: Solve for \( m \).
Using the condition \( P(X \leq \frac{1}{2}) = \frac{1}{2} \), we can solve for the value of \( m \), and then substitute into the formula for variance. Step 3: Calculate the variance.
After solving for \( m \), we find that the variance is \( \frac{1}{20} \). Step 4: Conclusion.
Thus, the correct answer is (B) \( \frac{1}{20} \).