Question

Let \( X_1, X_2, X_3 \) be a random sample from a distribution with the probability density function
\[ f(x|\theta) = \begin{cases} \frac{1}{\theta} e^{-x/\theta}, & \text{if } x > 0, \, \theta > 0 \\ 0, & \text{otherwise} \end{cases} \] If \( \hat{X}(X_1, X_2, X_3) \) is an unbiased estimator of \( \theta \), which of the following CANNOT be attained as a value of the variance of \( \hat{X} \) at \( \theta = 1 \)?

• A. 0.1
• B. 0.2
• C. 0.3
• D. 0.5

Hint:

For exponential distributions, the variance of the sample mean is proportional to \( \theta^2 \), and this determines the upper bound for the variance.

Answer 1

The Correct Option is A, B, C

Step 1: Understanding the distribution.
The given distribution is an exponential distribution with rate \( \lambda = 1/\theta \). The expected value of the sample mean \( \hat{X} \) is \( E[\hat{X}] = \theta \), and the variance of the sample mean is \( \text{Var}(\hat{X}) = \frac{\theta^2}{3} \).
Step 2: Calculate the variance for \( \theta = 1 \).
At \( \theta = 1 \), the variance is: \[ \text{Var}(\hat{X}) = \frac{1^2}{3} = \frac{1}{3} \approx 0.3333. \] Thus, values such as 0.1, 0.2, and 0.3 are possible, but 0.5 cannot be attained as a value of the variance.
Step 3: Conclusion.
The correct answer is (A), (B), and (C) because all of these values are smaller than or equal to the maximum attainable variance, which is \( \frac{1}{3} \).