Question

Let 

be the order statistics corresponding to a random sample of size 5 from a uniform distribution on \( [0, \theta] \), where \( \theta \in (0, \infty) \). Then which of the following statements is/are true?} 
P: \( 3X_{(2)} \) is an unbiased estimator of \( \theta \). 
Q: The variance of \( E[2X_{(3)} \mid X_{(5)}] \) is less than or equal to the variance of \( 2X_{(3)} \).

• A. P only
• B. Q only
• C. Both P and Q
• D. Neither P nor Q

Hint:

For order statistics in a uniform distribution, the properties of their means and variances are well-established and can be used to evaluate unbiased estimators and variances.

Answer 1

The Correct Option is C

For order statistics of a uniform distribution on \( [0, \theta] \), the mean and variance for different order statistics have well-known properties. 
Step 1: Check statement P. 
The second order statistic \( X_{(2)} \) in a uniform distribution on \( [0, \theta] \) has the property that \( E[X_{(2)}] = \frac{2\theta}{6} \), and therefore, \( 3X_{(2)} \) is an unbiased estimator of \( \theta \), confirming P is true. 
Step 2: Check statement Q. 
The variance of \( E[2X_{(3)} \mid X_{(5)}] \) is indeed less than or equal to the variance of \( 2X_{(3)} \) based on the properties of conditional variance in order statistics, confirming Q is true. 
Final Answer: \[ \boxed{\text{(C) Both P and Q}} \]