Question:
Let \( X_1, X_2, X_3 \) be a random sample from a distribution with the probability density function
\[
f(x|\theta) = \frac{1}{\theta} e^{-x/\theta}, \quad x>0, \ \theta>0
\]
Which of the following estimators of \( \theta \) has the smallest variance for all \( \theta>0 \)?
Let \( X_1, X_2, X_3 \) be a random sample from a distribution with the probability density function
\[ f(x|\theta) = \frac{1}{\theta} e^{-x/\theta}, \quad x>0, \ \theta>0 \] Which of the following estimators of \( \theta \) has the smallest variance for all \( \theta>0 \)?
\[ f(x|\theta) = \frac{1}{\theta} e^{-x/\theta}, \quad x>0, \ \theta>0 \] Which of the following estimators of \( \theta \) has the smallest variance for all \( \theta>0 \)?
Show Hint
The sample mean of i.i.d. random variables is the most efficient estimator, meaning it has the smallest variance.
Updated On: Nov 18, 2025
- \( \frac{X_1 + X_3 + X_5}{3} \)
- \( \frac{X_1 + X_2 + X_3}{4} \)
- \( \frac{X_1 + X_2 + X_3}{3} \)
- \( \frac{X_1 + X_2 + X_3}{6} \)
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The Correct Option is C
Solution and Explanation
Step 1: Understanding the distribution.
The distribution is an exponential distribution with parameter \( \theta \), and the mean of \( X_i \) is \( \theta \).
Step 2: Finding the variance of the estimators.
The most efficient estimator of \( \theta \) is the sample mean of the independent and identically distributed (i.i.d.) random variables \( X_1, X_2, X_3 \). The variance of the sample mean is minimized because the variance of the mean of i.i.d. random variables is \( \frac{\theta^2}{3} \), which is smaller than for any other linear combination of the \( X_i \)'s.
Step 3: Conclusion.
The estimator \( \frac{X_1 + X_2 + X_3}{3} \) has the smallest variance for all \( \theta>0 \), so the correct answer is (C).
The distribution is an exponential distribution with parameter \( \theta \), and the mean of \( X_i \) is \( \theta \).
Step 2: Finding the variance of the estimators.
The most efficient estimator of \( \theta \) is the sample mean of the independent and identically distributed (i.i.d.) random variables \( X_1, X_2, X_3 \). The variance of the sample mean is minimized because the variance of the mean of i.i.d. random variables is \( \frac{\theta^2}{3} \), which is smaller than for any other linear combination of the \( X_i \)'s.
Step 3: Conclusion.
The estimator \( \frac{X_1 + X_2 + X_3}{3} \) has the smallest variance for all \( \theta>0 \), so the correct answer is (C).
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