Question

Let \( \{X_n\}_{n \geq 1} \) be a sequence of i.i.d. random variables having common probability density function
\[ f(x) = \begin{cases} x e^{-x}, & x \geq 0 \\ 0, & \text{otherwise} \end{cases} \] Let \( \bar{X}_n = \frac{1}{n} \sum_{i=1}^{n} X_i \), \( n = 1, 2, \dots \). Then \[ \lim_{n \to \infty} P(\bar{X}_n = 2) \] equals

• A. 0
• B. \( \frac{1}{4} \)
• C. \( \frac{1}{2} \)
• D. 1

Hint:

The strong law of large numbers tells us that the sample mean converges almost surely to the expected value, meaning the probability of the sample mean equaling the expected value exactly is 0.

Answer 1

The Correct Option is A

Step 1: Understanding the distribution.
The random variables \( X_n \) are i.i.d. with a probability density function given by \( f(x) = x e^{-x} \), which is the gamma distribution with shape parameter 2 and rate parameter 1. The mean \( \mathbb{E}[X_n] \) is 2.
Step 2: Applying the law of large numbers.
By the strong law of large numbers, \( \bar{X}_n \) converges almost surely to the expected value \( \mathbb{E}[X_n] = 2 \). Therefore, the probability that \( \bar{X}_n \) equals exactly 2 becomes 0 as \( n \to \infty \).
Step 3: Conclusion.
Thus, the correct answer is (A) 0.