Question

Let \( X_1, X_2, \ldots, X_n \) be a random sample from an \( \text{Exp}(\lambda) \) distribution, where \( \lambda \in \{1, 2\} \). For testing \( H_0: \lambda = 1 \) against \( H_1: \lambda = 2 \), the most powerful test of size \( \alpha \), \( \alpha \in (0,1) \), will reject \( H_0 \) if and only if

• A. \( \sum_{i=1}^{n} X_i \leq \frac{1}{2} \chi^2_{2n,1-\alpha} \)
• B. \( \sum_{i=1}^{n} X_i \geq 2 \chi^2_{2n,1-\alpha} \)
• C. \( \sum_{i=1}^{n} X_i \leq \frac{1}{2} \chi^2_{n,1-\alpha} \)
• D. \( \sum_{i=1}^{n} X_i \geq 2 \chi^2_{n,1-\alpha} \)

Answer 1

The Correct Option is A

1. Likelihood Ratio Test: The likelihood ratio test is based on: \[ \Lambda = \frac{L(H_0)}{L(H_1)}, \] where the test statistic is proportional to \( \sum_{i=1}^n X_i \). 2. Critical Region: Under \( H_0 \), \( \sum_{i=1}^n X_i \sim \frac{1}{2} \chi^2_{2n} \). The most powerful test rejects \( H_0 \) if: \[ \sum_{i=1}^n X_i \leq \frac{1}{2} \chi^2_{2n, 1-\alpha}. \]