題組內容

1. (15%) Let \( X_i \) be an exponential random variable with the probability density function \[ f(x; \lambda_i) = \lambda_i exp(-\lambda_i x), x \ge 0; \lambda_i > 0 \] for \( i = 1, ..., n \). Assume \( X_i, i = 1, ..., n \), are independent. The detailed derivation has to be provided for all the following problem. No point is given when only answers are given.

(b) (5%) Assume \( n = 3 \), let \( X_{(i)}, i = 1, 2, 3 \), denote the order statistics of \( X_i, i = 1, 2, 3 \), where \( X_{(1)} \leq X_{(2)} \leq X_{(3)} \). Find the marginal distribution of \( X_{(3)} \).