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.



(a)(10%) Find \( P[X_1 < X_2 < X_3] \) and \( P[X_1 < X_2 | \max(X_1, X_2, X_3) = X_3] \).