題組內容

1. (50%) Let $X_1, X_2, \cdots, X_n$ be a random sample from an uniform distribution $$f(x; \theta) = \frac{1}{\theta_2 - \theta_1}, x \in (\theta_1, \theta_2); \theta_1 < \theta_2.$$ Let the distribution function of $f(x; \theta_1, \theta_2)$ be denoted as $F(x; \theta_1, \theta_2)$. The detailed derivation has to be provided for all the following problem. No point is given when only answers are given.

(e) (10%) Find the maximum likelihood estimate \( \hat{M} \) of the median of \( F(x; \theta_1, \theta_2) \) and the corresponding distribution of \( \hat{M} \).