題組內容

2. 設$X_1, X_2, \cdots, X_n, n \geq 1$, 為i.i.d. $\mathcal{U}(0, 1)$分佈之隨機變數, $X_{(1)} \leq X_{(2)} \leq \cdots \leq X_{(n)}$表其順序統計量。對1 < k < n, 試求

(i) 給定$X_{(k)} = a, 0 < a < 1, X_{(1)}, \cdots, X_{(k-1)}$之條件分佈,