← 返回数学题库
414概率困难multi partlong

Renyi Representation of Exponential Order-Statistic Spacings

题目

Let X1,…,XnX_1, \ldots, X_n be iid Exp⁡(λ)\operatorname{Exp}(\lambda) and let X(1)≤⋯≤X(n)X_{(1)} \le \cdots \le X_{(n)} be the order statistics. Define the normalized spacings Dk=(n−k+1)(X(k)−X(k−1))D_k = (n-k+1)(X_{(k)} - X_{(k-1)}) for k=1,…,nk = 1, \ldots, n, where X(0)=0X_{(0)} = 0.

分步问题

分步 a

Show that D1,D2,…,DnD_1, D_2, \ldots, D_n are independent, each with distribution Exp⁡(λ)\operatorname{Exp}(\lambda).

提交作答后加载提示与解析。

分步 b

Using part (a), derive a formula for E[X(k)]E[X_{(k)}] in terms of nn, kk, and λ\lambda.

提交作答后加载提示与解析。

解题计时

0:00

提交作答时记录，用于后续平均用时统计。