認為𝑌1,…,𝑌𝑛∼iidExp(1)Y1,…,Yn∼iidExp(1)Y_1, dots, Y_n overset{text{iid}}{sim} text{Exp}(1).展示∑𝑛𝑖=1(𝑌𝑖−𝑌(1))∼Gamma(𝑛−1,1)∑i=1n(Yi−Y(1))∼Gamma(n−1,1)sum_{i=1}^{n}(Y_i - Y_{(1)}) sim text{Gamma}(n-1, 1)

判斷以下陳述是否正確的最簡單方法是什麼*？*

認為. 展示.

注意.

經過， 這意味著.

很容易看出. 此外，我們還有參數化下

給出西安答案的解決方案：使用原始問題中的符號：

由此，我們得到.

證明在所有隨機生成書籍之母Devroye’s Non-uniform Random Variate Generation中給出，第 211 頁（這是一本非常優雅的書！）：

**定理 2.3 (Sukhatme, 1937)**如果我們定義然後歸一化指數間距

來自訂單統計一個 iid 指數樣本的大小 本身是獨立同分佈的指數變量

**證明。**自從

the joint density of the order statistic writes as

Setting , the change of variables from to has a constant Jacobian [incidentally equal to but this does not need to be computed] and hence the density of is proportional to which establishes the result. Q.E.D. An alternative suggested to me by Gérard Letac is to check that

has the same distribution as (by virtue of the memoryless property), which makes the derivation of straightforward.

引用自：https://stats.stackexchange.com/questions/272385