東京大学学際情報学府学際情報学専攻生物統計情報学コース 2020年度専門科目第2問

Author

Description

Kai

(2-1)

\[ \begin{aligned} E[X_1] &= \int_{- \infty}^\infty x f(x; \gamma) dx \\ &= \frac{1}{\gamma} \int_0^\infty x e^{- x / \gamma } dx \\ &= - \int_0^\infty x \left( e^{- x / \gamma } \right)' dx \\ &= - \left[ x e^{- x / \gamma } \right]_0^\infty + \int_0^\infty e^{- x / \gamma } dx \\ &= - \gamma \left[ e^{- x / \gamma } \right]_0^\infty \\ &= \gamma \end{aligned} \]

(2-2)

\[ \begin{aligned} P \left( X_{(1)} \leq x \right) &= 1 - P \left( X_{(1)} \gt x \right) \\ &= 1 - P \left( X_1 \gt x \text{ and } X_2 \gt x \text{ and } \cdots \text{ and } X_n \gt x \right) \\ &= 1 - P ( X_1 \gt x ) P ( X_2 \gt x ) \cdots P ( X_n \gt x ) \\ &= 1 - \left( \frac{1}{\gamma} \int_x^\infty e^{ - y / \gamma } dy \right)^n \\ &= 1 - \left( - \left[ e^{ - y / \gamma } \right]_x^\infty \right)^n \\ &= 1 - \left( e^{ - x / \gamma } \right)^n \\ &= 1 - e^{ - nx / \gamma } \end{aligned} \]

であるから、

\[ \begin{aligned} f_{(1)}(x; \gamma) &= \frac{d}{dx} \left( 1 - e^{ - nx / \gamma } \right) \\ &= \frac{n}{\gamma} e^{ - nx / \gamma } \\ E[X_{(1)}] &= \frac{\gamma}{n} \end{aligned} \]

(2-3)

(2-4)

\[ \begin{aligned} T_n &= \frac{1}{n-1} \sum_{i=2}^n X_{(i)} \\ &= \frac{1}{n-1} \sum_{i=1}^n X_{(i)} - \frac{1}{n-1} X_{(1)} \\ &= \frac{1}{n-1} \sum_{i=1}^n X_i - \frac{1}{n-1} X_{(1)} \end{aligned} \]