\contentsline {chapter}{\numberline {第1章}検定の前提となる事柄}{1}
\contentsline {section}{\numberline {1.1}確率変数}{1}
\contentsline {subsection}{\numberline {1.1.1}標本点、標本空間}{1}
\contentsline {subsection}{\numberline {1.1.2}$\sigma -$集合体}{1}
\contentsline {subsection}{\numberline {1.1.3}確率、確率空間}{2}
\contentsline {subsection}{\numberline {1.1.4}確率変数}{4}
\contentsline {subsection}{\numberline {1.1.5}確率分布}{5}
\contentsline {subsection}{\numberline {1.1.6}度数関数と密度関数}{5}
\contentsline {subsubsection}{離散的確率変数の場合}{6}
\contentsline {subsubsection}{連続的確率変数の場合}{6}
\contentsline {subsection}{\numberline {1.1.7}積率、平均および分散}{7}
\contentsline {subsection}{\numberline {1.1.8}積率母関数}{10}
\contentsline {section}{\numberline {1.2}多変数分布}{13}
\contentsline {subsection}{\numberline {1.2.1}離散的二変数分布}{14}
\contentsline {subsection}{\numberline {1.2.2}連続的二変数分布}{15}
\contentsline {subsection}{\numberline {1.2.3}二変数分布の積率}{17}
\contentsline {subsection}{\numberline {1.2.4}多変数分布の積率母関数}{19}
\contentsline {subsection}{\numberline {1.2.5}二変数分布の変数変換}{20}
\contentsline {subsubsection}{積率母関数による変数変換}{25}
\contentsline {section}{\numberline {1.3}標本分布}{25}
\contentsline {subsection}{\numberline {1.3.1}標本、確率標本}{25}
\contentsline {subsection}{\numberline {1.3.2}標本分布}{27}
\contentsline {subsection}{\numberline {1.3.3}統計量、統計量の分布}{27}
\contentsline {subsection}{\numberline {1.3.4}標本積率}{28}
\contentsline {section}{\numberline {1.4}正規分布}{31}
\contentsline {subsection}{\numberline {1.4.1}正規分布}{31}
\contentsline {subsubsection}{$E(X)=\intop \nolimits _{-\infty }^{\infty } xf(x)dx=\mu $ の証明}{31}
\contentsline {subsubsection}{$var(X)=\intop \nolimits _{-\infty }^{\infty } (x-\mu )^2 f(x) dx=\sigma ^2$の証明}{33}
\contentsline {subsection}{\numberline {1.4.2}正規分布を母集団とする標本集団の性質}{34}
\contentsline {section}{\numberline {1.5}ガンマ分布}{35}
\contentsline {section}{\numberline {1.6}カイ二乗分布}{37}
\contentsline {subsubsection}{正規変数の二乗の和の分布がカイ二乗分布になること}{38}
\contentsline {subsubsection}{$k \to \infty $ のときカイ二乗分布が正規分布に近付くことの証明}{39}
\contentsline {subsubsection}{母集団分布が正規分布の時、その標本分散はカイ二乗分布をすること}{40}
\contentsline {section}{\numberline {1.7}$t$ 分布}{41}
\contentsline {subsubsection}{$t$ 分布の平均と分散の証明}{43}
\contentsline {subsubsection}{$n \to \infty $ の時、$t$分布が正規分布に近付くことの証明}{44}
\contentsline {subsubsection}{標本平均値の分布}{45}
\contentsline {chapter}{\numberline {第2章}統計的決定理論}{46}
\contentsline {section}{\numberline {2.1}決定理論}{46}
\contentsline {subsection}{\numberline {2.1.1}決定理論の一般的手順}{46}
\contentsline {subsubsection}{(1) 推定の問題}{47}
\contentsline {subsubsection}{(2) 検定の問題}{48}
\contentsline {subsection}{\numberline {2.1.2}損失関数、危険関数}{48}
\contentsline {chapter}{\numberline {第3章}母数の推定}{51}
\contentsline {section}{\numberline {3.1}点推定}{51}
\contentsline {subsection}{\numberline {3.1.1}点推定量の条件}{53}
\contentsline {subsubsection}{不偏性}{53}
\contentsline {subsubsection}{一致性}{54}
\contentsline {subsubsection}{効率(有効性)}{55}
\contentsline {subsection}{\numberline {3.1.2}点推定の方法}{56}
\contentsline {subsection}{\numberline {3.1.3}積率法}{56}
\contentsline {subsection}{\numberline {3.1.4}最大尤度法}{58}
\contentsline {section}{\numberline {3.2}区間推定}{61}
\contentsline {subsubsection}{信頼区間の意味}{62}
\contentsline {subsection}{\numberline {3.2.1}正規母集団の平均値の区間推定}{63}
\contentsline {subsection}{\numberline {3.2.2}正規母集団の分散の区間推定}{64}
\contentsline {chapter}{\numberline {第4章}検定}{67}
\contentsline {section}{\numberline {4.1}検定の基礎事項}{67}
\contentsline {subsection}{\numberline {4.1.1}検定とは}{67}
\contentsline {subsection}{\numberline {4.1.2}検定で注意すべき点}{68}
\contentsline {subsection}{\numberline {4.1.3}検出力関数}{70}
\contentsline {subsection}{\numberline {4.1.4}尤度比検定}{71}
\contentsline {subsubsection}{尤度比検定に関する定理 1}{72}
\contentsline {subsubsection}{尤度比検定に関する定理 2}{72}
\contentsline {subsubsection}{尤度比検定についての定理 3}{76}
\contentsline {section}{\numberline {4.2}母平均の検定}{78}
\contentsline {subsection}{\numberline {4.2.1}正規母集団の母平均の検定の一般的手順}{78}
\contentsline {subsection}{\numberline {4.2.2}母分散$\sigma ^2$ が既知の場合}{79}
\contentsline {subsection}{\numberline {4.2.3}母分散$\sigma ^2$ が未知の場合}{79}
\contentsline {subsection}{\numberline {4.2.4}応用例}{79}
\contentsline {section}{\numberline {4.3}2つの正規母集団の母平均の差の検定}{80}
\contentsline {subsection}{\numberline {4.3.1}2つの母分散$\sigma _x^2,\sigma _y^2$ が既知の場合}{80}
\contentsline {subsection}{\numberline {4.3.2}2つの母分散$\sigma _x^2,\sigma _y^2$ は未知だが、$\sigma _1^2=\sigma _2^2$ と仮定できる場合}{81}
\contentsline {subsubsection}{応用例}{82}
\contentsline {subsection}{\numberline {4.3.3}母分散$\sigma _x^2,\sigma _y^2$ が共に未知の場合}{83}
\contentsline {section}{\numberline {4.4}無相関の検定}{83}
\contentsline {chapter}{\numberline {第5章}未解決な問題}{84}
\contentsline {chapter}{Appendix A. 正規分布の重要性とその応用}{85}
\contentsline {chapter}{Appendix B. ガンマ関数とその性質}{86}
\contentsline {chapter}{Appendix C. ベータ関数とその性質}{88}
\contentsline {chapter}{Appendix D. 用語}{90}
\contentsline {section}{\numberline {}{参考文献}}{90}