지수분포, 구간추정 (1)

\kern{10pt} f_{X_i}(x) = \lambda e^{-\lambda x}~I(x>0) \\[10pt] E(X_i) = {1\over\lambda}_, \kern{10pt} \hat\lambda = {1\over\overline{X}}_.

\begin{align*} 1-\alpha &= P\Big(a < \hat\lambda < b\Big) = P\bigg(a<{1\over\overline{X}}<b\bigg) \\[15pt] &= P\bigg({1\over b}<\overline{X} <{1\over a}\bigg) \\[15pt] &=P\bigg({n\over b}<\sum X_i< {n\over a}\bigg) \end{align*}

\begin{align*} &P\bigg({\lambda n \over b}< \lambda \sum X_i< {\lambda n\over a}\bigg) \\[20pt] &P\bigg({\lambda n\over b}<~U~< {\lambda n\over a}\bigg) \end{align*}

\begin{align*} &P\bigg(U_{1-{\alpha\over2}} < \lambda\sum X_i < U_{1-{\alpha\over2}}\bigg) \\[15pt] &P\Bigg( {U_{1-{\alpha\over2}} \over \sum X_i} < \lambda < {U_{{\alpha\over2}}\over\sum X_i}\Bigg) \end{align*}

\Bigg[ {U_{1-{\alpha\over2}} \over \sum X_{i}}_,\kern{5pt} {U_{{\alpha\over2}}\over\sum X_i}\Bigg]_. \kern{15pt} U \sim Gamma(n, 1)

P(U \le u_{1-{\alpha \over 2}}) = {\alpha \over 2}_. \kern{15pt} P(U \le u_{{\alpha \over 2}}) = {\alpha\over2}_.