지수분포, 구간추정 (2)

\kern{10pt} f_{X_i}(x) = \lambda e^{-\lambda x}~I(x>0) \\[10pt] \mu = {1\over\lambda}_, \kern{10pt} \sigma^2 = {1\over\lambda^2}

\sqrt{n}~\bigg(\overline{X}-{1\over n}\bigg) \sim N\bigg(0_,~~\frac{1}{\lambda^2}\bigg)

\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( \dfrac{{1\over b}-{1 \over\lambda}} {{1\over\lambda} \cdot{1\over\sqrt{n}}}< \dfrac{{\overline{X}}-{1 \over\lambda}} {{1\over\lambda} \cdot{1\over\sqrt{n}}} <\dfrac{{1\over a}-{1 \over\lambda}} {{1\over\lambda} \cdot{1\over\sqrt{n}}}\bigg) \\[20pt] &\kern{5pt}if.~n \to \infty \\[5pt] &=P\bigg( \dfrac{{1\over b}-{1 \over\lambda}} {{1\over\lambda} \cdot{1\over\sqrt{n}}}< Z <\dfrac{{1\over a}-{1 \over\lambda}} {{1\over\lambda} \cdot{1\over\sqrt{n}}}\bigg) \end{align*}

\begin{align*} P(a<\hat\lambda<b) &\backsimeq P\Big( -Z_{a\over2}<\sqrt{n}~\lambda~\big(\overline{X}-\textstyle{1\over \lambda}\big)< Z_{a\over2} \Big) \\[10pt] &= P\Big( -Z_{a\over2}<\sqrt{n}~\big(\lambda\overline{X}-1\big)< Z_{a\over2} \Big) \end{align*}

P\bigg(1-{Z_{a\over2}\over \sqrt{n}} < \lambda\overline{X} < 1+{Z_{a\over2}\over \sqrt{n}} \bigg)

P\Bigg({1\over \overline{X}} -{Z_{a\over2}\over \overline{X}\sqrt{n}} < \lambda < {1\over \overline{X}} +{Z_{a\over2}\over \overline{X}\sqrt{n}} \Bigg)

{1\over \overline{X}}~-~ Z_{a\over2} \cdot {1\over \overline{X}\sqrt{n}} \kern{15pt} \longrightarrow \kern{15pt} \hat\lambda~\pm~ Z_{a\over2} \cdot {\lambda \over \sqrt{n}}