正态总体参数的区间估计(应用概率统计陈魁清华大学出版社)

正态总体参数的区间估计

一、单个正态总体参数的区间估计
二、两个正态总体参数的区间估计

一、单个正态总体参数的区间估计

总体X∼N(μ,σ2),E(X)=μ,V(X)=σ2,X\sim N(\mu,\sigma^2),\ E(X)=\mu,\ V(X)=\sigma^2,X∼N(μ,σ2), E(X)=μ, V(X)=σ2,
样本为X1,X2,...,Xn,Xi∼N(μ,σ2),i=1,2,...,n.X_1,X_2,...,X_n,\ X_i\sim N(\mu,\sigma^2),\ i=1,2,...,n.X1,X2,...,Xn, Xi∼N(μ,σ2), i=1,2,...,n.
X‾=1n∑i=1nXi,X‾∼N(μ,σ2n),Xi−μσ∼N(0,1).\overline{X}=\frac{1}{n}\sum\limits_{i=1}^nX_i,\ \overline{X}\sim N\Big(\mu,\frac{\sigma^2}{n}\Big),\ \frac{X_i-\mu}{\sigma}\sim N(0,1). X=n1i=1∑nXi, X∼N(μ,nσ2), σXi−μ∼N(0,1).

μ\muμ的置信区间

σ2\sigma^2σ2	统计量及分布	置信度	置信区间(μ‾,μ‾)(\underline{\mu},\overline{\mu})(μ,μ)	单侧置信限
已知	Z＝X‾−μσ/n∼N(0,1)Z＝\frac{\overline{X}-\mu}{\sigma/\sqrt{n}}\sim N(0,1)Z＝σ/nX−μ∼N(0,1)	1−α1-\alpha1−α	(X‾∓δ),δ＝zα2σn(\overline{X}\mp \delta),\ \delta＝z_{\frac{\alpha}{2}}\frac{\sigma}{\sqrt{n}}(X∓δ), δ＝z2αnσ	μ‾=X‾−δ,μ‾=X‾+δ,δ＝zασn\underline{\mu}=\overline{X}-\delta,\overline{\mu}=\overline{X}+\delta,\ \delta＝z_{\alpha}\frac{\sigma}{\sqrt{n}}μ=X−δ,μ=X+δ, δ＝zαnσ
未知	T＝X‾−μS/n∼t(n−1)T＝\frac{\overline{X}-\mu}{S/\sqrt{n}}\sim t(n-1)T＝S/nX−μ∼t(n−1)	1−α1-\alpha1−α	(X‾∓δ),δ＝tα2(n−1)Sn(\overline{X}\mp \delta),\ \delta＝t_{\frac{\alpha}{2}}(n-1)\frac{S}{\sqrt{n}}(X∓δ), δ＝t2α(n−1)nS	μ‾=X‾−δ,μ‾=X‾+δ,δ＝tα(n−1)Sn\underline{\mu}=\overline{X}-\delta,\overline{\mu}=\overline{X}+\delta,\ \delta＝t_{\alpha}(n-1)\frac{S}{\sqrt{n}}μ=X−δ,μ=X+δ, δ＝tα(n−1)nS

σ2\sigma^2σ2的置信区间

μ\muμ	统计量及分布	置信度	置信区间(σ‾2,σ‾2)(\underline\sigma^2,\overline\sigma^2)(σ2,σ2)	置信区间(σ‾,σ‾)(\underline\sigma,\overline\sigma)(σ,σ)	单侧置信限
已知	χ2＝∑i＝1n(Xi−μσ)2∼χ2(n)\chi^2＝\sum\limits_{i＝1}^{n}\Big(\frac{X_i-\mu}{\sigma}\Big)^2\sim\chi^2(n)χ2＝i＝1∑n(σXi−μ)2∼χ2(n)	1−α1-\alpha1−α	(∑i=1n(Xi−μ)2χα22(n),∑i=1n(Xi−μ)2χ1−α22(n))\Bigg(\frac{\sum\limits_{i=1}^n(X_i-\mu)^2}{\chi^2_{\frac{\alpha}{2}}(n)},\frac{\sum\limits_{i=1}^n(X_i-\mu)^2}{\chi^2_{1-\frac{\alpha}{2}}(n)}\Bigg)(χ2α2(n)i=1∑n(Xi−μ)2,χ1−2α2(n)i=1∑n(Xi−μ)2)	(∑i=1n(Xi−μ)2χα22(n),∑i=1n(Xi−μ)2χ1−α22(n))\Bigg(\sqrt{\frac{\sum\limits_{i=1}^n(X_i-\mu)^2}{\chi^2_{\frac{\alpha}{2}}(n)}},\sqrt{\frac{\sum\limits_{i=1}^n(X_i-\mu)^2}{\chi^2_{1-\frac{\alpha}{2}}(n)}}\Bigg)(χ2α2(n)i=1∑n(Xi−μ)2,χ1−2α2(n)i=1∑n(Xi−μ)2)	σ2‾=∑i=1n(Xi−μ)2χα2(n),σ2‾=∑i=1n(Xi−μ)2χ1−α2(n)\underline{\sigma^2}=\frac{\sum\limits_{i=1}^n(X_i-\mu)^2}{\chi^2_{\alpha}(n)},\overline{\sigma^2}=\frac{\sum\limits_{i=1}^n(X_i-\mu)^2}{\chi^2_{1-\alpha}(n)}σ2=χα2(n)i=1∑n(Xi−μ)2,σ2=χ1−α2(n)i=1∑n(Xi−μ)2 σ‾=∑i=1n(Xi−μ)2χα2(n),\underline\sigma=\sqrt{\frac{\sum\limits_{i=1}^n(X_i-\mu)^2}{\chi^2_{\alpha}(n)}},σ=χα2(n)i=1∑n(Xi−μ)2, σ‾=∑i=1n(Xi−μ)2χ1−α2(n)\overline\sigma=\sqrt{\frac{\sum\limits_{i=1}^n(X_i-\mu)^2}{\chi^2_{1-\alpha}(n)}}σ=χ1−α2(n)i=1∑n(Xi−μ)2
未知	χ2＝(n−1)S2σ2∼χ2(n−1)\chi^2＝\frac{(n-1)S^2}{\sigma^2}\sim\chi^2(n-1)χ2＝σ2(n−1)S2∼χ2(n−1)	1−α1-\alpha1−α	((n−1)S2χα22(n−1),(n−1)S2χ1−α22(n−1))\Bigg(\frac{(n-1)S^2}{\chi^2_{\frac{\alpha}{2}}(n-1)},\frac{(n-1)S^2}{\chi^2_{1-\frac{\alpha}{2}}(n-1)}\Bigg)(χ2α2(n−1)(n−1)S2,χ1−2α2(n−1)(n−1)S2)	((n−1)χα22(n−1)S,(n−1)χ1−α22(n−1)S)\Bigg(\sqrt{\frac{(n-1)}{\chi^2_{\frac{\alpha}{2}}(n-1)}}S,\sqrt{\frac{(n-1)}{\chi^2_{1-\frac{\alpha}{2}}(n-1)}}S\Bigg)(χ2α2(n−1)(n−1)S,χ1−2α2(n−1)(n−1)S)	σ2‾=(n−1)S2χα2(n−1),σ2‾=(n−1)S2χ1−α2(n−1)\underline{\sigma^2}=\frac{(n-1)S^2}{\chi^2_{\alpha}(n-1)},\overline{\sigma^2}=\frac{(n-1)S^2}{\chi^2_{1-\alpha}(n-1)}σ2=χα2(n−1)(n−1)S2,σ2=χ1−α2(n−1)(n−1)S2 σ‾=(n−1)χα2(n−1)S,\underline\sigma=\sqrt{\frac{(n-1)}{\chi^2_{\alpha}(n-1)}}S,σ=χα2(n−1)(n−1)S, σ‾=(n−1)χ1−α2(n−1)S\overline\sigma=\sqrt{\frac{(n-1)}{\chi^2_{1-\alpha}(n-1)}}Sσ=χ1−α2(n−1)(n−1)S

二、两个正态总体参数的区间估计

总体X∼N(μ1,σ12),Y∼N(μ2,σ22)X\sim N(\mu_1,\sigma^2_1),Y\sim N(\mu_2,\sigma^2_2)X∼N(μ1,σ12),Y∼N(μ2,σ22)，且X,YX,YX,Y相互独立，E(X)=μ1,V(X)=σ12;E(Y)=μ2,V(Y)=σ22E(X)=\mu_1,V(X)=\sigma^2_1;E(Y)=\mu_2,V(Y)=\sigma^2_2E(X)=μ1,V(X)=σ12;E(Y)=μ2,V(Y)=σ22，μ1−μ2\mu_1-\mu_2μ1−μ2为期望差，σ12/σ22\sigma^2_1/ \sigma^2_2σ12/σ22为方差比。
XXX样本X1,X2,...,Xn1,Xi∼N(μ1,σ12),i=1,2,...,n1,X_1,X_2,...,X_{n_1},\ X_i\sim N(\mu_1,\sigma^2_1),\ i=1,2,...,n_1,X1,X2,...,Xn1, Xi∼N(μ1,σ12), i=1,2,...,n1,
X‾=1n1∑i=1n1Xi,X‾∼N(μ1,σ12n1),Xi‾−μ1σ1∼N(0,1).\overline{X}=\frac{1}{n_1}\sum\limits_{i=1}^{n_1}X_i,\ \overline{X}\sim N\Big(\mu_1,\frac{\sigma^2_1}{n_1}\Big),\ \frac{\overline{X_i}-\mu_1}{\sigma_1}\sim N(0,1). X=n11i=1∑n1Xi, X∼N(μ1,n1σ12), σ1Xi−μ1∼N(0,1).
YYY样本Y1,Y2,...,Yn2,Yi∼N(μ2,σ22),i=1,2,...,n2,Y_1,Y_2,...,Y_{n_2},\ Y_i\sim N(\mu_2,\sigma^2_2),\ i=1,2,...,n_2,Y1,Y2,...,Yn2, Yi∼N(μ2,σ22), i=1,2,...,n2,
Y‾=1n2∑i=1n2Yi,Y‾∼N(μ2,σ22n2),Yi‾−μ2σ2∼N(0,1).\overline{Y}=\frac{1}{n_2}\sum\limits_{i=1}^{n_2}Y_i,\ \overline{Y}\sim N\Big(\mu_2,\frac{\sigma^2_2}{n_2}\Big),\ \frac{\overline{Y_i}-\mu_2}{\sigma_2}\sim N(0,1). Y=n21i=1∑n2Yi, Y∼N(μ2,n2σ22), σ2Yi−μ2∼N(0,1).
X‾−Y‾∼N(μ1−μ2,σ12n1+σ22n2).\overline{X}-\overline{Y}\sim N\Big(\mu_1-\mu_2,\frac{\sigma^2_1}{n_1}+\frac{\sigma^2_2}{n_2}\Big). X−Y∼N(μ1−μ2,n1σ12+n2σ22).

μ1−μ2\mu_1-\mu_2μ1−μ2的置信区间

σ12,σ22\ \ \ \sigma^2_1,\sigma^2_2\ \ \ σ12,σ22	统计量及分布	置信度	置信区间(μ1−μ2‾,μ1−μ2‾)\ \ \ (\underline{\mu_1-\mu_2},\overline{\mu_1-\mu_2})\ \ \ (μ1−μ2,μ1−μ2)	单侧置信限
均已知	Z=(X‾−Y‾)−(μ1−μ2)σ12n1+σ22n2∼N(0,1)Z=\frac{(\overline{X}-\overline{Y})-(\mu_1-\mu_2)}{\sqrt{\frac{\sigma^2_1}{n_1}+\frac{\sigma^2_2}{n_2}}}\sim N(0,1)Z=n1σ12+n2σ22(X−Y)−(μ1−μ2)∼N(0,1)	1−α1-\alpha1−α	(X‾−Y‾∓δ),(\overline{X}-\overline{Y}\mp \delta),(X−Y∓δ), δ=zα2σ12n1+σ22n2\delta=z_{\frac{\alpha}{2}}\sqrt{\frac{\sigma^2_1}{n_1}+\frac{\sigma^2_2}{n_2}}δ=z2αn1σ12+n2σ22	μ1−μ2‾=X‾−Y‾−δ,\underline{\mu_1-\mu_2}=\overline{X}-\overline{Y}-\delta,μ1−μ2=X−Y−δ, μ1−μ2‾=X‾−Y‾+δ,\overline{\mu_1-\mu_2}=\overline{X}-\overline{Y}+\delta,μ1−μ2=X−Y+δ, δ=zασ12n1+σ22n2\delta=z_{\alpha}\sqrt{\frac{\sigma^2_1}{n_1}+\frac{\sigma^2_2}{n_2}}δ=zαn1σ12+n2σ22
均未知,但σ12=σ22\sigma^2_1=\sigma^2_2σ12=σ22	T=(X‾−Y‾)−(μ1−μ2)Sw1n1+1n2∼t(n1+n2−2),T=\frac{(\overline{X}-\overline{Y})-(\mu_1-\mu_2)}{S_w\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}\sim t(n_1+n_2-2),T=Swn11+n21(X−Y)−(μ1−μ2)∼t(n1+n2−2), Sw2=(n1−1)S12+(n2−1)S22n1+n2−2S_w^2=\frac{(n_1-1)S_1^2+(n_2-1)S_2^2}{n_1+n_2-2}Sw2=n1+n2−2(n1−1)S12+(n2−1)S22	1−α1-\alpha1−α	(X‾−Y‾∓δ),(\overline{X}-\overline{Y}\mp \delta),(X−Y∓δ), δ=tα2Sw1n1+1n2\delta=t_{\frac{\alpha}{2}}S_w\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}δ=t2αSwn11+n21	μ1−μ2‾=X‾−Y‾−δ,\underline{\mu_1-\mu_2}=\overline{X}-\overline{Y}-\delta,μ1−μ2=X−Y−δ, μ1−μ2‾=X‾−Y‾+δ,\overline{\mu_1-\mu_2}=\overline{X}-\overline{Y}+\delta,μ1−μ2=X−Y+δ, δ=tαSw1n1+1n2\delta=t_{\alpha}S_w\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}δ=tαSwn11+n21
均未知,但n1,n2n_1,n_2n1,n2很大(>50)	Z=(X‾−Y‾)−(μ1−μ2)S12n1+S22n2∼N(0,1)Z=\frac{(\overline{X}-\overline{Y})-(\mu_1-\mu_2)}{\sqrt{\frac{S_1^2}{n_1}+\frac{S_2^2}{n_2}}}\sim N(0,1)Z=n1S12+n2S22(X−Y)−(μ1−μ2)∼N(0,1) 可用S12,S22S_1^2,S_2^2S12,S22代替σ12,σ22\sigma^2_1,\sigma^2_2σ12,σ22，和σ12,σ22\sigma^2_1,\sigma^2_2σ12,σ22已知的情况一样处理	1−α1-\alpha1−α	(X‾−Y‾∓δ),(\overline{X}-\overline{Y}\mp \delta),(X−Y∓δ), δ=zα2S12n1+S22n2\delta=z_{\frac{\alpha}{2}}\sqrt{\frac{S_1^2}{n_1}+\frac{S_2^2}{n_2}}δ=z2αn1S12+n2S22	μ1−μ2‾=X‾−Y‾−δ,\underline{\mu_1-\mu_2}=\overline{X}-\overline{Y}-\delta,μ1−μ2=X−Y−δ,μ1−μ2‾=X‾−Y‾+δ,\overline{\mu_1-\mu_2}=\overline{X}-\overline{Y}+\delta,μ1−μ2=X−Y+δ, δ=zαS12n1+S22n2\delta=z_{\alpha}\sqrt{\frac{S_1^2}{n_1}+\frac{S_2^2}{n_2}}δ=zαn1S12+n2S22

σ12/σ22\sigma^2_1/ \sigma^2_2σ12/σ22的置信区间

μ1,μ2\mu_1,\mu_2μ1,μ2	统计量及分布	置信度	置信区间(σ12σ22‾,σ12σ22‾)\Big(\underline{\frac{\sigma_1^2}{\sigma_2^2}},\overline{\frac{\sigma_1^2}{\sigma_2^2}}\Big)(σ22σ12,σ22σ12)	单侧置信限
均未知	F=(n1−1)S12σ12/n1−1(n2−1)S22σ22/n2−1∼F(n1−1,n2−1)F=\frac{\frac{(n_1-1)S_1^2}{\sigma^2_1}\Big/n_1-1}{\frac{(n_2-1)S_2^2}{\sigma^2_2}\Big/n_2-1}\sim F(n_1-1,n_2-1)F=σ22(n2−1)S22/n2−1σ12(n1−1)S12/n1−1∼F(n1−1,n2−1)	1−α1-\alpha1−α	(S12/S22Fα2(n1−1,n2−1),S12S22Fα2(n2−1,n1−1))\Bigg(\frac{S_1^2/S_2^2}{F_\frac{\alpha}{2}(n_1-1,n_2-1)},\frac{S_1^2}{S_2^2}F_\frac{\alpha}{2}(n_2-1,n_1-1)\Bigg)(F2α(n1−1,n2−1)S12/S22,S22S12F2α(n2−1,n1−1))	σ12σ22‾=S12/S22Fα(n1−1,n2−1),\underline{\frac{\sigma_1^2}{\sigma_2^2}}=\frac{S_1^2/S_2^2}{F_\alpha(n_1-1,n_2-1)},σ22σ12=Fα(n1−1,n2−1)S12/S22, σ12σ22‾=S12S22Fα(n2−1,n1−1)\overline{\frac{\sigma_1^2}{\sigma_2^2}}=\frac{S_1^2}{S_2^2}F_{\alpha}(n_2-1,n_1-1)σ22σ12=S22S12Fα(n2−1,n1−1)
均已知	F=∑i=1n1(Xi−μ1σ1)2/n1∑i=1n2(Xi−μ2σ2)2/n2∼F(n1,n2)F=\frac{\sum\limits_{i=1}^{n_1}\Big(\frac{X_i-\mu_1}{\sigma_1}\Big)^2\Big/n1}{\sum\limits_{i=1}^{n_2}\Big(\frac{X_i-\mu_2}{\sigma_2}\Big)^2\Big/n2}\sim F(n_1,n_2)F=i=1∑n2(σ2Xi−μ2)2/n2i=1∑n1(σ1Xi−μ1)2/n1∼F(n1,n2)	1−α1-\alpha1−α	(n2∑i=1n1(Xi−μ1)2n1∑i=1n2(Yi−μ2)21Fα2(n1,n2),n2∑i=1n1(Xi−μ1)2n1∑i=1n2(Yi−μ2)2Fα2(n2,n1))\Bigg(\frac{n_2\sum\limits_{i=1}^{n_1}(X_i-\mu_1)^2}{n_1\sum\limits_{i=1}^{n_2}(Y_i-\mu_2)^2}\frac{1}{F_\frac{\alpha}{2}(n_1,n_2)},\frac{n_2\sum\limits_{i=1}^{n_1}(X_i-\mu_1)^2}{n_1\sum\limits_{i=1}^{n_2}(Y_i-\mu_2)^2}F_\frac{\alpha}{2}(n_2,n_1)\Bigg)(n1i=1∑n2(Yi−μ2)2n2i=1∑n1(Xi−μ1)2F2α(n1,n2)1,n1i=1∑n2(Yi−μ2)2n2i=1∑n1(Xi−μ1)2F2α(n2,n1))	σ12σ22‾=n2∑i=1n1(Xi−μ1)2n1∑i=1n2(Yi−μ2)21Fα(n1,n2),\underline{\frac{\sigma_1^2}{\sigma_2^2}}=\frac{n_2\sum\limits_{i=1}^{n_1}(X_i-\mu_1)^2}{n_1\sum\limits_{i=1}^{n_2}(Y_i-\mu_2)^2}\frac{1}{F_{\alpha}(n_1,n_2)},σ22σ12=n1i=1∑n2(Yi−μ2)2n2i=1∑n1(Xi−μ1)2Fα(n1,n2)1, σ12σ22‾=n2∑i=1n1(Xi−μ1)2n1∑i=1n2(Yi−μ2)2Fα(n2,n1)\overline{\frac{\sigma_1^2}{\sigma_2^2}}=\frac{n_2\sum\limits_{i=1}^{n_1}(X_i-\mu_1)^2}{n_1\sum\limits_{i=1}^{n_2}(Y_i-\mu_2)^2}F_{\alpha}(n_2,n_1)σ22σ12=n1i=1∑n2(Yi−μ2)2n2i=1∑n1(Xi−μ1)2Fα(n2,n1)