文中表格来自于《概率论与数理统计》，主要内容：

  • 常见分布的分布律、期望、方差
  • 正态总体均值、方差的置信区间、单侧置信限
  • 正态总体均值、方差检验，包含原假设、备择假设、检验统计量、拒绝域、P_P\_ 值
  • 方差分析检验与拟合优度检验
  • 方差分析表

几种常见的概率分布表

分布 参数范围 概率分布律或密度函数 数学期望 方差
(0−1)(0-1) 分布 0<p<10<p<1 P{X=k}=pk(1−p)1−k,k=0,1P\{X=k\}=p^k(1-p)^{1-k},k=0,1 pp p(1−p)p(1-p)
二项分布 n⩾10<p<1n\geqslant1 \\ 0<p<1 P{X=k}=Cnkpk(1−p)n−k,k=0,1,⋯ ,nP\{X=k\}=C_n^kp^k(1-p)^{n-k},k=0,1,\cdots,n npnp np(1−p)np(1-p)
负二项分布(巴斯卡分布) r⩾10<p<1r\geqslant1 \\ 0<p<1 P{X=k}=Ck−1r−1pr(1−p)k−r,k=r,r+1,⋯P\{X=k\}=C_{k-1}^{r-1}p^r(1-p)^{k-r},k=r,r+1,\cdots rp\cfrac{r}{p} r(1−p)p2\cfrac{r(1-p)}{p^2}
几何分布 0<p<10<p<1 P{X=k}=p(1−p)k−1,k=1,2,⋯P\{X=k\}=p(1-p)^{k-1},k=1,2,\cdots 1p\cfrac{1}{p} 1−pp2\cfrac{1-p}{p^2}
超几何分布 M⩽Nn⩽NM\leqslant N \\ n\leqslant N P{X=k}=CMkCN−Mn−kCNn,max(0,n−N+M)⩽k⩽min(n,M)P\{X=k\}=\cfrac{C^k_MC^{n-k}_{N-M}}{C^n_N},max(0,n-N+M)\leqslant k\leqslant min(n,M) nMN\cfrac{nM}{N} nMN(1−MN)N−nN−1\cfrac{nM}{N}(1-\cfrac{M}{N})\cfrac{N-n}{N-1}
泊松分布 λ>0\lambda >0 P{X=k}=λke−λk!,k=0,1,2,⋯P\{X=k\}=\cfrac{\lambda^ke^{-\lambda}}{k!},k=0,1,2,\cdots λ\lambda λ\lambda
均匀分布 a<ba<b f(x)={1b−a,a<x<b0,otherwisef(x)=\begin{cases}\cfrac{1}{b-a}&,a<x<b \\ 0&,\mathrm{otherwise}\end{cases} a+b2\cfrac{a+b}{2} (b−a)212\cfrac{(b-a)^2}{12}
正态分布 μ,σ>0\mu,\sigma>0 f(x)=12πσe−12(x−μσ)2f(x)=\cfrac{1}{\sqrt{2\pi}\sigma}e^{-\cfrac{1}{2}(\cfrac{x-\mu}{\sigma})^2} μ\mu σ2\sigma^2
标准正态分布 μ=0,σ=1\mu=0,\sigma=1 f(x)=12πe−12x2f(x)=\cfrac{1}{\sqrt{2\pi}}e^{-\cfrac{1}{2}x^2} 00 11
指数分布 λ>0\lambda >0 f(x)={λe−λx,x>00,otherwisef(x)=\begin{cases}\lambda e^{-\lambda x}&,x>0 \\ 0&,\mathrm{otherwise}\end{cases} 1λ\cfrac{1}{\lambda} 1λ2\cfrac{1}{\lambda^2}
Γ\Gamma 分布 α>0,β>0\alpha>0,\beta>0 f(x)={βαΓ(α)xα−1e−βx,x>00,otherwisef(x)=\begin{cases}\cfrac{\beta^\alpha}{\Gamma(\alpha)}x^{\alpha-1}e^{-\beta x}&,x>0 \\ 0&,\mathrm{otherwise}\end{cases} αβ\cfrac{\alpha}{\beta} αβ2\cfrac{\alpha}{\beta^2}
χ2\chi^2 分布 n⩾1n\geqslant 1 f(x)={12n2Γ(n2)xn2−1e−x2,x>00,otherwisef(x)=\begin{cases}\cfrac{1}{2^{\cfrac{n}{2}}\Gamma(\cfrac{n}{2})}x^{\cfrac{n}{2}-1}e^{-\cfrac{x}{2}}&,x>0 \\ 0&,\mathrm{otherwise}\end{cases} nn 2n2n
威布尔分布 η>0,β>0\eta>0,\beta>0 f(x)={βη(xη)β−1e−(xη)β,x>00,otherwisef(x)=\begin{cases}\cfrac{\beta}{\eta}(\cfrac{x}{\eta})^{\beta-1}e^{-(\cfrac{x}{\eta})^\beta}&,x>0 \\ 0&,\mathrm{otherwise}\end{cases} ηΓ(1β+1)\eta\Gamma(\cfrac{1}{\beta}+1) η2(Γ(2β+1)−Γ(1β+1)2)\eta^2(\Gamma(\cfrac{2}{\beta}+1)-\Gamma(\cfrac{1}{\beta}+1)^2)
瑞利分布 σ>0\sigma>0 f(x)={xσ2e−(x2σ)2,x>00,otherwisef(x)=\begin{cases}\cfrac{x}{\sigma^2}e^{-(\cfrac{x}{2\sigma})^2}&,x>0 \\ 0&,\mathrm{otherwise}\end{cases} π2σ\sqrt{\cfrac{\pi}{2}}\sigma 4−π2σ2\cfrac{4-\pi}{2}\sigma^2
β\beta 分布 α>0,β>0\alpha>0,\beta>0 f(x)={Γ(α+β)Γ(α)Γ(β)xα−1(1−x)β−1,0<x<10,otherwisef(x)=\begin{cases}\cfrac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}x^{\alpha-1}(1-x)^{\beta-1}&,0<x<1 \\ 0&,\mathrm{otherwise}\end{cases} αα+β\cfrac{\alpha}{\alpha+\beta} αβ(α+β)2(α+β+1)\cfrac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}
对数正态分布 μ,σ>0\mu,\sigma>0 f(x)={12πσxe−(lnx−μ2σ)2,x>00,otherwisef(x)=\begin{cases}\cfrac{1}{\sqrt{2\pi}\sigma x}e^{-(\cfrac{ln x-\mu}{2\sigma})^2}&,x>0 \\ 0&,\mathrm{otherwise}\end{cases} eμ+σ22e^{\mu+\cfrac{\sigma^2}{2}} e2μ+σ2(eσ2−1)e^{2\mu+\sigma^2}(e^{\sigma^2}-1)
柯西分布 a,λ>0a,\lambda>0 f(x)=1πλλ2+(x−a)2f(x)=\cfrac{1}{\pi}\cfrac{\lambda}{\lambda^2+(x-a)^2} nullnull nullnull
tt 分布 n⩾0n\geqslant 0 f(x)=Γ(n+12)nπΓ(n2)(1+x2n)−n+12f(x)=\cfrac{\Gamma(\cfrac{n+1}{2})}{\sqrt{n\pi}\Gamma(\cfrac{n}{2})}(1+\cfrac{x^2}{n})^{-\cfrac{n+1}{2}} 0,n>10,n>1 nn−2,n>2\cfrac{n}{n-2},n>2
FF 分布 n1,n2n_1,n_2 f(x)={Γ(n1+n22)Γ(n12)Γ(n22)(n1n2)(n1n2x)n12−1(1+n1n2x)−n1+n22,x>00,otherwisef(x)=\begin{cases}\cfrac{\Gamma(\cfrac{n_1+n_2}{2})}{\Gamma(\cfrac{n_1}{2})\Gamma(\cfrac{n_2}{2})}(\cfrac{n_1}{n_2})(\cfrac{n_1}{n_2}x)^{\cfrac{n_1}{2}-1}(1+\cfrac{n_1}{n_2}x)^{-\cfrac{n_1+n_2}{2}}&,x>0 \\ 0&,otherwise\end{cases} n2n2−2,n2>2\cfrac{n_2}{n_2-2},n_2>2 2n22(n1+n2−2)n1(n2−2)2(n2−4),n2>4\cfrac{2n_2^2(n_1+n_2-2)}{n_1(n_2-2)^2(n_2-4)},n_2>4

正态总体均值、方差的置信区间、单侧置信限

待估参数 其他参数 枢轴量的分布 置信区间 单侧置信限(U) 单侧置信限(L)
μ\mu σ2已知\sigma^2 \\ \text{已知} Z=X‾−μσ/n∼N(0,1)Z=\cfrac{\overline{X}-\mu}{\sigma/\sqrt{n}}\sim N(0,1) (X‾−σnzα/2,X‾+σnzα/2)(\overline{X}-\cfrac{\sigma}{\sqrt{n}}z_{\alpha/2} ,\\ \overline{X}+\cfrac{\sigma}{\sqrt{n}}z_{\alpha/2}) μ^U=X‾+σnzα\widehat{\mu}_U=\overline{X}+\cfrac{\sigma}{\sqrt{n}}z_{\alpha} μ^L=X‾−σnzα\widehat{\mu}_L=\overline{X}-\cfrac{\sigma}{\sqrt{n}}z_{\alpha}
μ\mu σ2未知\sigma^2 \\ \text{未知} t=X‾−μS/n∼t(n−1)t=\cfrac{\overline{X}-\mu}{S/\sqrt{n}}\sim t(n-1) (X‾−Sntα/2(n−1),X‾+Sntα/2(n−1))(\overline{X}-\cfrac{S}{\sqrt{n}}t_{\alpha/2}(n-1) ,\\ \overline{X}+\cfrac{S}{\sqrt{n}}t_{\alpha/2}(n-1)) μ^U=X‾+Sntα(n−1)\widehat{\mu}_U=\overline{X}+\cfrac{S}{\sqrt{n}}t_{\alpha(n-1)} μ^L=X‾−Sntα(n−1)\widehat{\mu}_L=\overline{X}-\cfrac{S}{\sqrt{n}}t_{\alpha(n-1)}
σ2\sigma^2 μ已知\mu \\ \text{已知} χ2=(n−1)S2σ2∼χ2(n−1)\chi^2=\cfrac{(n-1)S^2}{\sigma^2}\sim \chi^2(n-1) ((n−1)S2χα/22(n−1),(n−1)S2χ1−α/22(n−1))(\cfrac{(n-1)S^2}{\chi^2_{\alpha/2}(n-1)} ,\\ \cfrac{(n-1)S^2}{\chi^2_{1-\alpha/2}(n-1)}) σ^U2=(n−1)S2χ1−α2(n−1)\widehat{\sigma}^2_U=\cfrac{(n-1)S^2}{\chi^2_{1-\alpha}(n-1)} σ^L2=(n−1)S2χα2(n−1)\widehat{\sigma}^2_L=\cfrac{(n-1)S^2}{\chi^2_{\alpha}(n-1)}
μ1−μ2\mu_1-\mu_2 σ12,σ22已知\sigma_1^2,\sigma_2^2 \\ \text{已知} Z=(X‾−Y‾)−(μ1−μ2)σ12/n1+σ22/n2∼N(0,1)Z=\cfrac{(\overline{X}-\overline{Y})-(\mu_1-\mu_2)}{\sqrt{\sigma_1^2/n_1+\sigma_2^2/n2}}\sim N(0,1) (X‾−Y‾−zα/2σ12/n1+σ22/n2,X‾−Y‾+zα/2σ12/n1+σ22/n2)(\overline{X}-\overline{Y}-z_{\alpha/2}{\sqrt{\sigma_1^2/n_1+\sigma_2^2/n_2}} ,\\ \overline{X}-\overline{Y}+z_{\alpha/2}{\sqrt{\sigma_1^2/n_1+\sigma_2^2/n_2}}) μ1−μ2^U=X‾−Y‾+zασ12/n1+σ22/n2\widehat{\mu_1-\mu_2}_U=\overline{X}-\overline{Y}+z_{\alpha}{\sqrt{\sigma_1^2/n_1+\sigma_2^2/n2}} μ1−μ2^L=X‾−Y‾−zασ12/n1+σ22/n2\widehat{\mu_1-\mu_2}_L=\overline{X}-\overline{Y}-z_{\alpha}{\sqrt{\sigma_1^2/n_1+\sigma_2^2/n2}}
μ1−μ2\mu_1-\mu_2 σ12=σ22未知\sigma_1^2=\sigma_2^2 \\ \text{未知} t=(X‾−Y‾)−(μ1−μ2)Sw1/n1+1/n2∼t(n1+n2−2)Sw2=(n1−1)S12+(n2−1)S22n1+n2−2t=\cfrac{(\overline{X}-\overline{Y})-(\mu_1-\mu_2)}{S_w\sqrt{1/n_1+1/n2}}\sim t(n_1+n_2-2) \\ S^2_w=\cfrac{(n_1-1)S^2_1+(n_2-1)S^2_2}{n_1+n_2-2} (X‾−Y‾−tα/2(n1+n2−2)Sw1/n1+2/n2,X‾−Y‾+tα/2(n1+n2−2)Sw1/n1+2/n2)(\overline{X}-\overline{Y}-t_{\alpha/2}(n_1+n_2-2)S_w{\sqrt{1/n_1+2/n_2}} ,\\ \overline{X}-\overline{Y}+t_{\alpha/2}(n_1+n_2-2)S_w{\sqrt{1/n_1+2/n_2}}) μ1−μ2^U=X‾−Y‾+tα(n1+n2−2)Sw1/n1+2/n2\widehat{\mu_1-\mu_2}_U=\overline{X}-\overline{Y}+t_{\alpha}(n_1+n_2-2)S_w{\sqrt{1/n_1+2/n_2}} μ1−μ2^L=X‾−Y‾−tα(n1+n2−2)Sw1/n1+2/n2\widehat{\mu_1-\mu_2}_L=\overline{X}-\overline{Y}-t_{\alpha}(n_1+n_2-2)S_w{\sqrt{1/n_1+2/n_2}}
μ1−μ2\mu_1-\mu_2 σ12≠σ22未知\sigma_1^2\ne \sigma_2^2 \\ \text{未知} t=(X‾−Y‾)−(μ1−μ2)S12/n1+S22/n2{∼N(0,1),n1>50,n2>50∼t(k),k=min(n1−1,n2−1)t=\cfrac{(\overline{X}-\overline{Y})-(\mu_1-\mu_2)}{\sqrt{S_1^2/n_1+S_2^2/n2}} \\ \begin{cases}\sim N(0,1)&,n_1>50,n_2>50 \\ \sim t(k)&,k=min(n_1-1,n_2-1)\end{cases} (X‾−Y‾−qα/2S12/n1+S22/n2,X‾−Y‾+qα/2S12/n1+S22/n2)qp={zp,t∼N(0,1)tp(k),t∼t(k)(\overline{X}-\overline{Y}-q_{\alpha/2}\sqrt{S_1^2/n_1+S_2^2/n2} ,\\ \overline{X}-\overline{Y}+q_{\alpha/2}\sqrt{S_1^2/n_1+S_2^2/n2}) \\ q_p=\begin{cases}z_p&,t\sim N(0,1) \\ t_p(k)&,t\sim t(k)\end{cases} μ1−μ2^U=X‾−Y‾+qαS12/n1+S22/n2\widehat{\mu_1-\mu_2}_U=\overline{X}-\overline{Y}+q_{\alpha}\sqrt{S_1^2/n_1+S_2^2/n2} μ1−μ2^L=X‾−Y‾−qαS12/n1+S22/n2\widehat{\mu_1-\mu_2}_L=\overline{X}-\overline{Y}-q_{\alpha}\sqrt{S_1^2/n_1+S_2^2/n2}
σ12/σ22\sigma_1^2/\sigma_2^2 μ1,μ2未知\mu_1,\mu_2 \\ \text{未知} F=S12/S22σ12/σ22∼F(n1−1,n2−1)F=\cfrac{S_1^2/S_2^2}{\sigma_1^2/\sigma_2^2}\sim F(n_1-1,n_2-1) (S12/S22Fα/2(n1−1,n2−1),S12/S22F1−α/2(n1−1,n2−1))(\cfrac{S_1^2/S_2^2}{F_{\alpha/2}(n_1-1,n_2-1)} ,\\ \cfrac{S_1^2/S_2^2}{F_{1-\alpha/2}(n_1-1,n_2-1)}) σ12−σ22^U=S12/S22F1−α(n1−1,n2−1)\widehat{\sigma_1^2-\sigma_2^2}_U=\cfrac{S_1^2/S_2^2}{F_{1-\alpha}(n_1-1,n_2-1)} σ12−σ22^L=S12/S22Fα(n1−1,n2−1)\widehat{\sigma_1^2-\sigma_2^2}_L=\cfrac{S_1^2/S_2^2}{F_{\alpha}(n_1-1,n_2-1)}

正态总体均值、方差的检验法

原假设H0H_0 检验统计量 备择假设H1H_1 拒绝域 检验统计量的取值 P_P\_值
μ⩽μ0μ⩾μ0μ=μ0(σ2已知)\mu\leqslant\mu_0 \\ \mu\geqslant\mu_0 \\ \mu=\mu_0 \\ (\sigma^2\text{已知}) Z=X‾−μ0σ/nZ=\cfrac{\overline{X}-\mu_0}{\sigma/\sqrt{n}} μ>μ0μ<μ0μ≠μ0\mu>\mu_0 \\ \mu<\mu_0 \\ \mu\ne\mu_0 Z⩾zαZ⩽−zα∣Z∣⩾zα/2Z\geqslant z_\alpha \\ Z\leqslant-z_\alpha \\ \vert Z\vert\geqslant z_{\alpha/2} z0=x‾−μ0σ/nz_0=\cfrac{\overline{x}-\mu_0}{\sigma/\sqrt{n}} 1−Φ(z0)Φ(z0)2(1−Φ(∣z0∣))1-\Phi(z_0) \\ \Phi(z_0) \\ 2(1-\Phi(\vert z_0\vert))
μ⩽μ0μ⩾μ0μ=μ0(σ2未知)\mu\leqslant\mu_0 \\ \mu\geqslant\mu_0 \\ \mu=\mu_0 \\ (\sigma^2\text{未知}) T=X‾−μ0S/nT=\cfrac{\overline{X}-\mu_0}{S/\sqrt{n}} μ>μ0μ<μ0μ≠μ0\mu>\mu_0 \\ \mu<\mu_0 \\ \mu\ne\mu_0 T⩾tα(n−1)T⩽−tα(n−1)∣T∣⩾tα/2(n−1)T\geqslant t_\alpha(n-1) \\ T\leqslant-t_\alpha(n-1) \\ \vert T\vert\geqslant t_{\alpha/2}(n-1) t0=x‾−μ0s/nt_0=\cfrac{\overline{x}-\mu_0}{s/\sqrt{n}} P(t(n−1)⩾t0)P(t(n−1)⩽t0)2P(t(n−1)⩾∣t0∣)P(t(n-1)\geqslant t_0) \\ P(t(n-1)\leqslant t_0) \\ 2P(t(n-1)\geqslant\vert t_0\vert)
μ1−μ2⩽δμ1−μ2⩾δμ1−μ2=δ(σ12,σ22已知)\mu_1-\mu_2\leqslant\delta \\ \mu_1-\mu_2\geqslant\delta \\ \mu_1-\mu_2=\delta \\ (\sigma_1^2,\sigma_2^2\text{已知}) Z=X‾−Y‾−δσ12n1+σ22n2Z=\cfrac{\overline{X}-\overline{Y}-\delta}{\sqrt{\cfrac{\sigma_1^2}{n_1}+\cfrac{\sigma_2^2}{n_2}}} μ1−μ2>δμ1−μ2<δμ1−μ2≠δ\mu_1-\mu_2>\delta \\ \mu_1-\mu_2<\delta \\ \mu_1-\mu_2\ne\delta Z⩾zαZ⩽−zα∣Z∣⩾zα/2Z\geqslant z_\alpha \\ Z\leqslant-z_\alpha \\ \vert Z\vert\geqslant z_{\alpha/2} z0=x‾−y‾−δσ12n1+σ22n2z_0=\cfrac{\overline{x}-\overline{y}-\delta}{\sqrt{\cfrac{\sigma_1^2}{n_1}+\cfrac{\sigma_2^2}{n_2}}} 1−Φ(z0)Φ(z0)2(1−Φ(∣z0∣))1-\Phi(z_0) \\ \Phi(z_0) \\ 2(1-\Phi(\vert z_0\vert))
μ1−μ2⩽δμ1−μ2⩾δμ1−μ2=δ(σ12=σ22未知)\mu_1-\mu_2\leqslant\delta \\ \mu_1-\mu_2\geqslant\delta \\ \mu_1-\mu_2=\delta \\ (\sigma_1^2=\sigma_2^2\text{未知}) T=X‾−Y‾−δSw1n1+1n2Sw2=(n1−1)S12+(n2−1)S22n1+n2−2T=\cfrac{\overline{X}-\overline{Y}-\delta}{S_w\sqrt{\cfrac{1}{n_1}+\cfrac{1}{n_2}}} \\ S_w^2=\cfrac{(n_1-1)S_1^2+(n_2-1)S_2^2}{n_1+n_2-2} μ1−μ2>δμ1−μ2<δμ1−μ2≠δ\mu_1-\mu_2>\delta \\ \mu_1-\mu_2<\delta \\ \mu_1-\mu_2\ne\delta T⩾tα(n1+n2−2)T⩽−tα(n1+n2−2)∣T∣⩾tα/2(n1+n2−2)T\geqslant t_\alpha(n_1+n_2-2) \\ T\leqslant-t_\alpha(n_1+n_2-2) \\ \vert T\vert\geqslant t_{\alpha/2}(n_1+n_2-2) t0=x‾−y‾−δsw1n1+1n2t_0=\cfrac{\overline{x}-\overline{y}-\delta}{s_w\sqrt{\cfrac{1}{n_1}+\cfrac{1}{n_2}}} P(t(n1+n2−2)⩾t0)P(t(n1+n2−2)⩽t0)2P(t(n1+n2−2)⩾∣t0∣)P(t(n_1+n_2-2)\geqslant t_0) \\ P(t(n_1+n_2-2)\leqslant t_0) \\ 2P(t(n_1+n_2-2)\geqslant\vert t_0\vert)
μ1−μ2⩽δμ1−μ2⩾δμ1−μ2=δ(σ12≠σ22未知)\mu_1-\mu_2\leqslant\delta \\ \mu_1-\mu_2\geqslant\delta \\ \mu_1-\mu_2=\delta \\ (\sigma_1^2\ne\sigma_2^2\text{未知}) T=X‾−Y‾−δS12n1+S22n2T=\cfrac{\overline{X}-\overline{Y}-\delta}{\sqrt{\cfrac{S_1^2}{n_1}+\cfrac{S_2^2}{n_2}}} μ1−μ2>δμ1−μ2<δμ1−μ2≠δ\mu_1-\mu_2>\delta \\ \mu_1-\mu_2<\delta \\ \mu_1-\mu_2\ne\delta T⩾tα(k)T⩽−tα(k)∣T∣⩾tα/2(k)k=min(n1−1,n2−1)T\geqslant t_\alpha(k) \\ T\leqslant-t_\alpha(k) \\ \vert T\vert\geqslant t_{\alpha/2}(k) \\ k=min(n_1-1,n_2-1) t0=x‾−y‾−δs12n1+s22n2t_0=\cfrac{\overline{x}-\overline{y}-\delta}{\sqrt{\cfrac{s_1^2}{n_1}+\cfrac{s_2^2}{n_2}}} P(t(k)⩾t0)P(t(k)⩽t0)2P(t(k)⩾∣t0∣)P(t(k)\geqslant t_0) \\ P(t(k)\leqslant t_0) \\ 2P(t(k)\geqslant\vert t_0\vert)
σ2⩽σ02σ2⩾σ02σ2=σ02(μ未知)\sigma^2\leqslant\sigma_0^2 \\ \sigma^2\geqslant\sigma_0^2 \\ \sigma^2=\sigma_0^2 \\ (\mu\text{未知}) χ2=(n−1)S2σ02\chi^2=\cfrac{(n-1)S^2}{\sigma_0^2} σ2>σ02σ2<σ02σ2≠σ02\sigma^2>\sigma_0^2 \\ \sigma^2<\sigma_0^2 \\ \sigma^2\ne\sigma_0^2 χ2⩾χα2(n−1)χ2⩽χ1−α2(n−1)χ2⩾χα/22(n−1),orχ2⩽χ1−α/22(n−1)\chi^2\geqslant \chi^2_\alpha(n-1) \\ \chi^2\leqslant\chi^2_{1-\alpha}(n-1) \\ \chi^2\geqslant\chi^2_{\alpha/2}(n-1),or \\ \chi^2\leqslant\chi^2_{1-\alpha/2}(n-1) χ02=(n−1)s2σ02\chi_0^2=\cfrac{(n-1)s^2}{\sigma_0^2} P(χ2(n−1)⩾χ02)P(χ2(n−1)⩽χ02)2min(P(χ2(n−1)⩽χ02),1−P(χ2(n−1)⩽χ02))P(\chi^2(n-1)\geqslant\chi_0^2) \\ P(\chi^2(n-1)\leqslant\chi_0^2) \\ 2min(P(\chi^2(n-1)\leqslant\chi_0^2), \\ 1-P(\chi^2(n-1)\leqslant\chi_0^2))
σ12⩽σ22σ12⩾σ22σ12=σ22(μ1,μ2未知)\sigma_1^2\leqslant\sigma_2^2 \\ \sigma_1^2\geqslant\sigma_2^2 \\ \sigma_1^2=\sigma_2^2 \\ (\mu_1,\mu_2\text{未知}) F=S12S22F=\cfrac{S_1^2}{S_2^2} σ12>σ22σ12<σ22σ12≠σ22\sigma_1^2>\sigma_2^2 \\ \sigma_1^2<\sigma_2^2 \\ \sigma_1^2\ne\sigma_2^2 F⩾Fα(n1−1,n2−1)F⩽Fα(n1−1,n2−1)F⩾Fα/2(n1−1,n2−1),orF⩽F1−α/2(n1−1,n2−1)F\geqslant F_\alpha(n_1-1,n_2-1) \\ F\leqslant F_\alpha(n_1-1,n_2-1) \\ F\geqslant F_{\alpha/2}(n_1-1,n_2-1),or \\ F\leqslant F_{1-\alpha/2}(n_1-1,n_2-1) f0=s12s22f_0=\cfrac{s_1^2}{s_2^2} P(F(n1−1,n2−1)⩾f0)P(F(n1−1,n2−1)⩽f0)2min(P(F(n1−1,n2−1)⩽f0),1−P(F(n1−1,n2−1)⩽f0))P(F(n_1-1,n_2-1)\geqslant f_0) \\ P(F(n_1-1,n_2-1)\leqslant f_0) \\ 2min(P(F(n_1-1,n_2-1)\leqslant f_0), \\ 1-P(F(n_1-1,n_2-1)\leqslant f_0))
μD⩽0μD⩾0μD=0(成对数据)\mu_D\leqslant0 \\ \mu_D\geqslant0 \\ \mu_D=0 \\ (\text{成对数据}) T=D‾SD/nT=\cfrac{\overline{D}}{S_D/\sqrt{n}} μD>0μD<0μD≠0\mu_D>0 \\ \mu_D<0 \\ \mu_D\ne0 T⩾tα(n−1)T⩽tα(n−1)∣T∣⩾tα/2(n−1)T\geqslant t_\alpha(n-1) \\ T\leqslant t_\alpha(n-1) \\ \vert T\vert\geqslant t_{\alpha/2}(n-1) t0=d‾sd/nt_0=\cfrac{\overline{d}}{s_d/\sqrt{n}} P(t(n−1)⩾t0)P(t(n−1)⩽t0)2P(t(n−1)⩾∣t0∣)P(t(n-1)\geqslant t_0) \\ P(t(n-1)\leqslant t_0) \\ 2P(t(n-1)\geqslant \vert t_0\vert)

方差分析、拟合优度检验

原假设H0H_0 检验统计量 备择假设H1H_1 拒绝域 检验统计量的取值 P_P\_值
β1=⋯=βp=0\beta_1=\cdots=\beta_p=0 F=SSR/pSSE/(n−p−1)F=\cfrac{SS_R/p}{SS_E/(n-p-1)} β1,⋯ ,βp≢0\beta_1,\cdots,\beta_p ≢ 0 F⩾Fα(p,n−p−1)F\geqslant F_\alpha(p,n-p-1) f0=SSR/pSSE/(n−p−1)f_0=\cfrac{SS_R/p}{SS_E/(n-p-1)} 未知\text{未知}
βj=0\beta_j=0 Tj=βj^SSEn−p−1cj+1,j+1,j=1,2,⋯ ,pT_j=\cfrac{\widehat{\beta_j}}{\sqrt{\cfrac{SS_E}{n-p-1}}\sqrt{c_{j+1,j+1}}},j=1,2,\cdots,p βj≠0\beta_j\ne0 ∣T∣⩾tα/2(n−p−1)\vert T\vert\geqslant t_{\alpha/2}(n-p-1) tj=βj^SSEn−p−1cj+1,j+1,j=1,2,⋯ ,pt_j=\cfrac{\widehat{\beta_j}}{\sqrt{\cfrac{SS_E}{n-p-1}}\sqrt{c_{j+1,j+1}}},j=1,2,\cdots,p 未知\text{未知}
F(x)=F0(x)F(x)=F_0(x) χ2=∑i=1n(ni−npi^)2npi^n>50,npi^⩾5\chi^2=\sum_{i=1}^n\cfrac{(n_i-n\widehat{p_i})^2}{n\widehat{p_i}} \\ n>50, n\widehat{p_i} \geqslant 5 F(x)≠F0(x)F(x)\ne F_0(x) χ2⩾χα2(k−r−1)k为分类数,r为未知参数个数\chi^2\geqslant \chi^2_\alpha(k-r-1)\\ k\text{为分类数}, r\text{为\text{未知}参数个数} χ2=∑i=1n(ni−npi^)2npi^\chi^2=\sum_{i=1}^n\cfrac{(n_i-n\widehat{p_i})^2}{n\widehat{p_i}} P(χ2(k−r−1)⩾χ02)P(\chi^2(k-r-1)\geqslant \chi^2_0)

方差分析表

方差来源 自由度(df) 平方和(SS) 均方(MS) F
回归 pp SSR=∑i=1n(yi^−y‾)2SS_R=\sum_{i=1}^{n}(\widehat{y_i}-\overline{y})^2 SSR/pSS_R/p F=SSR/pSSE/(n−p−1)F=\cfrac{SS_R/p}{SS_E/(n-p-1)}
残差 n−p−1n-p-1 SSE=∑i=1n(yi−y^)2SS_E=\sum_{i=1}^{n}(y_i-\widehat{y})^2 SSE/(n−p−1)SS_E/(n-p-1)
总和 n−1n-1 SST=∑i=1n(yi−y‾)2SST=SSR+SSESS_T=\sum_{i=1}^{n}(y_i-\overline{y})^2 \\ SS_T=SS_R+SS_E