常用概率分布及其数学期望和方差
分布分布列 pkp_kpk 或分布密度 p(x)p(x)p(x)期望方差0−10-10−1 分布pk=pk(1−p)1−k,k=0,1p_k=p^k(1-p)^{1-k},\quad k=0,1pk=pk(1−p)1−k,k=0,1pppp(1−p)p(1-p)p(1−p)二项分布 b(n,p)b(n,p)b(n,p)pk=(nk)pk(1−p)n−k,k=0,1,⋯p_k=\binom{n}
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| 分布 | 分布列 pkp_kpk 或分布密度 p(x)p(x)p(x) | 期望 | 方差 |
|---|---|---|---|
| 0−10-10−1 分布 | pk=pk(1−p)1−k,k=0,1p_k=p^k(1-p)^{1-k},\quad k=0,1pk=pk(1−p)1−k,k=0,1 | ppp | p(1−p)p(1-p)p(1−p) |
| 二项分布 b(n,p)b(n,p)b(n,p) | pk=(nk)pk(1−p)n−k,k=0,1,⋯p_k=\binom{n}{k}p^k(1-p)^{n-k},\quad k=0,1,\cdotspk=(kn)pk(1−p)n−k,k=0,1,⋯ | npnpnp | np(1−p)np(1-p)np(1−p) |
| 泊松分布 P(λ)P(\lambda)P(λ) | pk=λkk!e−λ,k=0,1,⋯p_k=\frac{\lambda^k}{k!}e^{-\lambda},\quad k=0,1,\cdotspk=k!λke−λ,k=0,1,⋯ | λ\lambdaλ | λ\lambdaλ |
| 超几何分布 h(n,N,M)h(n,N,M)h(n,N,M) | pk=(Mk)(N−Mn−k)(Nn),k=0,1,⋯ ,r,r=min(M,n)p_k=\frac{\binom{M}{k}\binom{N-M}{n-k}}{\binom{N}{n}},\quad k=0,1,\cdots,r,r=\min{(M,n)}pk=(nN)(kM)(n−kN−M),k=0,1,⋯,r,r=min(M,n) | nMNn\frac{M}{N}nNM | nM(N−M)(N−n)N2(N−1)\frac{nM(N-M)(N-n)}{N^2(N-1)}N2(N−1)nM(N−M)(N−n) |
| 几何分布 Ge(p)Ge(p)Ge(p) | pk=(1−p)k−1p,k=1,2,⋯p_k=(1-p)^{k-1}p,\quad k=1,2,\cdotspk=(1−p)k−1p,k=1,2,⋯ | 1p\frac{1}{p}p1 | 1−pp2\frac{1-p}{p^2}p21−p |
| 负二项分布 Nb(r,p)Nb(r,p)Nb(r,p) | pk=(k−1r−1)(1−p)k−rpr,k=r,r+1,⋯p_k=\binom{k-1}{r-1}(1-p)^{k-r}p^r,\quad k=r,r+1,\cdotspk=(r−1k−1)(1−p)k−rpr,k=r,r+1,⋯ | rp\frac{r}{p}pr | r(1−p)p2\frac{r(1-p)}{p^2}p2r(1−p) |
| 正态分布 N(μ,σ2)N(\mu,\sigma^2)N(μ,σ2) | p(x)12πσexp{−(x−μ)22σ2},−∞<x<+∞p(x)\frac{1}{\sqrt{2\pi}\sigma}\exp\left\{-\frac{(x-\mu)^2}{2\sigma^2}\right\},\quad -\infin <x<+\infinp(x)2πσ1exp{−2σ2(x−μ)2},−∞<x<+∞ | μ\muμ | σ2\sigma^2σ2 |
| 均匀分布 U(a,b)U(a,b)U(a,b) | p(x)=1b−a,a<x<bp(x)=\frac{1}{b-a},\quad a<x<bp(x)=b−a1,a<x<b | a+b2\frac{a+b}{2}2a+b | (b−a)212\frac{(b-a)^2}{12}12(b−a)2 |
| 指数分布 Exp(λ)Exp(\lambda)Exp(λ) | p(x)=λe−λx,x⩾0p(x)=\lambda e^{-\lambda x},\quad x\geqslant 0p(x)=λe−λx,x⩾0 | 1λ\frac{1}{\lambda}λ1 | 1λ2\frac{1}{\lambda^2}λ21 |
| 伽马分布 Ga(α,λ)Ga(\alpha,\lambda)Ga(α,λ) | p(x)=λαΓ(α)xα−1e−λx,x⩾0p(x)=\frac{\lambda^\alpha}{\Gamma(\alpha)}x^{\alpha-1}e^{-\lambda x},\quad x\geqslant 0p(x)=Γ(α)λαxα−1e−λx,x⩾0 | αλ\frac{\alpha}{\lambda}λα | αλ2\frac{\alpha}{\lambda^2}λ2α |
| X2(n)\mathcal{X}^2(n)X2(n)分布 | p(x)=xn/2−1e−x/2Γ(n/2)2n/2,x⩾0p(x)=\frac{x^{n/2-1}e^{-x/2}}{\Gamma(n/2)2^{n/2}},\quad x\geqslant 0p(x)=Γ(n/2)2n/2xn/2−1e−x/2,x⩾0 | nnn | 2n2n2n |
| 贝塔分布 Be(a,b)Be(a,b)Be(a,b) | p(x)=Γ(a+b)Γ(a)Γ(b)xα−1(1−x)b−1,0<x<1p(x)=\frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}x^{\alpha-1}(1-x)^{b-1},\quad 0<x<1p(x)=Γ(a)Γ(b)Γ(a+b)xα−1(1−x)b−1,0<x<1 | aa+b\frac{a}{a+b}a+ba | ab(a+b)2(a+b+1)\frac{ab}{(a+b)^2(a+b+1)}(a+b)2(a+b+1)ab |
| 对数正态分布 LN(μ,σ2)LN(\mu,\sigma^2)LN(μ,σ2) | p(x)=12πσxexp{−(lnx−μ)22σ2},x>0p(x)=\frac{1}{\sqrt{2\pi}\sigma x}\exp\left\{-\frac{(\ln x-\mu)^2}{2\sigma^2}\right\},x>0p(x)=2πσx1exp{−2σ2(lnx−μ)2},x>0 | eμ+σ2/2e^{\mu+\sigma^2/2}eμ+σ2/2 | e2μ+σ2(eσ2−1)e^{2\mu+\sigma^2}(e^{\sigma^2}-1)e2μ+σ2(eσ2−1) |
| 柯西分布 Cau(μ,λ)Cau(\mu,\lambda)Cau(μ,λ) | p(x)=1πλλ2+(x−μ)2,−∞<x<+∞p(x)=\frac{1}{\pi}\frac{\lambda}{\lambda^2+(x-\mu)^2},\quad -\infty<x<+\inftyp(x)=π1λ2+(x−μ)2λ,−∞<x<+∞ | 不存在 | 不存在 |
| 韦布尔分布 | p(x)=F′(x),F(x)=1−exp{−(xη)m},x>0p(x)=F'(x),F(x)=1-\exp\left\{-(\frac{x}{\eta})^m\right\},x>0p(x)=F′(x),F(x)=1−exp{−(ηx)m},x>0 | ηΓ(1+1m)\eta\Gamma\left(1+\frac{1}{m}\right)ηΓ(1+m1) | η2[Γ(1+2m)−Γ2(1+1m)]\eta^2\left[\Gamma\left(1+\frac{2}{m}\right)-\Gamma^2\left(1+\frac{1}{m}\right)\right]η2[Γ(1+m2)−Γ2(1+m1)] |
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