本文综述了多种常见的概率分布，包括伯努利分布、二项分布、泊松分布、正态分布等，提供了每种分布的概率密度函数、期望、方差及特征函数等关键信息。
伯努利分布 Bernoulli Distribution
f(x∣p)={px(1−p)1−xx=0,1,0otherwise.
f(x|p)=\left\{
\begin{aligned}
&p^x(1-p)^{1-x}&&x=0,1,\\
&0&&\text{otherwise.}
\end{aligned}
\right.
f(x∣p)={px(1−p)1−x0x=0,1,otherwise.
E(X)=pVar(X)=p(1−p)ψ(t)=pet+1−p E(X)=p\\ Var(X)=p(1-p)\\ \psi(t)=pe^t+1-p E(X)=pVar(X)=p(1−p)ψ(t)=pet+1−p
二项分布 Binomial Distribution
f(x∣n,p)={(nx)px(1−p)n−xx=0,1,2,⋯ ,n,0otherwise.
f(x|n,p)=\left\{
\begin{aligned}
&\binom{n}{x}p^x(1-p)^{n-x}&&x=0,1,2,\cdots,n,\\
&0&&\text{otherwise.}
\end{aligned}
\right.
f(x∣n,p)=⎩⎪⎨⎪⎧(xn)px(1−p)n−x0x=0,1,2,⋯,n,otherwise.
E(X)=npVar(X)=np(1−p)ψ(t)=(pet+1−p)n E(X)=np\\ Var(X)=np(1-p)\\ \psi(t)=(pe^t+1-p)^n E(X)=npVar(X)=np(1−p)ψ(t)=(pet+1−p)n
泊松分布 Poisson Distribution
f(x∣λ)={e−λλxx!x=0,1,2,⋯ ,0otherwise.
f(x|\lambda)=\left\{
\begin{aligned}
&\frac{e^{-\lambda}\lambda^x}{x!}&&x=0,1,2,\cdots,\\
&0&&\text{otherwise.}
\end{aligned}
\right.
f(x∣λ)=⎩⎪⎨⎪⎧x!e−λλx0x=0,1,2,⋯,otherwise.
E(X)=λVar(X)=λψ(t)=eλ(et−1) E(X)=\lambda\\ Var(X)=\lambda\\ \psi(t)=e^{\lambda(e^t-1)} E(X)=λVar(X)=λψ(t)=eλ(et−1)
正态分布 Normal Distribution
f(x∣n,p)=12πσexp[−12(x−μσ)2] −∞<x<∞.
f(x|n,p)=\frac{1}{\sqrt{2\pi}\sigma}\exp[-\frac{1}{2}(\frac{x-\mu}{\sigma})^2]\ \ \ \ -\infin<x<\infin.
f(x∣n,p)=2πσ1exp[−21(σx−μ)2] −∞<x<∞.
E(X)=μVar(X)=σ2ψ(t)=exp(μt+12σ2t2) E(X)=\mu\\ Var(X)=\sigma^2\\ \psi(t)=\exp(\mu t+\frac{1}{2}\sigma^2t^2) E(X)=μVar(X)=σ2ψ(t)=exp(μt+21σ2t2)
伽马分布 Gamma Distribution
f(x∣α,β)={βαΓ(α)xα−1e−βxx>00otherwise.
f(x|\alpha,\beta)=\left\{
\begin{aligned}
&\frac{\beta^\alpha}{\Gamma(\alpha)}x^{\alpha-1}e^{-\beta x}&&x>0\\
&0&&\text{otherwise.}
\end{aligned}
\right.
f(x∣α,β)=⎩⎪⎨⎪⎧Γ(α)βαxα−1e−βx0x>0otherwise.
E(X)=αβVar(X)=αβ2ψ(t)=(ββ−t)α t<β E(X)=\frac{\alpha}{\beta}\\ Var(X)=\frac{\alpha}{\beta^2}\\ \psi(t)=(\frac{\beta}{\beta-t})^\alpha\ \ \ \ t<\beta E(X)=βαVar(X)=β2αψ(t)=(β−tβ)α t<β
指数分布 Exponential Distribution
f(x∣β)={βe−βxx>00otherwise.
f(x|\beta)=\left\{
\begin{aligned}
&\beta e^{-\beta x}&&x>0\\
&0&&\text{otherwise.}
\end{aligned}
\right.
f(x∣β)={βe−βx0x>0otherwise.
E(X)=1βVar(X)=1β2ψ(t)=ββ−t t<β E(X)=\frac{1}{\beta}\\ Var(X)=\frac{1}{\beta^2}\\ \psi(t)=\frac{\beta}{\beta-t}\ \ \ \ t<\beta E(X)=β1Var(X)=β21ψ(t)=β−tβ t<β
贝塔分布 Beta Distribution
f(x∣α,β)={Γ(α)Γ(β)Γ(α+β)xα−1(1−x)β−10<x<10otherwise.
f(x|\alpha,\beta)=\left\{
\begin{aligned}
&\frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)}x^{\alpha-1}(1-x)^{\beta-1}&&0<x<1\\
&0&&\text{otherwise.}
\end{aligned}
\right.
f(x∣α,β)=⎩⎪⎨⎪⎧Γ(α+β)Γ(α)Γ(β)xα−1(1−x)β−100<x<1otherwise.
E(X)=αα+βVar(X)=αβ(α+β)2(α+β+1)ψ(t)=unknown E(X)=\frac{\alpha}{\alpha+\beta}\\ Var(X)=\frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}\\ \psi(t)=\text{unknown} E(X)=α+βαVar(X)=(α+β)2(α+β+1)αβψ(t)=unknown
卡方分布 Chi-Square Distribution
f(x∣α,β)={12m2Γ(m2)xm2−1e−x2x>00otherwise.
f(x|\alpha,\beta)=\left\{
\begin{aligned}
&\frac{1}{2^{\frac{m}{2}}\Gamma(\frac{m}{2})}x^{\frac{m}{2}-1}e^{-\frac{x}{2}}&&x>0\\
&0&&\text{otherwise.}
\end{aligned}
\right.
f(x∣α,β)=⎩⎪⎨⎪⎧22mΓ(2m)1x2m−1e−2x0x>0otherwise.
E(X)=mVar(X)=2mψ(t)=(11−2t)m2 t<12 E(X)=m\\ Var(X)=2m\\ \psi(t)=(\frac{1}{1-2t})^\frac{m}{2}\ \ \ \ t<\frac{1}{2} E(X)=mVar(X)=2mψ(t)=(1−2t1)2m t<21
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