Probability Distribution
2022-06-21 10:11:15 0 举报AI智能生成
概率分布
作者其他创作
大纲/内容
连续分布
Uniform Distribution
PDF
<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)="><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\frac{1}{b-a} a<x<b"><span></span><span></span></span>
0 otherwise
CDF
<span class="equation-text" contenteditable="false" data-index="0" data-equation="F(x)="><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="0 x<a"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\frac{x-a}{b-a} a\leq x\leq b"><span></span><span></span></span>
Normal Ditribution
Two-tailed（双尾）
68%=1
90%=1.65
95%=1.96
99%=2.56
One-tailed（单尾）
90%=1.28
95%=1.65
99%=2.33
Standardize
<span class="equation-text" contenteditable="false" data-index="0" data-equation="X\sim{N(\mu,\sigma^2)}\implies{\frac{X-\mu}{\sigma}\sim{Z(0,1)}}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="P(X\geq{x})=P(\frac{x-\mu}{\sigma}\leq{\frac{x-\mu}{\sigma}})=\phi(\frac{x-\mu}{\sigma})"><span></span><span></span></span>
Lognormal Distribution
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\ln(X)\sim{N(\mu,\sigma^2)}\implies {X\sim{\log{N(\mu,\sigma^2)}}}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="X\sim{N(\mu,\sigma^2)}\implies{e^{x}\sim{\log N(\mu,\sigma^2)}}"><span></span><span></span></span>
非负右偏
Student's T-Distribution
PDF
kurtosis(矮峰但肥尾）df=n-1
Chi-Square Distribution
Z1,Z2,Z3...Zk iid 且～Z(0,1)
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\chi^2=Z_{1}^2+Z_{2}^2+....+Z_{k}^2\sim{\chi^2(K)}"><span></span><span></span></span>
非负右偏，自由度k越大，越接近正态分布
应用
Hypothesis Test
<span class="equation-text" contenteditable="false" data-index="0" data-equation="单个总体方差\sigma^2 = a"><span></span><span></span></span>
White Noise检验
F-Distribution
m个服从Z分布的随机变量Zm的平方和
<span class="equation-text" contenteditable="false" data-index="0" data-equation="X_{1}\sim{\chi^2(m)} X_{2}\sim{\chi^2(n)} 且X1，X2独立"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="F(m,n)=\frac{\chi^2(m)/m}{\chi^2(n)/n}"><span></span><span></span></span>
应用
Hypothesis Test
<span class="equation-text" contenteditable="false" data-index="0" data-equation="检验两个总体的标准差 \sigma^2_{1}=\sigma_{2}^2"><span></span><span></span></span>
抽样统计量服从的分布本质为连续分布
类型
Parameter Distribution（参数分布）
数学解析式
Nonparameter Ditribution（非参数分布）
无解析式
离散分布
Possion Distribution
<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)=p(x)=\frac{\lambda^x}{x!}e^{-\lambda}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\mu=\lambda"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\sigma^2=\lambda"><span></span><span></span></span>
度量单位时间内成功x次的概率<br>
Binomial Random Variable
<span class="equation-text" contenteditable="false" data-index="0" data-equation="p(x)=C_{n}^x(P)^x(1-P)^{n-x}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\mu=nP"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\sigma^2=nP(1-P)"><span></span><span></span></span>
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