常见的分布期望及其方差

一时失言乱红尘 2023-06-16 03:52 11阅读 0赞

<table> 
 <thead> 
 <tr> 
 <th>分布名称</th> 
 <th>表达式</th> 
 <th>期望</th> 
 <th>方差</th> 
 </tr> 
 </thead> 
 <tbody> 
 <tr> 
 <td>0-1分布</td> 
 <td> P i = P ( X = i ) = p i p n − i , ( i = 0 , 1 ) P_i=P({X=i})=p^ip^{n-i},(i=0,1) Pi=P(X=i)=pipn−i,(i=0,1)</td> 
 <td>p</td> 
 <td>p(1-p)</td> 
 </tr> 
 <tr> 
 <td>二项分布</td> 
 <td> P i = P ( X = i ) = C n i p i p n − i , ( i = 0 , 1 , 2 , 3... ) P_i=P({X=i})=C^i_np^ip^{n-i},(i=0,1,2,3...) Pi=P(X=i)=Cnipipn−i,(i=0,1,2,3...)</td> 
 <td>np</td> 
 <td>np(1-p)</td> 
 </tr> 
 <tr> 
 <td>泊松分布</td> 
 <td> P i = P ( X = i ) = λ i i ! e − λ x , ( i = 0 , 1 , 2 , 3... ) P_i=P({X=i})=\frac{\lambda ^i}{i!}e^{-\lambda x},(i=0,1,2,3...) Pi=P(X=i)=i!λie−λx,(i=0,1,2,3...)</td> 
 <td> λ \lambda λ</td> 
 <td> λ \lambda λ</td> 
 </tr> 
 <tr> 
 <td>指数分布</td> 
 <td> f ( x ) = { λ e − λ x , 0 &lt; x 0 , x &lt; = 0 f(x)=\begin{cases}\lambda e^{-\lambda x},\quad &amp;0&lt;x \\0, &amp;x&lt;=0\end{cases} f(x)={ λe−λx,0,0&lt;xx&lt;=0</td> 
 <td> 1 λ \frac{1}{\lambda} λ1</td> 
 <td> 1 λ 2 \frac{1}{\lambda^2} λ21</td> 
 </tr> 
 <tr> 
 <td>几何分布</td> 
 <td> P i = P ( X = i ) = p ( 1 − p ) i − 1 , ( i = 0 , 1 , 2 , 3... ) P_i=P({X=i})=p(1-p)^{i-1},(i=0,1,2,3...) Pi=P(X=i)=p(1−p)i−1,(i=0,1,2,3...)</td> 
 <td> 1 p \frac{1}{p} p1</td> 
 <td> 1 − p p 2 \frac{1-p}{p^2} p21−p</td> 
 </tr> 
 <tr> 
 <td>超几何分布</td> 
 <td> P i = P ( X = i ) = C M i C N − M n − i C N n , ( i = 0 , 1 , 2 , 3... ) P_i=P{(X=i)}=\frac{C_M^iC_{N-M}^{n-i}}{C_N^n},(i=0,1,2,3...) Pi=P(X=i)=CNnCMiCN−Mn−i,(i=0,1,2,3...)</td> 
 <td> n M N n\frac{M}{N} nNM</td> 
 <td> n M N ( 1 − M N ) N − n n − 1 n\frac{M}{N}(1-\frac{M}{N})\frac{N-n}{n-1} nNM(1−NM)n−1N−n</td> 
 </tr> 
 <tr> 
 <td>正态分布</td> 
 <td> f ( x ) = 1 σ 2 π e − ( x − u ) 2 2 σ 2 , ( − ∞ &lt; x &lt; + ∞ ) f(x)=\frac{1}{\sigma \sqrt{2 \pi}}e^{-\frac{(x-u)^2}{2 \sigma ^2}},(-\infty &lt;x &lt;+\infty) f(x)=σ2π 1e−2σ2(x−u)2,(−∞&lt;x&lt;+∞)</td> 
 <td>u</td> 
 <td> σ 2 \sigma ^2 σ2</td> 
 </tr> 
 <tr> 
 <td> χ 2 \chi^2 χ2分布</td> 
 <td> χ 2 = x 1 2 + x 2 2 + . . . . + x n 2 , ( X 1 , X 2 , . . . , X n 相 互 独 立 ， 且 都 服 从 标 准 正 态 分 布 N ( 0 , 1 ) ) \chi ^2 = x_1^2+x_2^2+....+x_n^2,(X_1,X_2,...,X_n相互独立，且都服从标准正态分布N(0,1)) χ2=x12+x22+....+xn2,(X1,X2,...,Xn相互独立，且都服从标准正态分布N(0,1))</td> 
 <td>n</td> 
 <td>2n</td> 
 </tr> 
 <tr> 
 <td>均匀分布</td> 
 <td> f ( x ) = 1 b − 1 f(x) = \frac{1}{b-1} f(x)=b−11</td> 
 <td> a + b 2 \frac{a+b}{2} 2a+b</td> 
 <td> ( b − a ) 2 12 \frac{(b-a)^2}{12} 12(b−a)2</td> 
 </tr> 
 </tbody> 
</table>