概率统计随机过程之两两独立与相互独立

概率统计随机过程之两两独立与相互独立

\(A,B,C\)互相独立，说明\(A,B,C\)间无关联，是互相独立的，但两两独立指\(A\)和\(B\)间独立，\(B\)和\(C\)之间独立，\(A\)和\(C\)间独立，但三者放在一起，并不能判断他们是无关的。

所以，两两独立不一定相互独立，相互独立必然两两独立。

例如：有三个随机变量\(A,B,C\)如果他们两两独立， 那么： \[P(AB)=P(A)P(B)\\ P(AC)=P(A)P(C)\\ P(BC)=P(B)P(C)\]

但是\(P(ABC)\)不一定等于\(P(A)P(B)P(C)\)，

如果相互独立的话，那么上式就是成立的。

请看下例:

例1：设有四张外型一样的卡片,上分别写 有数字\(2,3,5,30\)今从中任取一张观察其上数字: \[A=\{取到是2的倍数\};\\B=\{取到是3的倍数\};\\C=\{取到是5的倍数\};\] 则\(A,B,C\)是两两独立而不是相互独立.

因为: \[A=\{2,30\}; B=\{3,30\}; C=\{5,30\};\\AB=AC=BC=ABC=\{30\}\] 所以: \(P(A)=P(B)=P(C)=2/4=1/2;P(AB)=P(A)P(B)=(1/2)*(1/2)=0.25,\)

类似\(P(BC)=P(B)P(C))=0.25; P(AC)=P(A)P(C))=0.25\)

而\(P(ABC)=(1/2)*(1/2)*(1/2)=0.125,\)

但\(P(A)P(B)P(C)=0.5*0.5*0.5=0.125\)两者不等.

下面这个例子来自维基百科： >例2： Suppose \(X\) and \(Y\) are two independent tosses of a fair coin, where we designate 1 for heads and 0 for tails. Let the third random variable \(Z\) be equal to 1 if exactly one of those coin tosses resulted in "heads", and 0 otherwise(异或). Then jointly the triple \((X, Y, Z)\) has the following probability distribution: >\[(X,Y,Z)=\left\{{\begin{matrix}(0,0,0)&{\text{with probability}}\ 1/4,\\(0,1,1)&{\text{with probability}}\ 1/4,\\(1,0,1)&{\text{with probability}}\ 1/4,\\(1,1,0)&{\text{with probability}}\ 1/4.\end{matrix}}\right.\] >Here the marginal probability distributions are identical: \(f_{X}(0)=f_{Y}(0)=f_{Z}(0)=1/2\), and \(f_{X}(1)=f_{Y}(1)=f_{Z}(1)=1/2\). The bivariate distributions also agree: \(f_{{X,Y}}=f_{{X,Z}}=f_{{Y,Z}}\), where \(f_{{X,Y}}(0,0)=f_{{X,Y}}(0,1)=f_{{X,Y}}(1,0)=f_{{X,Y}}(1,1)=1/4\). > >Since each of the pairwise joint distributions equals the product of their respective marginal distributions, the variables are pairwise independent: > >- X and Y are independent, and >- X and Z are independent, and >- Y and Z are independent. > >However, \(X, Y\), and \(Z\) are not mutually independent, since \({\displaystyle f_{X,Y,Z}(x,y,z)\neq f_{X}(x)f_{Y}(y)f_{Z}(z),}\) the left side equalling for example 1/4 for \((x, y, z) = (0, 0, 0)\) while the right side equals 1/8 for \((x, y, z) = (0, 0, 0)\). In fact, any of \(\{X,Y,Z\}\) is completely determined by the other two (any of \(X, Y, Z\) is the sum (modulo(模) 2) of the others). That is as far from independence as random variables can get.

总的来说，有些随机变量可以由其他随机变量决定，因此，两两独立的随机变量之间可形成组合和其他随机变量形成联系。