定理 3.1.2 (随机变量的独立性条件与联合分布)

设 $\left(\Omega ,\mathcal{F},P\right)$ 为概率空间, $\left(X_1,X_2,\cdots ,X_n\right)$ 为其上的随机变量.

(1) 若 $X_1,X_2,\cdots ,X_n$ 都为离散型随机变量, 有分布列

$$P\left(X_i=a_j^{\left(i\right)}\right),j=1,2,\cdots ,i=1,2,\cdots ,n,$$

则 $X_1,X_2,\cdots ,X_n$ 相互独立的充分必要条件是

$$\begin{array}{rl}P\left(X_1=a_{l_1}^{(1)},X_2=a_{l_2}^{(2)},\cdots ,X_n=a_{l_n}^{(n)}\right)& =P\left(X_1=a_{l_1}^{(1)}\right)P\left(X_2=a_{l_2}^{(2)}\right)\cdots P\left(X_n=a_{l_n}^{(n)}\right),\end{array}\text{\hspace{1em}(3.1.7)}$$

其中 $l_1,l_2,\cdots ,l_n$ 取值于 $\{1,2,\cdots \}$.

对于任何区域 $D\subset {\mathbf{R}}^n$, 有

$$\begin{array}{rl}P\left((X_1,X_2,\cdots ,X_n)\in D\right)& =\sum _{\left(a_{l_1}^{(1)},a_{l_2}^{(2)},\cdots ,a_{l_n}^{(n)}\right)\in D}P\left(X_1=a_{l_1}^{(1)},X_2=a_{l_2}^{(2)},\cdots ,X_n=a_{l_n}^{(n)}\right),\end{array}\text{\hspace{1em}(3.1.8)}$$

(2) 若 $X_1,X_2,\cdots ,X_n$, 都为连续型随机变量, 联合分布密度函数为 $f_{X_1,X_2,\cdots ,X_n}$, 边缘分布密度函数为 $f_{X_i}\left(i=1,2,\cdots ,n\right)$, 则 $X_1,X_2,\cdots ,X_n$ 相互独立的充分必要条件是

$$\begin{array}{rl}{f}_{X_1,X_2,\cdots ,X_n}(x_1,x_2,\cdots ,x_n)& =f_{X_1}(x_1)f_{X_2}(x_2)\cdots f_{X_n}(x_n),\forall (x_1,x_2,\cdots ,x_n)\in {\mathbb{R}}^n,\end{array}\text{\hspace{1em}(3.1.9)}$$

对于任何区域 $D\subset {\mathbf{R}}^n$, 有

$$P\left(\left(X_1,X_2,\cdots ,X_n\right)\in D\right)=\int \cdots {\int }_D{f}_{X_1,X_2,\cdots ,X_n}\left(x_1,x_2,\cdots ,x_n\right)\mathrm{d}{x}_1\mathrm{d}{x}_2\cdots \mathrm{d}{x}_n.\text{\hspace{1em}(3.1.10)}$$