第二章 随机变量与分布函数
独立随机变量
独立性
设$\mathbf{x}=(x_1,x_2,\cdots,x_m)$是随机向量,联合分布为$F_\mathbf{X}(\mathbf{x})$,边际分布为$F_{X_i}(x_i)$,那么如果对所有的$\mathbf{x}$,都有
\[F_\mathbf{X}(\mathbf{x})=\prod_{i=1}^m F_{X_i}(x_i)\]那么称$X_1,X_2,\cdots,X_m$相互独立。
随机变量的运算
加减
考虑$Z=X+Y$的分布:
\[\begin{aligned} F_Z(z)&=P(X+Y\leq z) \\ &=\int_{(x,y):x+y\leq z}p(x,y)dxdy \\ &=\int_{-\infty}^\infty \int_{-\infty}^{z-x} p(x,y)dydx \\ &=\int_{-\infty}^z\int_{-\infty}^\infty p(x,z^*-x)dxdz^* (令z^*=x+y) \\ p_Z(z)&=\int_{-\infty}^\infty p(x,z-x)dx \end{aligned}\]同理,$Z=X-Y$的分布:
\[\begin{aligned} F_Z(z)&=P(Y-X\leq z) \\ &=\int_{(x,y):y-x\leq z}p(x,y)dxdy \\ &=\int_{-\infty}^\infty \int_{-\infty}^{x+z} p(x,y)dydx \\ &=\int_{-\infty}^z\int_{-\infty}^\infty p(x,z^*+x)dxdz^* (令z^*=y-x) \\ p_Z(z)&=\int_{-\infty}^\infty p(x,z+x)dx \end{aligned}\]乘除
$Z=X*Y$的分布:
\[\begin{aligned} p_Z(z)&=\int_{-\infty}^\infty \frac{1}{|x|}p(x,\frac{z}{x})dx \end{aligned}\]$Z=\frac{Y}{X}$的分布:
\[\begin{aligned} p_Z(z)&=\int_{-\infty}^\infty |x|p(x,zx)dx \end{aligned}\]一般的变换
假设基于$(X,Y)$的变换如下:
\[\begin{cases} &U=f_1(X,Y) \\ &V=f_2(X,Y) \end{cases}\]假设存在逆变换:
\[\begin{cases} &X=g_1(U,V) \\ &Y=g_2(U,V) \end{cases}\]且$g_1,g_2$可导,Jacobi变换存在,行列式为:
\[J=det\Bigg(\begin{matrix} \frac{\partial{g_1}}{\partial{u}} & \frac{\partial{g_1}}{\partial{v}} \\ \frac{\partial{g_2}}{\partial{u}} & \frac{\partial{g_2}}{\partial{v}} \end{matrix} \Bigg)\]则有
\[p_{(U,V)}(u,v)=p_{(X,Y)}(g_1(u,v),g_2(u,v))|J|\]次序统计量
设$X_1,X_2,\cdots,X_n$是独立随机变量,则按从小到大排序得到$X_{(1)},X_{(2)},\cdots,X_{(n)}$,称为次序统计量,则
\[\begin{aligned} F_{X_{(n)}}(x)&=[F(x)]^n \\ F_{X_{(1)}}(x)&=1-(1-F(x))^n \\ F_{X_{(k)}}(x)&=(n-k+1)C_{n}^{k-1}F^{k-1}(x)p(x)(1-F(x))^{n-k} \end{aligned}\]