多维随机变量函数的分布

Distribution of Functions of Multivariate Random Variables 随机变量函数的分布 多维随机变量

graph LR
1[["多维随机变量
函数的分布"]]--> 连续型
1----> 离散型
连续型--->一般情形 & 和 & 商与乘积 & 极值
和-->不相互独立
和-->相互独立-.->卷积
极值--> min & max

主要分类

1. 多维离散型随机变量函数的分布

多维离散型随机变量的函数依然是离散型随机变量 只需要结合题意，明确随机变量的取值，熟悉基本的离散分布即可

• 找到函数的取值范围 (离散的值)
• 计算每一个取值的概率

2. 多维连续型随机变量函数的分布

注意：多维连续型随机变量的函数 不一定是连续型随机变量

• 如果离散，找所有离散的值及其概率即可
• 如果连续，按照下面来计算

注意

以下的章节主要谈论：多维连续型随机变量的函数为连续型随机变量的情况

一、一般情形

已知二维随机变量 $(X,Y)$ 的联合密度函数 $f(x,y)$ 求随机变量 $Z=g(X,Y)$ 的分布函数 $F_Z(z)$

分布函数法：

F_{Z}(z)&=P\left\{Z\leq z \right\} \\ &=P\left\{g(X,Y) \leq z\right\} \\ & =\iint \limits_{g(x,y)\leq z}f(x,y)dxdy\\ f_{Z}(z)&=F'_{Z}(z) \end{align}$$ 本质上就是求[[二重积分\|二重积分]]，关键是找好积分区域 $g(x,y)\leq z$ 因为积分区域是关于 $z$ 的函数（积分限也可能为 $z$ 的函数，可以将 $z$ 暂时看为常数，只考虑 $x,y$） 所以对 $x,y$ 进行二重积分时，会将 $x,y$ 变量消去，最终得到 $z$ 的函数，也即 $z$ 的分布函数，再进行求导，得到 $z$ 的概率密度函数 ### 二、和的分布 二维随机变量 $(X,Y)$ 的联合密度函数 $f(x,y)$ $Z=X+Y$，求 $f_{Z}(z)$ $$\begin{align} F_{Z}(z)&=P\left\{Z\leq z \right\} \\ &=P\left\{X+Y\leq z \right\} \\ &=\iint \limits_{x+y\leq z}f(x,y)dxdy \\ &=\int _{-\infty}^{+\infty} \, dx \int _{-\infty}^{z-x} f(x,y)\, dy \\ &=\int _{-\infty}^{+\infty} \, dx \int _{-\infty}^{z} f(x,u-x)\, du \quad (y=u-x) \\ & =\int _{-\infty}^{z} \, du \int _{-\infty}^{+\infty} f(x,u-x)\, dx \end{align}$$ 所以 $Z=X+Y$ 为连续型随机变量，且密度函数为： $$\begin{align} f_{Z}(z)=\int _{-\infty}^{+\infty} f(x,z-x)\, dx \\ \end{align}$$ 同理： $$\begin{align} f_{Z}(z)=\int _{-\infty}^{+\infty} f(z-y,y)\, dy \end{align}$$ >[!important] 确定实际的积分区域！ > 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y="14.720312499999999" font-family="Helvetica, Segoe UI Emoji" font-size="16px" fill="#1e1e1e" text-anchor="start" style="white-space: pre;" direction="ltr" dominant-baseline="alphabetic">z=2</text></g></svg> 特别地：如果随机变量 $X,Y$ [[随机变量的独立性\|相互独立]]，则有 $f(x,y)=f_{X}(x)f_{Y}(y)$ $$\begin{align} f_{Z}(z)&=\int _{-\infty}^{+\infty} f_{X}(x)f_{Y}(z-y)\, dx \\ &=\int _{-\infty}^{+\infty} f_{X}(z-y)f_{Y}(y)\, dy \\ &=f_{X}(x)*f_{Y}(y) \end{align}$$ 两个随机变量相互独立，则它们和的密度函数等于 X 与 Y密度函数的[[卷积\|卷积]] 正态分布 $X\sim N(a,b^{2})$ $Y\sim N(c,d^{2})$ ### 三、商和乘积的分布 二维随机变量 $(X,Y)$ 的联合密度函数 $f(x,y)$ #### 商的分布 $$\begin{align} Z=\dfrac{Y}{X} \quad \Rightarrow \quad f_{Z}(z)=\int _{-\infty}^{+\infty} |x|f(x,xz)\, dx \end{align}$$ #### 乘积的分布 $$\begin{align} Z=XY \quad \Rightarrow \quad f_{Z}(z)=\int _{-\infty}^{+\infty} \dfrac{1}{|x|} f(x,\dfrac{z}{x})\, dx \end{align}$$ <svg xmlns="http://www.w3.org/2000/svg" version="1.1" viewBox="0 0 602.5309244791663 264" width="602.5309244791663" height="264"> <!-- 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C141.83 -33.07, 210.46 -5.51, 231.92 0 M0 0 C17.2 -5.51, 64.52 -33.07, 103.17 -33.07 C141.83 -33.07, 210.46 -5.51, 231.92 0" stroke="#e03131" stroke-width="2" fill="none"></path></g><g transform="translate(129.5167875744046 74.95309297821404) rotate(0 115.96118745349713 -16.53438023158492)"><path d="M206.94 1 C216.28 0.62, 225.62 0.25, 231.92 0 M206.94 1 C212.8 0.76, 218.66 0.53, 231.92 0" stroke="#e03131" stroke-width="2" fill="none"></path></g><g transform="translate(129.5167875744046 74.95309297821404) rotate(0 115.96118745349713 -16.53438023158492)"><path d="M212.15 -15.29 C219.54 -9.58, 226.94 -3.86, 231.92 0 M212.15 -15.29 C216.78 -11.71, 221.42 -8.12, 231.92 0" stroke="#e03131" stroke-width="2" fill="none"></path></g></g><mask></mask></svg> >[!important] 注意 > >实际计算时，注意 $x$ 的积分上下限！！！ >利用题目信息，明确积分范围 >常常因为变量 $z$ 取值的不同，积分的表达式有所区别 >当进行变量代换时，要找好 $z$ 与 $x$ 的关系 ### 四、极值分布 $X_{1},X_{2},\cdots,X_{n}$ 是相互[[随机变量的独立性\|独立]]的随机变量 $X_{i}$ 的分布函数为 $F_{i}(x)$ $$\begin{align} X_{(1)}=min\left\{X_{1},X_{2},\cdots,X_{n} \right\} \quad 最小顺序统计量\\ X_{(n)}=max\left\{X_{1},X_{2},\cdots,X_{n} \right\} \quad 最大顺序统计量 \end{align}$$ $$\begin{align} F_{(n)}(x)&=P\left\{X_{(n)}\leq x \right\} \\ &=P\left\{max\left\{X_{1},X_{2},\cdots,X_{n} \right\} \leq x\right\} \\ &=P\left\{X_{1}\leq x \right\}P\left\{X_{2}\leq x \right\}\cdots P\left\{X_{n}\leq x \right\}\\ &=\prod\limits_{i=1}^{n}F_{i}(x)\\ \\ F_{(1)}(x) &=P\left\{X_{(1)} \leq x\right\} \\ &=P\left\{min\left\{X_{1},X_{2},\cdots,X_{n} \right\}\leq x \right\} \\ &=1-P\left\{ min\left\{X_{1},X_{2},\cdots,X_{n} \right\}>x\right\} \\ &=1-P\left\{X_{1}>x \right\}P\left\{X_{2}>x \right\}\cdots P\left\{X_{n}>x \right\} \\ &=1-(1-P\left\{X_{1}\leq x \right\})\cdots(1-P\left\{X_{n}\leq x \right\})\\ &=1-\prod\limits_{i=1}^{n}(1-F_{i}(x)) \end{align}$$ 最小值的分布,转为对立事件讨论 $$\begin{align} f_{(1)}(x)=\dfrac{\mathrm{d} }{\mathrm{d} x} \left[F_{(1)}(x)\right] = \sum\limits_{i=1}^{n}f_{i}(x)\prod\limits_{j=1,j\neq i}^{n}\left[1-F_{j}(x)\right] \end{align}$$ $$\begin{align} f_{n}(x)=\dfrac{\mathrm{d} }{\mathrm{d} x} \left[F_{n}(x)\right] =\sum\limits_{i=1}^{n}f_{i}(x)\prod\limits_{j=1,j\neq i}^{n}F_{j}(x) \end{align}$$ 如果随机变量服从相同的分布，[[独立同分布\|独立同分布]]，也可简单记为： $$\begin{align} f_{(1)}(x)&=nf(x)[1-F(x)]^{n-1} \\ f_{(n)}(x)&=nf(x)[F(x)]^{n-1} \end{align}$$