# 随机变量本文重定向自 连续随机变量

${\displaystyle S=\lbrace (i,j)|i=1,\ldots ,6,;j=1,\ldots ,6\rbrace }$

${\displaystyle X(i,j):=i+j,x=2,3,\ldots ,12}$
${\displaystyle Y(i,j):=\mid i-j\mid ,y=0,1,2,3,4,5.}$

${\displaystyle X=\lbrace x_{1},x_{2},x_{3},\ldots ,\rbrace }$,

${\displaystyle X=\lbrace x|a\leq x\leq b\rbrace }$, ${\displaystyle -\infty

## 随机变量的函数

${\displaystyle F_{Y}(y)=\operatorname {P} (g(X)\leq y).}$

${\displaystyle F_{Y}(y)=\operatorname {P} (g(X)\leq y)={\begin{cases}\operatorname {P} (X\leq g^{-1}(y))=F_{X}(g^{-1}(y)),&{\text{if }}g^{-1}{\text{ increasing}},\\\\\operatorname {P} (X\geq g^{-1}(y))=1-F_{X}(g^{-1}(y)),&{\text{if }}g^{-1}{\text{ decreasing}}.\end{cases}}}$

${\displaystyle f_{Y}(y)=f_{X}(g^{-1}(y))\left|{\frac {dg^{-1}(y)}{dy}}\right|.}$

### 例子

${\displaystyle F_{Y}(y)=\operatorname {P} (X^{2}\leq y).}$

${\displaystyle F_{Y}(y)=0\qquad {\hbox{if}}\quad y<0.}$

${\displaystyle \operatorname {P} (X^{2}\leq y)=\operatorname {P} (|X|\leq {\sqrt {y}})=\operatorname {P} (-{\sqrt {y}}\leq X\leq {\sqrt {y}}),}$

${\displaystyle F_{Y}(y)=F_{X}({\sqrt {y}})-F_{X}(-{\sqrt {y}})\qquad {\hbox{if}}\quad y\geq 0.}$