# 三角形分布

参数 概率密度函數 累積分布函數 ${\displaystyle a:~a\in (-\infty ,\infty )}$${\displaystyle b:~b>a\,}$${\displaystyle c:~a\leq c\leq b\,}$ ${\displaystyle a\leq x\leq b\!}$ ${\displaystyle \left\{{\begin{matrix}{\frac {2(x-a)}{(b-a)(c-a)}}&\mathrm {for\ } a\leq x\leq c\\&\\{\frac {2(b-x)}{(b-a)(b-c)}}&\mathrm {for\ } c ${\displaystyle \left\{{\begin{matrix}{\frac {(x-a)^{2}}{(b-a)(c-a)}}&\mathrm {for\ } a\leq x\leq c\\&\\1-{\frac {(b-x)^{2}}{(b-a)(b-c)}}&\mathrm {for\ } c ${\displaystyle {\frac {a+b+c}{3}}}$ ${\displaystyle \left\{{\begin{matrix}a+{\frac {\sqrt {(b-a)(c-a)}}{\sqrt {2}}}&\mathrm {for\ } c\!\geq \!{\frac {b\!-\!a}{2}}\\&\\b-{\frac {\sqrt {(b-a)(b-c)}}{\sqrt {2}}}&\mathrm {for\ } c\!\leq \!{\frac {b\!-\!a}{2}}\end{matrix}}\right.}$ ${\displaystyle c\,}$ ${\displaystyle {\frac {a^{2}+b^{2}+c^{2}-ab-ac-bc}{18}}}$ ${\displaystyle {\frac {{\sqrt {2}}(a\!+\!b\!-\!2c)(2a\!-\!b\!-\!c)(a\!-\!2b\!+\!c)}{5(a^{2}\!+\!b^{2}\!+\!c^{2}\!-\!ab\!-\!ac\!-\!bc)^{\frac {3}{2}}}}}$ ${\displaystyle {\frac {12}{5}}}$ ${\displaystyle {\frac {1}{2}}+\ln \left({\frac {b-a}{2}}\right)}$ ${\displaystyle 2{\frac {(b\!-\!c)e^{at}\!-\!(b\!-\!a)e^{ct}\!+\!(c\!-\!a)e^{bt}}{(b-a)(c-a)(b-c)t^{2}}}}$ ${\displaystyle -2{\frac {(b\!-\!c)e^{iat}\!-\!(b\!-\!a)e^{ict}\!+\!(c\!-\!a)e^{ibt}}{(b-a)(c-a)(b-c)t^{2}}}}$

${\displaystyle f(x|a,b,c)=\left\{{\begin{matrix}{\frac {2(x-a)}{(b-a)(c-a)}}&\mathrm {for\ } a\leq x\leq c\\&\\{\frac {2(b-x)}{(b-a)(b-c)}}&\mathrm {for\ } c

## 特例

### 已知两点

c=a 或者 c=b，分布就可以进行简化。例如，如果 a=0、b=1 并且 c=1，那么上面的方程简化为：

${\displaystyle \left.{\begin{matrix}f(x)&=&2x\\\\F(x)&=&x^{2}\end{matrix}}\right\}\mathrm {for\ } 0\leq x\leq 1}$
${\displaystyle {\begin{matrix}E(X)&=&{\frac {2}{3}}\\&&\\\mathrm {Var} (X)&=&{\frac {1}{18}}\end{matrix}}}$

### 两个标准一致变量的分布

a=0、b=1 且 c=0.5 的分布为 ${\displaystyle X={\frac {X_{1}+X_{2}}{2}}}$，其中 ${\displaystyle X_{1},X_{2}}$ 是两个連續型均勻分佈随机变量

${\displaystyle f(x)=\left\{{\begin{matrix}4x&\mathrm {for\ } 0\leq x<{\frac {1}{2}}\\\\4-4x&\mathrm {for\ } {\frac {1}{2}}\leq x\leq 1\end{matrix}}\right.}$
${\displaystyle F(x)=\left\{{\begin{matrix}2x^{2}&\mathrm {for\ } 0\leq x<{\frac {1}{2}}\\\\1-2(1-x)^{2}&\mathrm {for\ } {\frac {1}{2}}\leq x\leq 1\end{matrix}}\right.}$
${\displaystyle {\begin{matrix}E(X)&=&{\frac {1}{2}}\\\\\mathrm {Var} (X)&=&{\frac {1}{24}}\end{matrix}}}$