# 連續型均勻分布

参数 概率密度函數 累積分布函數 ${\displaystyle a,b\in (-\infty ,\infty )\,\!}$ ${\displaystyle a\leq x\leq b\,\!}$ ${\displaystyle {\begin{matrix}{\frac {1}{b-a}}&{\mbox{for }}a\leq x\leq b\\\\0&\mathrm {for} \ xb\end{matrix}}\,\!}$ ${\displaystyle {\begin{matrix}0&{\mbox{for }}x ${\displaystyle {\frac {a+b}{2}}\,\!}$ ${\displaystyle {\frac {a+b}{2}}\,\!}$ 任何${\displaystyle [a,b]\,\!}$内的值 ${\displaystyle {\frac {(b-a)^{2}}{12}}\,\!}$ ${\displaystyle 0\,\!}$ ${\displaystyle -{\frac {6}{5}}\,\!}$ ${\displaystyle \ln(b-a)\,\!}$ ${\displaystyle {\frac {e^{tb}-e^{ta}}{t(b-a)}}\,\!}$ ${\displaystyle {\frac {e^{itb}-e^{ita}}{it(b-a)}}\,\!}$

## 定义

${\displaystyle f(x)=\left\{{\begin{matrix}{\frac {1}{b-a}}&\ \ \ {\mbox{for }}a\leq x\leq b\\0&{\mbox{elsewhere}}\end{matrix}}\right.}$
${\displaystyle F(x)=\left\{{\begin{matrix}0&{\mbox{for }}x

MGF：

${\displaystyle M_{X}(t)=E(e^{tx})={\frac {e^{tb}-e^{ta}}{t(b-a)}}}$

## 公式

${\displaystyle E[X]={\frac {a+b}{2}}}$

${\displaystyle VAR[X]={\frac {(b-a)^{2}}{12}}}$

${\displaystyle P(c\leq x\leq d)=F(d)-F(c)=\int _{c}^{d}{\frac {1}{b-a}}\,dx={\frac {d-c}{b-a}}}$