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連續型均勻分布 .
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连续型均匀分布
概率密度函數
累積分布函數
参数
a
,
b
∈
(
−
∞
,
∞
)
{\displaystyle a,b\in (-\infty ,\infty )\,\!}
值域
a
≤
x
≤
b
{\displaystyle a\leq x\leq b\,\!}
概率密度函数
1
b
−
a
for
a
≤
x
≤
b
0
f
o
r
x
<
a
o
r
x
>
b
{\displaystyle {\begin{matrix}{\frac {1}{b-a))&{\mbox{for ))a\leq x\leq b\\\\0&\mathrm {for} \ x<a\ \mathrm {or} \ x>b\end{matrix))\,\!}
累積分布函數
0
for
x
<
a
x
−
a
b
−
a
for
a
≤
x
<
b
1
for
x
≥
b
{\displaystyle {\begin{matrix}0&{\mbox{for ))x<a\\{\frac {x-a}{b-a))&~~~~~{\mbox{for ))a\leq x<b\\1&{\mbox{for ))x\geq b\end{matrix))\,\!}
期望值
a
+
b
2
{\displaystyle {\frac {a+b}{2))\,\!}
中位數
a
+
b
2
{\displaystyle {\frac {a+b}{2))\,\!}
眾數
任何
[
a
,
b
]
{\displaystyle [a,b]\,\!}
方差
(
b
−
a
)
2
12
{\displaystyle {\frac {(b-a)^{2)){12))\,\!}
偏度
0
{\displaystyle 0\,\!}
峰度
−
6
5
{\displaystyle -{\frac {6}{5))\,\!}
熵
ln
(
b
−
a
)
{\displaystyle \ln(b-a)\,\!}
矩生成函数
e
t
b
−
e
t
a
t
(
b
−
a
)
{\displaystyle {\frac {e^{tb}-e^{ta)){t(b-a)))\,\!}
特徵函数
e
i
t
b
−
e
i
t
a
i
t
(
b
−
a
)
{\displaystyle {\frac {e^{itb}-e^{ita)){it(b-a)))\,\!}
連續型均匀分布 （英語：continuous uniform distribution ）或矩形分布 （rectangular distribution ）的随机变量
X
{\displaystyle {\mathit {X))}
概率密度函数 在該變量的值域內為常數。若
X
{\displaystyle X}
[
a
,
b
]
{\displaystyle [a,b]}
X
∼
U
[
a
,
b
]
{\displaystyle X\sim U[a,b]}
定义
一个均匀分布在区间[a,b]上的连续型随机变量
X
{\displaystyle X}
概率密度函数 ：
f
(
x
)
=
{
1
b
−
a
for
a
≤
x
≤
b
0
elsewhere
{\displaystyle f(x)=\left\((\begin{matrix}{\frac {1}{b-a))&\ \ \ {\mbox{for ))a\leq x\leq b\\0&{\mbox{elsewhere))\end{matrix))\right.}
累积分布函数 ：
F
(
x
)
=
{
0
for
x
<
a
x
−
a
b
−
a
for
a
≤
x
<
b
1
for
x
≥
b
{\displaystyle F(x)=\left\((\begin{matrix}0&{\mbox{for ))x<a\\{\frac {x-a}{b-a))&\ \ \ {\mbox{for ))a\leq x<b\\1&{\mbox{for ))x\geq b\end{matrix))\right.}
MGF：
M
X
(
t
)
=
E
(
e
t
x
)
=
e
t
b
−
e
t
a
t
(
b
−
a
)
{\displaystyle M_{X}(t)=E(e^{tx})={\frac {e^{tb}-e^{ta)){t(b-a)))}
公式
期望值 和中值 ：
是指连续型均匀分布函数的期望值和中值等于区间[a,b]上的中间点。
E
[
X
]
=
a
+
b
2
{\displaystyle E[X]={\frac {a+b}{2))}
方差 ：
V
A
R
[
X
]
=
(
b
−
a
)
2
12
{\displaystyle VAR[X]={\frac {(b-a)^{2)){12))}
均匀分布具有下属意义的等可能性。若
X
∼
U
[
a
,
b
]
{\displaystyle X\sim U[a,b]}
概率 ：
P
(
c
≤
x
≤
d
)
=
F
(
d
)
−
F
(
c
)
=
∫
c
d
1
b
−
a
d
x
=
d
−
c
b
−
a
{\displaystyle P(c\leq x\leq d)=F(d)-F(c)=\int _{c}^{d}{\frac {1}{b-a))\,dx={\frac {d-c}{b-a))}
只与区间[c,d]的长度有关，而与它的位置无关。
離散單變量
有限支集 無限支集
beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Flory–Schulz Gauss–Kuzmin 幾何分佈 对数分布 mixed Poisson 负二项分布 Panjer parabolic fractal 卜瓦松分布 Skellam Yule–Simon zeta
連續單變量
混合單變量
联合分布
Discrete: Ewens multinomial Continuous: 狄利克雷分布
multivariate Laplace 多元正态分布 multivariate stable multivariate t normal-gamma 随机矩阵 LKJ 矩阵正态分布 matrix t matrix gamma 威沙特分佈
定向統計
循環單變量定向統計 圆均匀分布 univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy 球形雙變量 Kent 環形雙變量 bivariate von Mises 多變量 von Mises–Fisher Bingham 退化分布 和奇異分佈 其它
Circular 复合泊松分布 elliptical exponential natural exponential location–scale Maximum entropy Mixture Pearson Tweedie Wrapped
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![{\displaystyle [a,b]\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/08268f073d76e500a09a309a45bcc9f19e599db4)
![{\displaystyle [a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935)
![{\displaystyle X\sim U[a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5d509fb56e608a786132dbf75361145b770d13b)
![{\displaystyle E[X]={\frac {a+b}{2))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6aac0f969513cf4f324ea28113acaf9956661d2)
![{\displaystyle VAR[X]={\frac {(b-a)^{2)){12))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9230e9d71d8cf078f2ef7e47ab5ff3a458e3ea6d)
