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User:Yydl
Distribution
f
(
x
)
{\displaystyle f(x)}
μ
{\displaystyle \mu }
σ
2
{\displaystyle \sigma ^{2}}
M
x
(
t
)
{\displaystyle M_{x}(t)}
Binomial
(
n
x
)
(
p
x
)
(
1
−
p
)
n
−
x
{\displaystyle {\binom {n}{x}}(p^{x})(1-p)^{n-x}}
n
p
{\displaystyle np}
n
p
q
{\displaystyle npq}
(
p
e
t
+
q
)
n
{\displaystyle (pe^{t}+q)^{n}}
Poisson
e
−
λ
λ
x
x
!
x
=
0
,
1
,
2
…
{\displaystyle {\frac {e^{-\lambda }\lambda ^{x}}{x!}}\quad x=0,1,2\dots }
λ
{\displaystyle \lambda }
λ
{\displaystyle \lambda }
e
λ
(
e
t
−
1
)
{\displaystyle e^{\lambda (e^{t}-1)}}
Hypergeometric
(
a
x
)
(
b
n
−
x
)
(
a
+
b
n
)
x
=
0
,
1
,
…
n
{\displaystyle {\frac {{\binom {a}{x}}{\binom {b}{n-x}}}{\binom {a+b}{n}}}\quad x=0,1,\dots n}
n
a
a
+
b
{\displaystyle {\frac {na}{a+b}}}
n
a
b
(
a
+
b
−
n
)
(
a
+
b
)
2
(
a
+
b
−
1
)
{\displaystyle {\frac {nab(a+b-n)}{(a+b)^{2}(a+b-1)}}}
Geometric
p
(
1
−
p
)
x
−
1
x
=
1
,
2
,
…
{\displaystyle p(1-p)^{x-1}\quad x=1,2,\dots }
1
p
{\displaystyle {\frac {1}{p}}}
1
−
p
p
2
{\displaystyle {\frac {1-p}{p^{2}}}}
p
e
t
1
−
p
e
t
{\displaystyle {\frac {pe^{t}}{1-pe^{t}}}}
Discrete Uniform
1
n
x
=
1
,
2
,
…
n
{\displaystyle {\frac {1}{n}}\quad x=1,2,\dots n}
n
+
1
2
{\displaystyle {\frac {n+1}{2}}}
n
2
−
1
12
{\displaystyle {\frac {n^{2}-1}{12}}}
e
t
(
1
−
e
n
t
)
n
(
1
−
e
t
)
{\displaystyle {\frac {e^{t}(1-e^{nt})}{n(1-e^{t})}}}
(continuous) Uniform
{
1
b
−
a
a
<
x
<
b
0
elsewhere
{\displaystyle {\begin{cases}{\frac {1}{b-a}}&a<x<b\\0&{\text{elsewhere}}\end{cases}}}
a
+
b
2
{\displaystyle {\frac {a+b}{2}}}
(
b
−
a
)
2
12
{\displaystyle {\frac {(b-a)^{2}}{12}}}
e
b
t
−
e
a
t
t
(
b
−
a
)
{\displaystyle {\frac {e^{bt}-e^{at}}{t(b-a)}}}
Exponential
{
β
e
−
β
x
x
>
0
0
elsewhere
{\displaystyle {\begin{cases}\beta e^{-\beta x}&x>0\\0&{\text{elsewhere}}\end{cases}}}
1
β
{\displaystyle {\frac {1}{\beta }}}
1
β
2
{\displaystyle {\frac {1}{\beta ^{2}}}}
β
β
−
t
{\displaystyle {\frac {\beta }{\beta -t}}}
Normal
1
2
π
σ
e
−
1
2
(
x
−
μ
σ
)
2
−
∞
<
0
<
∞
{\displaystyle {\frac {1}{{\sqrt {2\pi }}\sigma }}e^{-{\frac {1}{2}}({\frac {x-\mu }{\sigma }})^{2}}\quad -\infty <0<\infty }
μ
{\displaystyle \mu }
σ
2
{\displaystyle \sigma ^{2}}
e
μ
t
+
1
2
σ
2
t
2
{\displaystyle e^{\mu t+{\frac {1}{2}}\sigma ^{2}t^{2}}}
Beta
{
Γ
(
α
+
β
)
Γ
(
α
)
Γ
(
β
)
x
α
−
1
(
1
−
x
)
β
−
1
0
<
x
<
1
0
elsewhere
{\displaystyle {\begin{cases}{\frac {\Gamma (\alpha +\beta )}{\Gamma (\alpha )\Gamma (\beta )}}x^{\alpha -1}(1-x)^{\beta -1}&0<x<1\\0&{\text{elsewhere}}\end{cases}}}
α
α
+
β
{\displaystyle {\frac {\alpha }{\alpha +\beta }}}
α
β
(
α
+
β
)
2
(
α
+
β
+
1
)
{\displaystyle {\frac {\alpha \beta }{(\alpha +\beta )^{2}(\alpha +\beta +1)}}}
Gamma
{
1
Γ
(
α
)
β
α
x
α
−
1
e
−
x
β
x
>
0
0
elsewhere
{\displaystyle {\begin{cases}{\frac {1}{\Gamma (\alpha )\beta ^{\alpha }}}x^{\alpha -1}e^{-{\frac {x}{\beta }}}&x>0\\0&{\text{elsewhere}}\end{cases}}}
α
β
{\displaystyle \alpha \beta }
α
β
2
{\displaystyle \alpha \beta ^{2}}
(
1
−
β
t
)
−
α
{\displaystyle (1-\beta t)^{-\alpha }}