Distribution
Fonction de distribution de probabilités
Entropie
Loi uniforme continue
f
(
x
)
=
1
b
−
a
1
1
[
a
,
b
]
{\displaystyle f(x)={\frac {1}{b-a}}1\!\!1_{[a,b]}}
ln
(
b
−
a
)
{\displaystyle \ln(b-a)\,}
Loi normale
f
(
x
)
=
1
2
π
σ
2
exp
(
−
(
x
−
μ
)
2
2
σ
2
)
{\displaystyle f(x)={\frac {1}{\sqrt {2\pi \sigma ^{2}}}}\exp \left(-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}\right)}
ln
(
σ
2
π
e
)
{\displaystyle \ln \left(\sigma {\sqrt {2\pi \,{\rm {e}}}}\right)}
Loi exponentielle
f
(
x
)
=
λ
exp
(
−
λ
x
)
{\displaystyle f(x)=\lambda \exp \left(-\lambda x\right)}
1
−
ln
λ
{\displaystyle 1-\ln \lambda \,}
Loi de Cauchy
f
(
x
)
=
λ
π
1
λ
2
+
x
2
{\displaystyle f(x)={\frac {\lambda }{\pi }}{\frac {1}{\lambda ^{2}+x^{2}}}}
ln
(
4
π
λ
)
{\displaystyle \ln(4\pi \lambda )\,}
Loi du χ²
f
(
x
)
=
1
2
n
/
2
σ
n
Γ
(
n
/
2
)
x
n
2
−
1
exp
(
−
x
2
σ
2
)
{\displaystyle f(x)={\frac {1}{2^{n/2}\sigma ^{n}\Gamma (n/2)}}x^{{\frac {n}{2}}-1}\exp \left(-{\frac {x}{2\sigma ^{2}}}\right)}
ln
2
σ
2
Γ
(
n
2
)
−
(
1
−
n
2
)
ψ
(
n
2
)
+
n
2
{\displaystyle \ln 2\sigma ^{2}\Gamma \left({\frac {n}{2}}\right)-\left(1-{\frac {n}{2}}\right)\psi \left({\frac {n}{2}}\right)+{\frac {n}{2}}}
Distribution Gamma
f
(
x
)
=
x
α
−
1
exp
(
−
x
β
)
β
α
Γ
(
α
)
{\displaystyle f(x)={\frac {x^{\alpha -1}\exp(-{\frac {x}{\beta }})}{\beta ^{\alpha }\Gamma (\alpha )}}}
ln
(
β
Γ
(
α
)
)
+
(
1
−
α
)
ψ
(
α
)
+
α
{\displaystyle \ln(\beta \Gamma (\alpha ))+(1-\alpha )\psi (\alpha )+\alpha \,}
Loi logistique
f
(
x
)
=
e
−
x
(
1
+
e
−
x
)
2
{\displaystyle f(x)={\frac {{\rm {e}}^{-x}}{(1+{\rm {e}}^{-x})^{2}}}}
2
{\displaystyle 2\,}
Statistique de Maxwell-Boltzmann
f
(
x
)
=
4
π
−
1
2
β
3
2
x
2
exp
(
−
β
x
2
)
{\displaystyle f(x)=4\pi ^{-{\frac {1}{2}}}\beta ^{\frac {3}{2}}x^{2}\exp(-\beta x^{2})}
1
2
ln
π
β
+
γ
−
1
2
{\displaystyle {\frac {1}{2}}\ln {\frac {\pi }{\beta }}+\gamma -{\frac {1}{2}}}
Distribution de Pareto
f
(
x
)
=
a
k
a
x
a
+
1
{\displaystyle f(x)={\frac {ak^{a}}{x^{a+1}}}}
ln
k
a
+
1
+
1
a
{\displaystyle \ln {\frac {k}{a}}+1+{\frac {1}{a}}}
Loi de Student
f
(
x
)
=
(
1
+
x
2
/
n
)
−
n
+
1
2
n
B
e
t
a
(
1
2
,
n
2
)
{\displaystyle f(x)={\frac {(1+x^{2}/n)^{-{\frac {n+1}{2}}}}{{\sqrt {n}}\,Beta({\frac {1}{2}},{\frac {n}{2}})}}}
n
+
1
2
ψ
(
n
+
1
2
)
−
ψ
(
n
2
)
+
ln
n
B
(
1
2
,
n
2
)
{\displaystyle {\frac {n+1}{2}}\psi \left({\frac {n+1}{2}}\right)-\psi \left({\frac {n}{2}}\right)+\ln {\sqrt {n}}\,\mathrm {B} \left({\frac {1}{2}},{\frac {n}{2}}\right)}
Distribution de Weibull
f
(
x
)
=
c
α
x
c
−
1
exp
(
−
x
c
α
)
{\displaystyle f(x)={\frac {c}{\alpha }}x^{c-1}\exp \left(-{\frac {x^{c}}{\alpha }}\right)}
(
c
−
1
)
γ
c
+
ln
α
1
/
c
c
+
1
{\displaystyle {\frac {(c-1)\gamma }{c}}+\ln {\frac {\alpha ^{1/c}}{c}}+1}
Loi normale multidimensionnelle
f
X
(
x
1
,
…
,
x
N
)
=
{\displaystyle f_{X}(x_{1},\dots ,x_{N})=}
1
(
2
π
)
N
/
2
|
Σ
|
1
/
2
exp
(
−
1
2
(
x
−
μ
)
⊤
Σ
−
1
(
x
−
μ
)
)
{\displaystyle {\frac {1}{(2\pi )^{N/2}\left|\Sigma \right|^{1/2}}}\exp \left(-{\frac {1}{2}}(x-\mu )^{\top }\Sigma ^{-1}(x-\mu )\right)}
1
2
ln
[
(
2
π
e
)
N
|
Σ
|
]
{\displaystyle {\frac {1}{2}}\ln \left[(2\pi {\rm {e}})^{N}|\Sigma |\right]}
![{\displaystyle f(x)={\frac {1}{b-a}}1\!\!1_{[a,b]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/792f886da1565070453ea160b627f8e0bf3b86e0)
![{\displaystyle {\frac {1}{2}}\ln \left[(2\pi {\rm {e}})^{N}|\Sigma |\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/be6ad50708fe09866888447f7a4abc7475f8d10e)
