modifica
Funció de densitat de probabilitat
Funció de distribució de probabilitat
Paràmetres
k
∈
(
0
,
∞
)
{\displaystyle k\in (0,\infty )}
forma
θ
∈
(
0
,
∞
)
{\displaystyle \theta \in (0,\infty )}
escala Suport
x
∈
(
0
,
∞
)
{\displaystyle x\in (0,\infty )}
fdp
f
(
x
)
=
1
θ
k
Γ
(
k
)
x
k
−
1
e
−
x
/
θ
{\displaystyle f(x)={\frac {1}{\theta ^{k}\,\Gamma (k)}}\,x^{k-1}e^{-x/\theta }}
FD
F
(
x
)
=
1
Γ
(
k
)
γ
(
k
,
x
θ
)
{\displaystyle F(x)={\frac {1}{\Gamma (k)}}\gamma \left(k,{\frac {x}{\theta }}\right)}
Esperança matemàtica
k
θ
{\displaystyle k\,\theta }
Mediana No té expressió tancada Moda
(
k
−
1
)
θ
,
si
k
≥
1
{\displaystyle (k-1)\theta ,\ {\text{si}}\ k\geq 1}
Variància
k
θ
2
{\displaystyle k\,\theta ^{2}}
Coeficient de simetria
2
k
{\displaystyle {\frac {2}{\sqrt {k}}}}
Curtosi
6
k
{\displaystyle {\frac {6}{k}}}
Entropia
k
+
ln
θ
+
ln
Γ
(
k
)
+
(
1
−
k
)
ψ
(
k
)
{\displaystyle {\begin{aligned}k&+\ln \theta +\ln \Gamma (k)\\&+(1-k)\psi (k)\end{aligned}}}
FGM
(
1
−
θ
t
)
−
k
,
per a
t
<
1
θ
{\displaystyle (1-\theta t)^{-k},\ {\text{per a }}t<{\frac {1}{\theta }}}
FC
(
1
−
θ
i
t
)
−
k
{\displaystyle (1-\theta it)^{-k}}
Informació de Fisher
I
(
k
,
θ
)
=
(
ψ
(
1
)
(
k
)
θ
−
1
θ
−
1
k
θ
−
2
)
{\displaystyle I(k,\theta )={\begin{pmatrix}\psi ^{(1)}(k)&\theta ^{-1}\\\theta ^{-1}&k\theta ^{-2}\end{pmatrix}}}
Hi ha dues parametritzacions diferents de la distribució gamma. La primera [1] paràmetre d'escala
θ
∈
(
0
,
∞
)
{\displaystyle \theta \in (0,\infty )}
paràmetre de forma
k
∈
(
0
,
∞
)
{\displaystyle k\in (0,\infty )}
[2]
f
(
x
;
k
,
θ
)
=
{
1
θ
k
Γ
(
k
)
x
k
−
1
e
−
x
/
θ
,
si
x
>
0
,
0
,
si
x
≤
0
,
{\displaystyle f(x;k,\theta )={\begin{cases}{\dfrac {1}{\theta ^{k}\,\Gamma (k)}}\,x^{k-1}\,e^{-x/\theta },&{\text{si}}\ x>0,\\0,&{\text{si}}\ x\leq 0,\end{cases}}}
on
Γ
(
k
)
{\displaystyle \Gamma (k)}
funció gamma . Si
X
{\displaystyle X}
X
∼
Γ
(
k
,
θ
)
{\displaystyle X\sim \Gamma (k,\theta )}
X
∼
Gamma
(
k
,
θ
)
{\displaystyle X\sim {\text{Gamma}}(k,\theta )}
paràmetre de taxa (rate parameter ),
β
∈
(
0
,
∞
)
{\displaystyle \beta \in (0,\infty )}
β
=
1
/
θ
{\displaystyle \beta =1/\theta }
α
=
k
∈
(
0
,
∞
)
{\displaystyle \alpha =k\in (0,\infty )}
[3] [4]
f
(
x
;
α
,
β
)
=
{
β
α
Γ
(
α
)
x
α
−
1
e
−
β
x
,
si
x
>
0
,
0
,
si
x
≤
0.
{\displaystyle f(x;\alpha ,\beta )={\begin{cases}{\dfrac {\beta ^{\alpha }}{\Gamma (\alpha )}}\,x^{\alpha -1}\,e^{-\beta x},&{\text{si}}\ x>0,\\0,&{\text{si}}\ x\leq 0.\end{cases}}}
F
(
x
;
k
,
θ
)
=
{
1
Γ
(
k
)
γ
(
k
,
x
θ
)
,
si
x
≥
0
0
,
si
x
<
0
,
{\displaystyle F(x;k,\theta )={\begin{cases}{\dfrac {1}{\Gamma (k)}}\gamma {\big (}k,{\dfrac {x}{\theta }}{\big )},&{\text{si}}\ x\geq 0\\0,&{\text{si}}\ x<0,\end{cases}}}
γ
(
k
,
x
)
{\displaystyle \gamma (k,x)}
funció gamma incompleta inferior.
La distribució gamma té moments de tots els ordres. Si
X
∼
Γ
(
k
,
θ
)
{\displaystyle X\sim \Gamma (k,\theta )}
n
≥
1
{\displaystyle n\geq 1}
E
[
X
n
]
=
θ
n
k
(
k
+
1
)
⋯
(
k
+
n
−
1
)
.
{\displaystyle E[X^{n}]=\theta ^{n}k(k+1)\cdots (k+n-1).}
E
[
X
]
=
θ
k
i
E
[
X
2
]
=
θ
2
k
(
k
+
1
)
,
{\displaystyle E[X]=\theta k\quad {\text{i}}\quad E[X^{2}]=\theta ^{2}k(k+1),}
Var
(
X
)
=
E
[
X
2
]
−
(
E
[
X
]
)
2
=
θ
2
k
.
{\displaystyle {\text{Var}}(X)=E[X^{2}]-{\big (}E[X]{\big )}^{2}=\theta ^{2}k.}
E
[
X
]
=
θ
k
i
Var
(
X
)
=
θ
2
k
.
{\displaystyle E[X]=\theta k\quad {\text{i}}\quad {\text{Var}}(X)=\theta ^{2}k.}
Prova
Reduirem la integral que intervé el càlcul dels moments a la funció gamma:
E
[
X
n
]
=
1
θ
k
Γ
(
k
)
∫
0
∞
x
n
+
k
−
1
e
−
x
/
θ
d
x
=
(
∗
)
θ
n
Γ
(
n
+
k
)
Γ
(
k
)
=
(
∗
∗
)
θ
n
(
k
+
n
−
1
)
(
k
−
n
−
2
)
⋯
k
,
{\displaystyle E[X^{n}]={\frac {1}{\theta ^{k}\,\Gamma (k)}}\int _{0}^{\infty }x^{n+k-1}e^{-x/\theta }\,dx\,{\underset {(*)}{=}}\,{\frac {\theta ^{n}\,\Gamma (n+k)}{\Gamma (k)}}\,{\underset {(**)}{=}}\,\theta ^{n}(k+n-1)(k-n-2)\cdots k,}
on a la igualtat (*) hem fet el canvi de variables
x
/
θ
=
y
{\displaystyle x/\theta =y}
per retrobar una funció gamma, i a la igualtat (**) hem utilitzat l'equació funcional de la funció gamma
Γ
(
t
+
1
)
=
t
Γ
(
t
)
,
t
>
0
{\displaystyle \Gamma (t+1)=t\,\Gamma (t),\ t>0}
.
modifica
La distribució gamma
Γ
(
k
,
θ
)
{\displaystyle \Gamma (k,\theta )}
M
(
t
)
=
1
(
1
−
θ
t
)
k
,
t
∈
(
−
∞
,
1
θ
)
.
{\displaystyle M(t)={\frac {1}{(1-\theta t)^{k}}},\quad t\in (-\infty ,{\frac {1}{\theta }}).}
Prova
De nou, utiitzarem aquí la funció gamma:
M
(
t
)
=
θ
k
Γ
(
k
)
∫
0
∞
e
t
x
x
k
−
1
e
−
x
/
θ
d
x
=
θ
k
Γ
(
k
)
∫
0
∞
e
−
x
(
1
θ
−
t
)
x
k
−
1
d
x
.
{\displaystyle M(t)={\frac {\theta ^{k}}{\Gamma (k)}}\int _{0}^{\infty }e^{tx}x^{k-1}e^{-x/\theta }\,dx={\frac {\theta ^{k}}{\Gamma (k)}}\int _{0}^{\infty }e^{-x({\frac {1}{\theta }}-t)}x^{k-1}\,dx.}
Si
t
<
1
/
θ
{\displaystyle t<1/\theta }
, fem el canvi
x
=
y
(
1
θ
−
t
)
−
1
{\displaystyle x=y\ ({\frac {1}{\theta }}-t)^{-1}}
i tenim
M
(
t
)
=
θ
k
Γ
(
k
)
(
1
θ
−
t
)
−
k
∫
0
∞
e
−
y
y
k
−
1
d
y
=
1
(
1
−
θ
t
)
k
.
{\displaystyle M(t)={\frac {\theta ^{k}}{\Gamma (k)}}{\Big (}{\frac {1}{\theta }}-t{\Big )}^{-k}\int _{0}^{\infty }e^{-y}y^{k-1}\,dy={\frac {1}{(1-\theta t)^{k}}}.}
La funció característica és [3]
φ
(
t
)
=
1
(
1
−
i
θ
t
)
k
=
exp
{
−
k
log
(
1
−
i
θ
t
)
}
,
t
∈
R
,
{\displaystyle \varphi (t)={\frac {1}{(1-i\theta t)^{k}}}=\exp {\Big \{}-k\log(1-i\theta t){\Big \}},\quad t\in \mathbb {R} ,}
log
{\displaystyle \log }
(
−
π
,
π
)
{\displaystyle (-\pi ,\pi )}
Si
X
1
∼
Γ
(
k
1
,
θ
)
{\displaystyle X_{1}\sim \Gamma (k_{1},\theta )}
X
2
∼
Γ
(
k
2
,
θ
)
{\displaystyle X_{2}\sim \Gamma (k_{2},\theta )}
X
1
+
X
2
∼
Γ
(
k
1
+
k
2
,
θ
)
{\displaystyle X_{1}+X_{2}\sim \Gamma (k_{1}+k_{2},\theta )}
Γ
(
k
,
θ
)
{\displaystyle \Gamma (k,\theta )}
[5]
k
{\displaystyle k}
X
1
{\displaystyle X_{1}}
X
2
{\displaystyle X_{2}}
X
1
,
…
,
X
n
{\displaystyle X_{1},\dots ,X_{n}}
X
i
∼
Γ
(
k
i
,
θ
)
{\displaystyle X_{i}\sim \Gamma (k_{i},\theta )}
∑
i
=
1
n
X
i
∼
Γ
(
∑
i
=
1
n
k
i
,
θ
)
.
{\displaystyle \sum _{i=1}^{n}X_{i}\sim \Gamma {\big (}\sum _{i=1}^{n}k_{i},\theta {\big )}.}
modifica
La distribució gamma
Γ
(
k
,
θ
)
{\displaystyle \Gamma (k,\theta )}
[6]
X
∼
Γ
(
k
,
θ
)
{\displaystyle X\sim \Gamma (k,\theta )}
n
≥
1
{\displaystyle n\geq 1}
X
1
,
…
,
X
n
{\displaystyle X_{1},\dots ,X_{n}}
X
=
D
X
1
+
⋯
+
X
n
,
{\displaystyle X\ {\overset {\mathcal {D}}{=}}\ X_{1}+\cdots +X_{n},}
=
D
{\displaystyle {\overset {\mathcal {D}}{=}}}
igualtat en distribució o llei. Aquí cal prendre
X
i
∼
Γ
(
k
/
n
,
θ
)
,
i
=
1
,
…
,
n
{\displaystyle X_{i}\sim \Gamma (k/n,\theta ),\ i=1,\dots ,n}
La representació de Lévy-Khintchine [7]
φ
(
t
)
=
exp
{
k
∫
0
∞
(
e
i
t
x
−
1
)
e
−
x
/
θ
x
d
x
}
.
{\displaystyle \varphi (t)=\exp {\Big \{}k\int _{0}^{\infty }{\big (}e^{itx}-1{\big )}\,{\frac {e^{-x/\theta }}{x}}\,dx{\Big \}}.}
g
(
x
)
=
k
e
−
x
/
θ
x
,
x
>
0
,
{\displaystyle g(x)=k\,{\frac {e^{-x/\theta }}{x}},\quad x>0,}
drift ) són zero (vegeu Sato [8]
Quan
k
≥
1
{\displaystyle k\geq 1}
Γ
(
k
,
θ
)
{\displaystyle \Gamma (k,\theta )}
θ
(
k
−
1
)
{\displaystyle \theta (k-1)}
moda de la distribució i que la distribució és unimodal. El valor del màxim és
(
k
−
1
)
k
−
1
e
−
(
k
−
1
)
/
(
θ
Γ
(
k
)
)
{\displaystyle (k-1)^{k-1}e^{-(k-1)}/(\theta \,\Gamma (k))}
fórmula de Stirling , per a valors grans de
k
{\displaystyle k}
1
/
(
θ
2
π
(
k
−
1
)
)
{\displaystyle 1/{\Big (}\theta {\sqrt {2\pi (k-1)}}{\Big )}}
[9]
Quan
k
∈
(
0
,
1
)
{\displaystyle k\in (0,1)}
lim
x
→
0
+
f
(
x
;
k
,
θ
)
=
∞
.
{\displaystyle \lim _{x\to 0^{+}}f(x;k,\theta )=\infty .}
modifica
Johnson et al. introdueixen la distribució gamma amb tres paràmetres: a més dels paràmetres de forma i escala
k
,
θ
∈
(
0
,
∞
)
{\displaystyle k,\theta \in (0,\infty )}
γ
∈
R
{\displaystyle \gamma \in \mathbb {R} }
f
(
x
;
k
,
θ
,
γ
)
=
{
1
θ
k
Γ
(
k
)
(
x
−
γ
)
k
−
1
e
−
(
x
−
γ
)
/
θ
,
si
x
>
γ
,
0
,
si
x
≤
γ
.
{\displaystyle f(x;k,\theta ,\gamma )={\begin{cases}{\dfrac {1}{\theta ^{k}\,\Gamma (k)}}\,(x-\gamma )^{k-1}\,e^{-(x-\gamma )/\theta },&{\text{si}}\ x>\gamma ,\\0,&{\text{si}}\ x\leq \gamma .\end{cases}}}
γ
=
0
{\displaystyle \gamma =0}
↑ Forbes , C; Evans , M.; Hastings , N.; Peacock , B. Statistical distributions. ISBN 978-0-470-62724-2 ↑ «The R project for statistical computing ». [Consulta: 9 febrer 2023]. ↑ 3,0 3,1 Sato , Ken-iti. Lévy processes and infinitely divisible distributions ISBN 0-521-55302-4 ↑ Bernardo , J. M.; Smith , A. F. M.. Bayesian theory ISBN 0-471-92416-4 ↑ Wilks , S. S.. Mathematical statistics ISBN 0-471-94644-3 ↑ Loeve , Michel. Teoría de la probabilidad ISBN 84-309-0663-0 ↑ Sato , Ken-iti. Lévy processes and infinitely divisible distributions ISBN 0-521-55302-4 ↑ Sato , Ken-iti. Lévy processes and infinitely divisible distributions ISBN 0-521-55302-4 ↑ Feller , William. Introducción a ls probabilidades y sus aplicaciones, vol. 2 . Mexico: Editorial Limua, 1978, p. 76. 
![{\displaystyle E[X^{n}]=\theta ^{n}k(k+1)\cdots (k+n-1).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a94cdda07c83387b9423cae6fb5c712056cc42b2)
![{\displaystyle E[X]=\theta k\quad {\text{i}}\quad E[X^{2}]=\theta ^{2}k(k+1),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ebe69b10bd994a5993e3117c3dbbc19ee1c9c90e)
![{\displaystyle {\text{Var}}(X)=E[X^{2}]-{\big (}E[X]{\big )}^{2}=\theta ^{2}k.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1a881b13c1811a9e8c306a9044fb9ea480f32a2)
![{\displaystyle E[X]=\theta k\quad {\text{i}}\quad {\text{Var}}(X)=\theta ^{2}k.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bbc01c86bcb9b10c7f3dfb62fcecebc8e08b4651)
![{\displaystyle E[X^{n}]={\frac {1}{\theta ^{k}\,\Gamma (k)}}\int _{0}^{\infty }x^{n+k-1}e^{-x/\theta }\,dx\,{\underset {(*)}{=}}\,{\frac {\theta ^{n}\,\Gamma (n+k)}{\Gamma (k)}}\,{\underset {(**)}{=}}\,\theta ^{n}(k+n-1)(k-n-2)\cdots k,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7a169ce2d423d60c3ab8bd91c99d9e931d76bb7)
