La funzione di densità di probabilità associata alla distribuzione paretiana è
f
(
x
)
=
α
β
α
x
α
+
1
{\displaystyle \ f(x)={\frac {\alpha \beta ^{\alpha }}{x^{\alpha +1}}}}
f
(
x
)
=
{
α
β
α
x
α
+
1
per
x
≥
β
0
per
x
<
β
{\displaystyle f(x)={\begin{cases}{\dfrac {\alpha \beta ^{\alpha }}{x^{\alpha +1}}}&{\text{per }}x\geq \beta \\0&{\text{per }}x<\beta \end{cases}}}
La distribuzione paretiana è caratterizzata da due parametri: uno di posizione
β
{\displaystyle \beta }
X
{\displaystyle X}
α
{\displaystyle \alpha }
La variabile casuale paretiana è spesso utilizzata per modellizzare la distribuzione del reddito; in tal caso,
β
{\displaystyle \beta }
reddito minimo .
Integrando la funzione densità tra
β
{\displaystyle \beta }
x
∈
(
β
;
+
∞
)
{\displaystyle x\in (\beta ;+\infty )}
funzione di distribuzione :
F
X
(
x
)
=
α
β
α
∫
β
+
∞
ξ
−
(
α
+
)
d
ξ
=
α
β
α
⋅
−
1
α
ξ
−
α
|
β
+
∞
=
−
β
α
⋅
1
ξ
α
|
β
x
=
−
β
α
(
1
x
α
−
1
β
α
)
=
1
−
(
β
x
)
α
{\displaystyle {\begin{alignedat}{2}F_{X}(x)=\alpha \beta ^{\alpha }\int _{\beta }^{+\infty }\xi ^{-(\alpha +)}d\xi &=\alpha \beta ^{\alpha }\cdot -{\frac {1}{\alpha }}\xi ^{-\alpha }{\bigg |}_{\beta }^{+\infty }=-\beta ^{\alpha }\cdot {\dfrac {1}{\xi ^{\alpha }}}{\bigg \vert }_{\beta }^{x}\\[2ex]&=-\beta ^{\alpha }\left({\dfrac {1}{x^{\alpha }}}-{\dfrac {1}{\beta ^{\alpha }}}\right)=1-\left({\dfrac {\beta }{x}}\right)^{\alpha }\end{alignedat}}}
F
(
x
)
=
{
1
−
(
β
x
)
α
per
x
≥
β
0
per
x
<
β
{\displaystyle F(x)={\begin{cases}1-{\bigg (}{\dfrac {\beta }{x}}{\bigg )}^{\alpha }&{\text{per }}x\geq \beta \\0&{\text{per }}x<\beta \end{cases}}}
I suoi principali parametri sono:
Momenti ordinari
μ
1
=
∫
β
+
∞
x
α
β
α
x
α
+
1
d
x
=
α
β
α
∫
β
+
∞
x
−
α
d
x
=
α
β
α
⋅
x
1
−
α
1
−
α
|
β
+
∞
=
α
β
α
1
−
α
⋅
1
x
α
−
1
|
β
+
∞
=
α
β
α
1
−
α
(
0
−
1
β
α
−
1
)
=
α
β
α
−
1
{\displaystyle {\begin{aligned}\mu _{1}&=\displaystyle \int _{\beta }^{+\infty }x{\dfrac {\alpha \beta ^{\alpha }}{x^{\alpha +1}}}\,dx=\alpha \beta ^{\alpha }\displaystyle \int _{\beta }^{+\infty }x^{-\alpha }\,dx=\alpha \beta ^{\alpha }\cdot {\dfrac {x^{1-\alpha }}{1-\alpha }}{\bigg \vert }_{\beta }^{+\infty }\\[2ex]&={\dfrac {\alpha \beta ^{\alpha }}{1-\alpha }}\cdot {\dfrac {1}{x^{\alpha -1}}}{\bigg \vert }_{\beta }^{+\infty }={\dfrac {\alpha \beta ^{\alpha }}{1-\alpha }}\left(0-{\dfrac {1}{\beta ^{\alpha -1}}}\right)={\dfrac {\alpha \beta }{\alpha -1}}\\\end{aligned}}}
Da cui si ottiene:
μ
1
=
{
α
β
x
−
1
per
α
>
1
∞
per
α
≥
1
{\displaystyle \mu _{1}={\begin{cases}{\dfrac {\alpha \beta }{x-1}}&{\text{per }}\alpha >1\\\infty &{\text{per }}\alpha \geq 1\end{cases}}}
μ
2
=
α
β
α
∫
β
+
∞
x
2
x
α
+
1
d
x
=
α
β
α
∫
β
+
∞
x
1
−
α
d
x
=
α
β
α
⋅
1
2
−
α
x
2
−
α
|
β
+
∞
=
α
β
α
⋅
1
2
−
α
1
x
α
−
2
|
β
+
∞
=
α
β
α
⋅
1
2
−
α
(
0
−
1
β
α
−
2
)
=
−
α
β
α
2
−
α
⋅
1
β
α
−
2
=
α
β
2
α
−
2
{\displaystyle {\begin{aligned}\mu _{2}&=\alpha \beta ^{\alpha }\int _{\beta }^{+\infty }{\dfrac {x^{2}}{x^{\alpha +1}}}\,dx=\alpha \beta ^{\alpha }\int _{\beta }^{+\infty }x^{1-\alpha }\,dx=\alpha \beta ^{\alpha }\cdot {\dfrac {1}{2-\alpha }}x^{2-\alpha }{\bigg \vert }_{\beta }^{+\infty }\\[2ex]&=\alpha \beta ^{\alpha }\cdot {\dfrac {1}{2-\alpha }}{\dfrac {1}{x^{\alpha -2}}}{\bigg \vert }_{\beta }^{+\infty }=\alpha \beta ^{\alpha }\cdot {\dfrac {1}{2-\alpha }}\left(0-{\dfrac {1}{\beta ^{\alpha -2}}}\right)\\[2ex]&=-{\dfrac {\alpha \beta ^{\alpha }}{2-\alpha }}\cdot {\dfrac {1}{\beta ^{\alpha -2}}}={\dfrac {\alpha \beta ^{2}}{\alpha -2}}\end{aligned}}}
Da cui si ricava:
μ
2
=
{
α
β
2
x
−
2
per
α
>
2
∞
per
α
≥
2
{\displaystyle \mu _{2}={\begin{cases}{\dfrac {\alpha \beta ^{2}}{x-2}}&{\text{per }}\alpha >2\\\infty &{\text{per }}\alpha \geq 2\end{cases}}}
In generale un momento di ordine
n
{\displaystyle n}
μ
n
=
{
α
β
n
x
−
n
per
0
<
n
<
α
∞
per
n
≥
α
{\displaystyle \mu _{n}={\begin{cases}{\dfrac {\alpha \beta ^{n}}{x-n}}&{\text{per }}0<n<\alpha \\\infty &{\text{per }}n\geq \alpha \end{cases}}}
M
(
θ
)
=
E
[
e
θ
x
]
=
α
(
−
β
θ
)
α
Γ
(
−
α
,
−
β
θ
)
{\displaystyle M(\theta )=E[e^{\theta x}]=\alpha (-\beta \theta )^{\alpha }\Gamma (-\alpha ,-\beta \theta )}
dove
Γ
(
−
α
,
−
β
θ
)
{\displaystyle \Gamma (-\alpha ,-\beta \theta )}
funzione gamma incompleta .
La funzione generatrice di momenti è definita solo per valori non positivi di
θ
{\displaystyle \theta }
Varianza
σ
2
=
μ
2
−
μ
1
2
=
α
β
2
α
−
2
−
(
α
β
α
−
1
)
2
=
α
β
2
α
−
2
−
α
2
β
2
(
α
−
1
)
2
=
α
β
2
(
α
−
1
)
2
−
α
2
β
2
(
α
−
2
)
(
α
−
2
)
(
α
−
1
)
2
=
α
3
β
2
+
α
β
2
−
2
α
2
β
2
−
α
3
β
2
+
2
α
2
β
2
(
α
−
2
)
(
α
−
1
)
2
=
α
β
2
(
α
−
2
)
(
α
−
1
)
2
{\displaystyle {\begin{aligned}\sigma ^{2}&=\mu _{2}-\mu _{1}^{2}={\dfrac {\alpha \beta ^{2}}{\alpha -2}}-\left({\dfrac {\alpha \beta }{\alpha -1}}\right)^{2}={\dfrac {\alpha \beta ^{2}}{\alpha -2}}-{\dfrac {\alpha ^{2}\beta ^{2}}{(\alpha -1)^{2}}}\\[2ex]&={\dfrac {\alpha \beta ^{2}(\alpha -1)^{2}-\alpha ^{2}\beta ^{2}(\alpha -2)}{(\alpha -2)(\alpha -1)^{2}}}\\[2ex]&={\dfrac {\alpha ^{3}\beta ^{2}+\alpha \beta ^{2}-2\alpha ^{2}\beta ^{2}-\alpha ^{3}\beta ^{2}+2\alpha ^{2}\beta ^{2}}{(\alpha -2)(\alpha -1)^{2}}}\\[2ex]&={\dfrac {\alpha \beta ^{2}}{(\alpha -2)(\alpha -1)^{2}}}\end{aligned}}}
Da cui ricaviamo:
σ
2
=
{
α
β
2
(
α
−
2
)
(
α
−
1
)
2
per
α
>
2
∞
per
α
∈
(
1
;
2
]
{\displaystyle \sigma ^{2}={\begin{cases}{\dfrac {\alpha \beta ^{2}}{(\alpha -2)(\alpha -1)^{2}}}&{\text{per }}\alpha >2\\\infty &{\text{per }}\alpha \in (1;2]\end{cases}}}
Si noti che per
α
≤
1
{\displaystyle \alpha \leq 1}
Mediana
1
−
(
β
ξ
0.5
)
α
=
1
2
→
(
β
ξ
0.5
)
α
=
1
2
{\displaystyle 1-{\bigg (}{\frac {\beta }{\xi _{0.5}}}{\bigg )}^{\alpha }={\frac {1}{2}}\rightarrow {\bigg (}{\frac {\beta }{\xi _{0.5}}}{\bigg )}^{\alpha }={\frac {1}{2}}}
ξ
0.5
=
2
α
β
{\displaystyle \xi _{0.5}={\sqrt[{\alpha }]{2}}\beta }
Simmetria
β
1
=
4
(
α
−
2
)
(
α
+
1
)
2
α
(
α
−
3
)
2
{\displaystyle \beta _{1}={\frac {4(\alpha -2)(\alpha +1)^{2}}{\alpha (\alpha -3)^{2}}}}
α
>
3
{\displaystyle \alpha >3}
Curtosi
β
2
=
3
(
α
−
2
)
(
3
α
2
+
α
+
2
)
α
(
α
−
3
)
(
α
−
4
)
{\displaystyle \beta _{2}={\frac {3(\alpha -2)(3\alpha ^{2}+\alpha +2)}{\alpha (\alpha -3)(\alpha -4)}}}
α
>
4
{\displaystyle \alpha >4}
La variabile casuale paretiana ha elasticità costante (negativa):
ε(x) = df / f / dx / x = -(α+1) che può essere interpretato nel senso che, qualunque sia il reddito x0
se
per il reddito x0 abbiamo y0 persone che lo guadagnano
allora
per il reddito x0 +1% ci saranno y0 -(α+1)% persone Collegamenti esterni
modifica
![{\displaystyle {\begin{alignedat}{2}F_{X}(x)=\alpha \beta ^{\alpha }\int _{\beta }^{+\infty }\xi ^{-(\alpha +)}d\xi &=\alpha \beta ^{\alpha }\cdot -{\frac {1}{\alpha }}\xi ^{-\alpha }{\bigg |}_{\beta }^{+\infty }=-\beta ^{\alpha }\cdot {\dfrac {1}{\xi ^{\alpha }}}{\bigg \vert }_{\beta }^{x}\\[2ex]&=-\beta ^{\alpha }\left({\dfrac {1}{x^{\alpha }}}-{\dfrac {1}{\beta ^{\alpha }}}\right)=1-\left({\dfrac {\beta }{x}}\right)^{\alpha }\end{alignedat}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2323585cd1b2ee56785e5647c78d2d6bb86b942e)
![{\displaystyle {\begin{aligned}\mu _{1}&=\displaystyle \int _{\beta }^{+\infty }x{\dfrac {\alpha \beta ^{\alpha }}{x^{\alpha +1}}}\,dx=\alpha \beta ^{\alpha }\displaystyle \int _{\beta }^{+\infty }x^{-\alpha }\,dx=\alpha \beta ^{\alpha }\cdot {\dfrac {x^{1-\alpha }}{1-\alpha }}{\bigg \vert }_{\beta }^{+\infty }\\[2ex]&={\dfrac {\alpha \beta ^{\alpha }}{1-\alpha }}\cdot {\dfrac {1}{x^{\alpha -1}}}{\bigg \vert }_{\beta }^{+\infty }={\dfrac {\alpha \beta ^{\alpha }}{1-\alpha }}\left(0-{\dfrac {1}{\beta ^{\alpha -1}}}\right)={\dfrac {\alpha \beta }{\alpha -1}}\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6693a5619d7eedb6e754d7158b09c805f03a1668)
![{\displaystyle {\begin{aligned}\mu _{2}&=\alpha \beta ^{\alpha }\int _{\beta }^{+\infty }{\dfrac {x^{2}}{x^{\alpha +1}}}\,dx=\alpha \beta ^{\alpha }\int _{\beta }^{+\infty }x^{1-\alpha }\,dx=\alpha \beta ^{\alpha }\cdot {\dfrac {1}{2-\alpha }}x^{2-\alpha }{\bigg \vert }_{\beta }^{+\infty }\\[2ex]&=\alpha \beta ^{\alpha }\cdot {\dfrac {1}{2-\alpha }}{\dfrac {1}{x^{\alpha -2}}}{\bigg \vert }_{\beta }^{+\infty }=\alpha \beta ^{\alpha }\cdot {\dfrac {1}{2-\alpha }}\left(0-{\dfrac {1}{\beta ^{\alpha -2}}}\right)\\[2ex]&=-{\dfrac {\alpha \beta ^{\alpha }}{2-\alpha }}\cdot {\dfrac {1}{\beta ^{\alpha -2}}}={\dfrac {\alpha \beta ^{2}}{\alpha -2}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e3e714daf3302be715660fa074a5d02ad6c77240)
![{\displaystyle M(\theta )=E[e^{\theta x}]=\alpha (-\beta \theta )^{\alpha }\Gamma (-\alpha ,-\beta \theta )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a128700dd0a1726879f996fb5fabed3a4ad8d71)
![{\displaystyle {\begin{aligned}\sigma ^{2}&=\mu _{2}-\mu _{1}^{2}={\dfrac {\alpha \beta ^{2}}{\alpha -2}}-\left({\dfrac {\alpha \beta }{\alpha -1}}\right)^{2}={\dfrac {\alpha \beta ^{2}}{\alpha -2}}-{\dfrac {\alpha ^{2}\beta ^{2}}{(\alpha -1)^{2}}}\\[2ex]&={\dfrac {\alpha \beta ^{2}(\alpha -1)^{2}-\alpha ^{2}\beta ^{2}(\alpha -2)}{(\alpha -2)(\alpha -1)^{2}}}\\[2ex]&={\dfrac {\alpha ^{3}\beta ^{2}+\alpha \beta ^{2}-2\alpha ^{2}\beta ^{2}-\alpha ^{3}\beta ^{2}+2\alpha ^{2}\beta ^{2}}{(\alpha -2)(\alpha -1)^{2}}}\\[2ex]&={\dfrac {\alpha \beta ^{2}}{(\alpha -2)(\alpha -1)^{2}}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7df380621bbbddffbef1ee168ea65afb8a84ccf6)
![{\displaystyle \sigma ^{2}={\begin{cases}{\dfrac {\alpha \beta ^{2}}{(\alpha -2)(\alpha -1)^{2}}}&{\text{per }}\alpha >2\\\infty &{\text{per }}\alpha \in (1;2]\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/97b1f6295bd1b8f78fbebd6683c8f59c3c465ca6)
![{\displaystyle \xi _{0.5}={\sqrt[{\alpha }]{2}}\beta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/11dfcc7defd876ad44a1698cd2e1053d6d791623)
Wikimedia Commons contiene immagini o altri file su distribuzione paretiana
