Schema di una distribuzione paretiana In teoria delle probabilità la distribuzione paretiana (o distribuzione di Pareto , così chiamata in onore di Vilfredo Pareto ) è una distribuzione di probabilità continua , utilizzata in particolar modo per descrivere la distribuzione dei redditi .
Metodologia
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La funzione di densità di probabilità associata alla distribuzione paretiana è:
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{\displaystyle \ f(x)={\frac {\alpha \beta ^{\alpha }}{x^{\alpha +1}}}}
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{\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
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{\displaystyle \beta }
x
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{\displaystyle x\in (\beta ;+\infty )}
funzione di distribuzione :
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{\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
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{\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
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{\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
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{\displaystyle \mu _{1}={\begin{cases}{\dfrac {\alpha \beta }{\alpha -1}}&{\text{per }}\alpha >1\\\infty &{\text{per }}\alpha \geq 1\end{cases}}}
μ
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{\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:
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{\displaystyle \mu _{2}={\begin{cases}{\dfrac {\alpha \beta ^{2}}{\alpha -2}}&{\text{per }}\alpha >2\\\infty &{\text{per }}\alpha \geq 2\end{cases}}}
In generale un momento di ordine
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{\displaystyle n}
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{\displaystyle \mu _{n}={\begin{cases}{\dfrac {\alpha \beta ^{n}}{x-n}}&{\text{per }}0<n<\alpha \\\infty &{\text{per }}n\geq \alpha \end{cases}}}
Funzione generatrice dei momenti
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{\displaystyle M(\theta )=E[e^{\theta x}]=\alpha (-\beta \theta )^{\alpha }\Gamma (-\alpha ,-\beta \theta )}
dove
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{\displaystyle \Gamma (-\alpha ,-\beta \theta )}
funzione gamma incompleta .
La funzione generatrice di momenti è definita solo per valori non positivi di
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{\displaystyle \theta }
Varianza
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{\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:
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{\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
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{\displaystyle \alpha \leq 1}
Mediana
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{\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
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{\displaystyle \xi _{0.5}={\sqrt[{\alpha }]{2}}\beta }
Simmetria
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{\displaystyle \beta _{1}={\frac {4(\alpha -2)(\alpha +1)^{2}}{\alpha (\alpha -3)^{2}}}}
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{\displaystyle \alpha >3}
Curtosi
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{\displaystyle \beta _{2}={\frac {3(\alpha -2)(3\alpha ^{2}+\alpha +2)}{\alpha (\alpha -3)(\alpha -4)}}}
α
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{\displaystyle \alpha >4}
Caratteristiche
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
Note
Voci correlate
Altri progetti
Collegamenti esterni
![{\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)
