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| {{Probability distribution | | {{Probability distribution |
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− | {{Probability distribution
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− | {概率分布
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− | | name =Pareto Type I
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| | name =Pareto Type I | | | name =Pareto Type I |
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− | | name = Pareto Type i
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| | type =density | | | type =density |
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− | | type =density
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− | 类型 = 密度
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| | pdf_image =[[File:Probability density function of Pareto distribution.svg|325px|Pareto Type I probability density functions for various ''α'']]<br />Pareto Type I probability density functions for various <math>\alpha</math> with <math>x_\mathrm{m} = 1.</math> As <math>\alpha \rightarrow \infty,</math> the distribution approaches <math>\delta(x - x_\mathrm{m}),</math> where <math>\delta</math> is the [[Dirac delta function]]. | | | pdf_image =[[File:Probability density function of Pareto distribution.svg|325px|Pareto Type I probability density functions for various ''α'']]<br />Pareto Type I probability density functions for various <math>\alpha</math> with <math>x_\mathrm{m} = 1.</math> As <math>\alpha \rightarrow \infty,</math> the distribution approaches <math>\delta(x - x_\mathrm{m}),</math> where <math>\delta</math> is the [[Dirac delta function]]. |
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− | | pdf_image =Pareto Type I probability density functions for various α<br />Pareto Type I probability density functions for various \alpha with x_\mathrm{m} = 1. As \alpha \rightarrow \infty, the distribution approaches \delta(x - x_\mathrm{m}), where \delta is the Dirac delta function.
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− | | pdf _ image = 各种 α < br/> Pareto i 型概率密度函数的 Pareto i 型概率密度函数。在 α 向右下方,分布趋近于 δ (x-x _ mathrm { m }) ,其中 δ 是狄拉克δ函数。
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| | cdf_image =[[File:Cumulative distribution function of Pareto distribution.svg|325px|Pareto Type I cumulative distribution functions for various ''α'']]<br />Pareto Type I cumulative distribution functions for various <math>\alpha</math> with <math>x_\mathrm{m} = 1.</math> | | | cdf_image =[[File:Cumulative distribution function of Pareto distribution.svg|325px|Pareto Type I cumulative distribution functions for various ''α'']]<br />Pareto Type I cumulative distribution functions for various <math>\alpha</math> with <math>x_\mathrm{m} = 1.</math> |
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− | | cdf_image =Pareto Type I cumulative distribution functions for various α<br />Pareto Type I cumulative distribution functions for various \alpha with x_\mathrm{m} = 1.
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− | 不同 α < br/> Pareto i 型累积分布函数的 Pareto i 型累积分布函数。
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| | parameters =<math>x_\mathrm{m} > 0</math> [[scale parameter|scale]] ([[real number|real]])<br /><math>\alpha > 0</math> [[shape parameter|shape]] (real) | | | parameters =<math>x_\mathrm{m} > 0</math> [[scale parameter|scale]] ([[real number|real]])<br /><math>\alpha > 0</math> [[shape parameter|shape]] (real) |
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− | | parameters =x_\mathrm{m} > 0 scale (real)<br />\alpha > 0 shape (real)
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− | 0 scale (real) < br/> alpha > 0 shape (real)
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| | support =<math>x \in [x_\mathrm{m}, \infty)</math> | | | support =<math>x \in [x_\mathrm{m}, \infty)</math> |
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− | | support =x \in [x_\mathrm{m}, \infty)
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− | | support = x in [ x _ mathrm { m } ,infty)
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| | pdf =<math>\frac{\alpha x_\mathrm{m}^\alpha}{x^{\alpha+1}}</math> | | | pdf =<math>\frac{\alpha x_\mathrm{m}^\alpha}{x^{\alpha+1}}</math> |
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− | | pdf =\frac{\alpha x_\mathrm{m}^\alpha}{x^{\alpha+1}}
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− | 1}{ x ^ { alpha + 1}}
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| | cdf =<math>1-\left(\frac{x_\mathrm{m}}{x}\right)^\alpha</math> | | | cdf =<math>1-\left(\frac{x_\mathrm{m}}{x}\right)^\alpha</math> |
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− | | cdf =1-\left(\frac{x_\mathrm{m}}{x}\right)^\alpha
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− | 1-left (frac { x _ mathrm { m }{ x } right) ^ alpha
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− | | mean =<math>\begin{cases}
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| | mean =<math>\begin{cases} | | | mean =<math>\begin{cases} |
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− | 开始{ cases }
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− | \infty & \text{for }\alpha\le 1 \\
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| \infty & \text{for }\alpha\le 1 \\ | | \infty & \text{for }\alpha\le 1 \\ |
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− | 1.1
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| \dfrac{\alpha x_\mathrm{m}}{\alpha-1} & \text{for }\alpha>1 | | \dfrac{\alpha x_\mathrm{m}}{\alpha-1} & \text{for }\alpha>1 |
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− | \dfrac{\alpha x_\mathrm{m}}{\alpha-1} & \text{for }\alpha>1
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− | 1 & text { for } alpha > 1
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| \end{cases}</math> | | \end{cases}</math> |
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− | \end{cases}</math>
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− | 结束{ cases } </math >
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| | median =<math>x_\mathrm{m} \sqrt[\alpha]{2}</math> | | | median =<math>x_\mathrm{m} \sqrt[\alpha]{2}</math> |
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− | | median =x_\mathrm{m} \sqrt[\alpha]{2}
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− | 中位数 = x _ mathrm { m } sqrt [ alpha ]{2}
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| | mode =<math>x_\mathrm{m}</math> | | | mode =<math>x_\mathrm{m}</math> |
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− | | mode =x_\mathrm{m}
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− | 2009年10月11日
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− | | variance =<math>\begin{cases}
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| | variance =<math>\begin{cases} | | | variance =<math>\begin{cases} |
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− | | 方差 = < math > begin { cases }
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− | \infty & \text{for }\alpha\le 2 \\
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| \infty & \text{for }\alpha\le 2 \\ | | \infty & \text{for }\alpha\le 2 \\ |
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− | 2.1.1.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2
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− | \dfrac{x_\mathrm{m}^2\alpha}{(\alpha-1)^2(\alpha-2)} & \text{for }\alpha>2
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| \dfrac{x_\mathrm{m}^2\alpha}{(\alpha-1)^2(\alpha-2)} & \text{for }\alpha>2 | | \dfrac{x_\mathrm{m}^2\alpha}{(\alpha-1)^2(\alpha-2)} & \text{for }\alpha>2 |
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− | 1) ^ 2(alpha-2)} & text { for } alpha > 2
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− | \end{cases}</math>
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| \end{cases}</math> | | \end{cases}</math> |
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− | 结束{ cases } </math >
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| | skewness =<math>\frac{2(1+\alpha)}{\alpha-3}\sqrt{\frac{\alpha-2}{\alpha}}\text{ for }\alpha>3</math> | | | skewness =<math>\frac{2(1+\alpha)}{\alpha-3}\sqrt{\frac{\alpha-2}{\alpha}}\text{ for }\alpha>3</math> |
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− | | skewness =\frac{2(1+\alpha)}{\alpha-3}\sqrt{\frac{\alpha-2}{\alpha}}\text{ for }\alpha>3
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− | | skewness = frac {2(1 + alpha)}{ alpha-3} sqrt { frac { alpha-2}{ alpha }}} text { for } alpha > 3
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| | kurtosis =<math>\frac{6(\alpha^3+\alpha^2-6\alpha-2)}{\alpha(\alpha-3)(\alpha-4)}\text{ for }\alpha>4</math> | | | kurtosis =<math>\frac{6(\alpha^3+\alpha^2-6\alpha-2)}{\alpha(\alpha-3)(\alpha-4)}\text{ for }\alpha>4</math> |
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− | | kurtosis =\frac{6(\alpha^3+\alpha^2-6\alpha-2)}{\alpha(\alpha-3)(\alpha-4)}\text{ for }\alpha>4
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− | | 峭度 = frac {6(alpha ^ 3 + alpha ^ 2-6 alpha-2)}{ alpha (alpha-3)(alpha-4)} text { for } alpha > 4
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| | entropy =<math>\log\left(\left(\frac{x_\mathrm{m}}{\alpha}\right)\,e^{1+\tfrac{1}{\alpha}}\right) </math> | | | entropy =<math>\log\left(\left(\frac{x_\mathrm{m}}{\alpha}\right)\,e^{1+\tfrac{1}{\alpha}}\right) </math> |
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− | | entropy =\log\left(\left(\frac{x_\mathrm{m}}{\alpha}\right)\,e^{1+\tfrac{1}{\alpha}}\right)
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− | | 熵 = log left (left (left (frac { x _ mathrm { m }{ alpha }右) ,e ^ {1 + tfrac {1}{ alpha }右)
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| | mgf =<math>\alpha(-x_\mathrm{m}t)^\alpha\Gamma(-\alpha,-x_\mathrm{m}t)\text{ for }t<0</math> | | | mgf =<math>\alpha(-x_\mathrm{m}t)^\alpha\Gamma(-\alpha,-x_\mathrm{m}t)\text{ for }t<0</math> |
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− | | mgf =\alpha(-x_\mathrm{m}t)^\alpha\Gamma(-\alpha,-x_\mathrm{m}t)\text{ for }t<0
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− | | mgf = alpha (- x _ mathrm { m } t) ^ alpha Gamma (- alpha,-x _ mathrm { m } t) text { for } t < 0
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| | char =<math>\alpha(-ix_\mathrm{m}t)^\alpha\Gamma(-\alpha,-ix_\mathrm{m}t)</math> | | | char =<math>\alpha(-ix_\mathrm{m}t)^\alpha\Gamma(-\alpha,-ix_\mathrm{m}t)</math> |
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− | | char =\alpha(-ix_\mathrm{m}t)^\alpha\Gamma(-\alpha,-ix_\mathrm{m}t)
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− | | char = alpha (- ix _ mathrm { m } t) ^ alpha Gamma (- alpha,-ix _ mathrm { m } t)
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| | fisher =<math>\mathcal{I}(x_\mathrm{m},\alpha) = \begin{bmatrix} | | | fisher =<math>\mathcal{I}(x_\mathrm{m},\alpha) = \begin{bmatrix} |
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− | | fisher =<math>\mathcal{I}(x_\mathrm{m},\alpha) = \begin{bmatrix}
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− | | fisher = < math > mathcal { i }(x _ mathrm { m } ,alpha) = begin { bmatrix }
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| \dfrac{\alpha}{x_\mathrm{m}^2} & -\dfrac{1}{x_\mathrm{m}} \\ | | \dfrac{\alpha}{x_\mathrm{m}^2} & -\dfrac{1}{x_\mathrm{m}} \\ |
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− | \dfrac{\alpha}{x_\mathrm{m}^2} & -\dfrac{1}{x_\mathrm{m}} \\
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− | 2} &-dfrac {1}{ x mathrm { m }
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− | -\dfrac{1}{x_\mathrm{m}} & \dfrac{1}{\alpha^2}
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| -\dfrac{1}{x_\mathrm{m}} & \dfrac{1}{\alpha^2} | | -\dfrac{1}{x_\mathrm{m}} & \dfrac{1}{\alpha^2} |
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− | - dfrac {1}{ x _ mathrm { m } & dfrac {1}{ alpha ^ 2}
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− | \end{bmatrix}</math>
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| \end{bmatrix}</math> | | \end{bmatrix}</math> |
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− | 结束{ bmatrix } </math >
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− | Right: <math>\mathcal{I}(x_\mathrm{m},\alpha) = \begin{bmatrix}
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| Right: <math>\mathcal{I}(x_\mathrm{m},\alpha) = \begin{bmatrix} | | Right: <math>\mathcal{I}(x_\mathrm{m},\alpha) = \begin{bmatrix} |
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− | 右: < math > mathcal { i }(x _ mathrm { m } ,alpha) = begin { bmatrix }
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− | \dfrac{\alpha^2}{x_\mathrm{m}^2} & 0 \\
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| \dfrac{\alpha^2}{x_\mathrm{m}^2} & 0 \\ | | \dfrac{\alpha^2}{x_\mathrm{m}^2} & 0 \\ |
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− | 2} & 0
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− | 0 & \dfrac{1}{\alpha^2}
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| 0 & \dfrac{1}{\alpha^2} | | 0 & \dfrac{1}{\alpha^2} |
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− | 0 & dfrac {1}{ alpha ^ 2}
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− | \end{bmatrix}</math>
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| \end{bmatrix}</math> | | \end{bmatrix}</math> |
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− | 结束{ bmatrix } </math >
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− | }}
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| }} | | }} |
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− | }}
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