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第2行： − − {{Probability distribution − − {概率分布 − − | name =Pareto Type I − − − | name = Pareto Type i − − − | type =density − − 类型 = 密度 − − − | 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. − − | 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。 − − − | 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. − − 不同 α < br/> Pareto i 型累积分布函数的 Pareto i 型累积分布函数。 − − − | parameters =x_\mathrm{m} > 0 scale (real)<br />\alpha > 0 shape (real) − − 0 scale (real) < br/> alpha > 0 shape (real) − − − | support =x \in [x_\mathrm{m}, \infty) − − | support = x in [ x _ mathrm { m } ，infty) − − − | pdf =\frac{\alpha x_\mathrm{m}^\alpha}{x^{\alpha+1}} − − 1}{ x ^ { alpha + 1}} − − − | cdf =1-\left(\frac{x_\mathrm{m}}{x}\right)^\alpha − − 1-left (frac { x _ mathrm { m }{ x } right) ^ alpha − − | mean =<math>\begin{cases} − − − 开始{ cases } − − \infty & \text{for }\alpha\le 1 \\ − − − 1.1 − − − \dfrac{\alpha x_\mathrm{m}}{\alpha-1} & \text{for }\alpha>1 − − 1 & text { for } alpha > 1 − − − \end{cases}</math> − − 结束{ cases } </math > − − − | median =x_\mathrm{m} \sqrt[\alpha]{2} − − 中位数 = x _ mathrm { m } sqrt [ alpha ]{2} − − − | mode =x_\mathrm{m} − − 2009年10月11日 − − | variance =<math>\begin{cases} − − − | 方差 = < math > begin { cases } − − \infty & \text{for }\alpha\le 2 \\ − − − 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 − − \dfrac{x_\mathrm{m}^2\alpha}{(\alpha-1)^2(\alpha-2)} & \text{for }\alpha>2 − − − 1) ^ 2(alpha-2)} & text { for } alpha > 2 − − \end{cases}</math> − − − 结束{ cases } </math > − − − | skewness =\frac{2(1+\alpha)}{\alpha-3}\sqrt{\frac{\alpha-2}{\alpha}}\text{ for }\alpha>3 − − | skewness = frac {2(1 + alpha)}{ alpha-3} sqrt { frac { alpha-2}{ alpha }}} text { for } alpha > 3 − − − | kurtosis =\frac{6(\alpha^3+\alpha^2-6\alpha-2)}{\alpha(\alpha-3)(\alpha-4)}\text{ for }\alpha>4 − − | 峭度 = frac {6(alpha ^ 3 + alpha ^ 2-6 alpha-2)}{ alpha (alpha-3)(alpha-4)} text { for } alpha > 4 − − − | entropy =\log\left(\left(\frac{x_\mathrm{m}}{\alpha}\right)\,e^{1+\tfrac{1}{\alpha}}\right) − − | 熵 = log left (left (left (frac { x _ mathrm { m }{ alpha }右) ，e ^ {1 + tfrac {1}{ alpha }右) − − − | mgf =\alpha(-x_\mathrm{m}t)^\alpha\Gamma(-\alpha,-x_\mathrm{m}t)\text{ for }t<0 − − | mgf = alpha (- x _ mathrm { m } t) ^ alpha Gamma (- alpha,-x _ mathrm { m } t) text { for } t < 0 − − − | char =\alpha(-ix_\mathrm{m}t)^\alpha\Gamma(-\alpha,-ix_\mathrm{m}t) − − | char = alpha (- ix _ mathrm { m } t) ^ alpha Gamma (- alpha,-ix _ mathrm { m } t) − − − | fisher =<math>\mathcal{I}(x_\mathrm{m},\alpha) = \begin{bmatrix} − − | fisher = < math > mathcal { i }(x _ mathrm { m } ，alpha) = begin { bmatrix } − − − \dfrac{\alpha}{x_\mathrm{m}^2} & -\dfrac{1}{x_\mathrm{m}} \\ − − 2} &-dfrac {1}{ x mathrm { m } − − -\dfrac{1}{x_\mathrm{m}} & \dfrac{1}{\alpha^2} − − − - dfrac {1}{ x _ mathrm { m } & dfrac {1}{ alpha ^ 2} − − \end{bmatrix}</math> − − − 结束{ bmatrix } </math > − − − − Right: <math>\mathcal{I}(x_\mathrm{m},\alpha) = \begin{bmatrix} − − 右: < math > mathcal { i }(x _ mathrm { m } ，alpha) = begin { bmatrix } − − \dfrac{\alpha^2}{x_\mathrm{m}^2} & 0 \\ − − − 2} & 0 − − 0 & \dfrac{1}{\alpha^2} − − − 0 & dfrac {1}{ alpha ^ 2} − − \end{bmatrix}</math> − − − 结束{ bmatrix } </math > − − }} − − − }} −
无编辑摘要
{{Probability distribution
{{Probability distribution
| name =Pareto Type I
| name =Pareto Type I
| type =density
| type =density
| 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]].
| 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>
| 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)
| support =<math>x \in [x_\mathrm{m}, \infty)</math>
| support =<math>x \in [x_\mathrm{m}, \infty)</math>
| pdf =<math>\frac{\alpha x_\mathrm{m}^\alpha}{x^{\alpha+1}}</math>
| pdf =<math>\frac{\alpha x_\mathrm{m}^\alpha}{x^{\alpha+1}}</math>
| cdf =<math>1-\left(\frac{x_\mathrm{m}}{x}\right)^\alpha</math>
| cdf =<math>1-\left(\frac{x_\mathrm{m}}{x}\right)^\alpha</math>
| mean =<math>\begin{cases}
| mean =<math>\begin{cases}
\infty & \text{for }\alpha\le 1 \\
\infty & \text{for }\alpha\le 1 \\
\dfrac{\alpha x_\mathrm{m}}{\alpha-1} & \text{for }\alpha>1
\dfrac{\alpha x_\mathrm{m}}{\alpha-1} & \text{for }\alpha>1
\end{cases}</math>
\end{cases}</math>
| median =<math>x_\mathrm{m} \sqrt[\alpha]{2}</math>
| median =<math>x_\mathrm{m} \sqrt[\alpha]{2}</math>
| mode =<math>x_\mathrm{m}</math>
| mode =<math>x_\mathrm{m}</math>
| variance =<math>\begin{cases}
| variance =<math>\begin{cases}
\infty & \text{for }\alpha\le 2 \\
\infty & \text{for }\alpha\le 2 \\
\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
\end{cases}</math>
\end{cases}</math>
| 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>
| 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>
| 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>
| 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>
| 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>
| fisher =<math>\mathcal{I}(x_\mathrm{m},\alpha) = \begin{bmatrix}
| fisher =<math>\mathcal{I}(x_\mathrm{m},\alpha) = \begin{bmatrix}
\dfrac{\alpha}{x_\mathrm{m}^2} & -\dfrac{1}{x_\mathrm{m}} \\
\dfrac{\alpha}{x_\mathrm{m}^2} & -\dfrac{1}{x_\mathrm{m}} \\
-\dfrac{1}{x_\mathrm{m}} & \dfrac{1}{\alpha^2}
-\dfrac{1}{x_\mathrm{m}} & \dfrac{1}{\alpha^2}
\end{bmatrix}</math>
\end{bmatrix}</math>
Right: <math>\mathcal{I}(x_\mathrm{m},\alpha) = \begin{bmatrix}
Right: <math>\mathcal{I}(x_\mathrm{m},\alpha) = \begin{bmatrix}
\dfrac{\alpha^2}{x_\mathrm{m}^2} & 0 \\
\dfrac{\alpha^2}{x_\mathrm{m}^2} & 0 \\
0 & \dfrac{1}{\alpha^2}
0 & \dfrac{1}{\alpha^2}
\end{bmatrix}</math>
\end{bmatrix}</math>
}}
}}