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We say that a random variable X has a Student t distribution with ν > 0 degrees of freedom, a scaling parameter δ > 0 and a location parameter μ ∈ R 1, denoted T( ν, δ, μ), if its probability density function (pdf) is \(\displaystyle f_X(x)=\frac {\Gamma \big (\frac 12(\nu +1)\big )}{\sqrt {\pi }\delta \Gamma \big (\frac 12\nu \big )} \bigg [1+\bigg (\frac {x-\mu }\delta \bigg )^2\bigg ]^{-\frac {\nu +1}2},\quad x\in R^1, \) where Γ( z) is the Euler’s gamma function. T(1, δ, μ) is the Cauchy distribution. T( ν, δ, μ) is heavy tailed and for an integer r \(\displaystyle E(X-\mu )^{2r}=\left \{ \begin {array}{ll} \frac {\delta ^{2r-1}\nu ^r\Gamma \big (\frac r2+1\big )\Gamma \big (\frac \nu 2-r\big )}{\sqrt {\pi }\Gamma \big (\frac 12\nu \big )}, & \mbox{if } 2r<\nu ,\\ +\infty , &\mbox{if } 2r\geq \nu . \end {array}\right . \)

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Student T Distribution, The

  • 工具书词条

  • 发表日期：

    2025年

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  • Bronius Grigelionis

摘要

We say that a random variable X has a Student t distribution with ν > 0 degrees of freedom, a scaling parameter δ > 0 and a location parameter μ ∈ R 1, denoted T( ν, δ, μ), if its probability density function (pdf) is \(\displaystyle f_X(x)=\frac {\Gamma \big (\frac 12(\nu +1)\big )}{\sqrt {\pi }\delta \Gamma \big (\frac 12\nu \big )} \bigg [1+\bigg (\frac {x-\mu }\delta \bigg )^2\bigg ]^{-\frac {\nu +1}2},\quad x\in R^1, \) where Γ( z) is the Euler’s gamma function. T(1, δ, μ) is the Cauchy distribution. T( ν, δ, μ) is heavy tailed and for an integer r \(\displaystyle E(X-\mu )^{2r}=\left \{ \begin {array}{ll} \frac {\delta ^{2r-1}\nu ^r\Gamma \big (\frac r2+1\big )\Gamma \big (\frac \nu 2-r\big )}{\sqrt {\pi }\Gamma \big (\frac 12\nu \big )}, & \mbox{if } 2r<\nu ,\\ +\infty , &\mbox{if } 2r\geq \nu . \end {array}\right . \)