Statistics/Distributions/Student-t

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Student-t Distribution[edit]

Student’s t
Probability density function
Student t pdf.svg
Cumulative distribution function
Student t cdf.svg
Parameters ν > 0 degrees of freedom (real)
Support x ∈ (−∞; +∞)
PDF \textstyle\frac{\Gamma \left(\frac{\nu+1}{2} \right)} {\sqrt{\nu\pi}\,\Gamma \left(\frac{\nu}{2} \right)} \left(1+\frac{x^2}{\nu} \right)^{-\frac{\nu+1}{2}}\!
CDF \begin{matrix}
     \frac{1}{2} + x \Gamma \left( \frac{\nu+1}{2} \right)  \cdot\\[0.5em]
     \frac{\,_2F_1 \left ( \frac{1}{2},\frac{\nu+1}{2};\frac{3}{2};
           -\frac{x^2}{\nu} \right)}
     {\sqrt{\pi\nu}\,\Gamma \left(\frac{\nu}{2}\right)}
     \end{matrix}
where 2F1 is the hypergeometric function
Mean 0 for ν > 1, otherwise undefined
Median 0
Mode 0
Variance \textstyle\frac{\nu}{\nu-2} for ν > 2, ∞ for 1 < ν ≤ 2, otherwise undefined
Skewness 0 for ν > 3, otherwise undefined
Ex. kurtosis \textstyle\frac{6}{\nu-4} for ν > 4, ∞ for 2 < ν ≤ 4, otherwise undefined
Entropy ...
MGF undefined
CF \textstyle\frac{K_{\nu/2} \left(\sqrt{\nu}|t|)(\sqrt{\nu}|t| \right)^{\nu/2}}{\Gamma(\nu/2)2^{\nu/2-1}} for ν > 0

Student t-distribution (or just t-distribution for short) is derived from the chi-square and normal distributions. We divide the standard normally distributed value of one variable over the root of a chi-square value over its r degrees of freedom. Mathematically, this appears as:


t = \frac{\mbox{Z}}{ \sqrt{ \chi_{r}^2 /r }}

where  Z = \frac{X - \bar{X}}{\sigma} and  \chi_{r}^2 = \chi_{r_1}^2 + ... + \chi_{r_n}^2 .

External links[edit]