Student’s t-distribution

Student's t
Probability density function
Cumulative distribution function
Parameters	$\nu >0$ 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	${\begin{matrix}{\frac {1}{2}}+x\Gamma \left({\frac {\nu +1}{2}}\right)\times \\[0.5em]{\frac {\,_{2}F_{1}\left({\frac {1}{2}},{\frac {\nu +1}{2}};{\frac {3}{2}};-{\frac {x^{2}}{\nu }}\right)}{{\sqrt {\pi \nu }}\,\Gamma \left({\frac {\nu }{2}}\right)}}\end{matrix}}$ where ₂F₁ is the hypergeometric function
Mean	0 for $\nu >1$ , otherwise undefined
Median	0
Mode	0
Variance	$\textstyle {\frac {\nu }{\nu -2}}$ for $\nu >2$ , ∞ for $1<\nu \leq 2$ , otherwise undefined
Skewness	0 for $\nu >3$ , otherwise undefined
Ex. kurtosis	$\textstyle {\frac {6}{\nu -4}}$ for $\nu >4$ , ∞ for $2<\nu \leq 4$ , otherwise undefined
Entropy	${\begin{matrix}{\frac {\nu +1}{2}}\left[\psi \left({\frac {1+\nu }{2}}\right)-\psi \left({\frac {\nu }{2}}\right)\right]\\[0.5em]+\ln {\left[{\sqrt {\nu }}B\left({\frac {\nu }{2}},{\frac {1}{2}}\right)\right]}\,{\scriptstyle {\text{(nats)}}}\end{matrix}}$ ψ: digamma function, B: beta function
MGF	undefined
CF	$\textstyle {\frac {K_{\nu /2}\left({\sqrt {\nu }}\|t\|\right)\cdot \left({\sqrt {\nu }}\|t\|\right)^{\nu /2}}{\Gamma (\nu /2)2^{\nu /2-1}}}$ for $\nu >0$ $K_{\nu }(x)$ : Modified Bessel function of the second kind

${\begin{matrix}{\frac {1}{2}}+x\Gamma \left({\frac {\nu +1}{2}}\right)\times \\[0.5em]{\frac {\,_{2}F_{1}\left({\frac {1}{2}},{\frac {\nu +1}{2}};{\frac {3}{2}};-{\frac {x^{2}}{\nu }}\right)}{{\sqrt {\pi \nu }}\,\Gamma \left({\frac {\nu }{2}}\right)}}\end{matrix}}$

${\begin{matrix}{\frac {\nu +1}{2}}\left[\psi \left({\frac {1+\nu }{2}}\right)-\psi \left({\frac {\nu }{2}}\right)\right]\\[0.5em]+\ln {\left[{\sqrt {\nu }}B\left({\frac {\nu }{2}},{\frac {1}{2}}\right)\right]}\,{\scriptstyle {\text{(nats)}}}\end{matrix}}$

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Wikipedia