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students_t_distribution [2015/11/30 11:26] nikolaj |
students_t_distribution [2015/11/30 11:44] nikolaj |
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| | @#FF9944: definition | @#FF9944: $f:$ ?? | | | @#FF9944: definition | @#FF9944: $f:$ ?? | | ||
| | @#FF9944: definition | @#FF9944: $f(t, \nu) := \dfrac{1}{\nu^{\frac{1}{2}}{\mathrm B}(\frac{1}{2},\frac{\nu}{2})}\cdot\left(1+\dfrac{t^2}{\nu}\right)^{-\frac{\nu+1}{2}}$ | | | @#FF9944: definition | @#FF9944: $f(t, \nu) := \dfrac{1}{\nu^{\frac{1}{2}}{\mathrm B}(\frac{1}{2},\frac{\nu}{2})}\cdot\left(1+\dfrac{t^2}{\nu}\right)^{-\frac{\nu+1}{2}}$ | | ||
| + | |||
| + | >todo | ||
| ----- | ----- | ||
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| $\lim_{n\to\infty}\left(1+\dfrac{x}{n}\right)^{a\,n+b}=\lim_{n\to\infty}\left(1+\dfrac{a\,x}{n}\right)^n={\mathrm e}^{a\,x}$. | $\lim_{n\to\infty}\left(1+\dfrac{x}{n}\right)^{a\,n+b}=\lim_{n\to\infty}\left(1+\dfrac{a\,x}{n}\right)^n={\mathrm e}^{a\,x}$. | ||
| - | here the normalization: | + | Here the normalization: |
| <code> | <code> | ||
| Plot[Sqrt[n/(2*Pi)]*Beta[1/2,n/2],{n,-6,6}] | Plot[Sqrt[n/(2*Pi)]*Beta[1/2,n/2],{n,-6,6}] | ||
| </code> | </code> | ||
| + | |||
| + | == Relation to the Cauchy distribution == | ||
| + | >todo | ||
| === Theorems === | === Theorems === | ||
| + | >todo | ||
| + | |||
| === Reference === | === Reference === | ||
| Wikipedia: | Wikipedia: | ||
