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HP17bII+ Programming t-distribution
07-25-2015, 10:12 AM
Post: #9
RE: HP17bII+ Programming t-distribution
(07-23-2015 09:52 PM)Don Shepherd Wrote:  according to the formula given at Wikipedia

\(\frac{\Gamma \left(\frac{\nu+1}{2} \right)} {\sqrt{\nu\pi}\,\Gamma \left(\frac{\nu}{2} \right)} \left(1+\frac{x^2}{\nu} \right)^{-\frac{\nu+1}{2}}\)

We can use the following identities:
  • \(\Gamma(n)=(n-1)!\) for \(n\epsilon \mathbb{N}\)
  • \(\Gamma(x+1)=x \Gamma(x)\)
  • \(\Gamma\left(\tfrac{1}{2}\right)=\sqrt{\pi}\)

For \(\nu\) even:

\(\begin{align}
\Gamma(\frac{\nu+1}{2})&=\frac{\nu-1}{2}\cdot\frac{\nu-3}{2}\cdots\frac{1}{2}\cdot\Gamma\left(\tfrac{1}{2}\right) \\
&=\frac{\nu-1}{2}\cdot\frac{\nu-2}{\nu-2}\cdot\frac{\nu-3}{2}\cdot\frac{\nu-4}{\nu-4}\cdots\frac{2}{2}\cdot\frac{1}{2}\cdot\Gamma\left(\tfrac{1}{2}\right) \\
&=\frac{\nu-1}{2}\cdot\frac{\nu-2}{2\cdot(\frac{\nu}{2}-1)}\cdot\frac{\nu-3}{2}\cdot\frac{\nu-4}{2\cdot(\frac{\nu}{2}-2)}\cdots\frac{2}{2\cdot1}\cdot\frac{1}{2}\cdot\Gamma\left(\tfrac{1}{2}\right) \\
&=\frac{(\nu-1)!}{2^{\nu-1}\,(\frac{\nu}{2}-1)!}\,\sqrt{\pi} \\
\end{align}\)

\(\Gamma(\frac{\nu}{2})=(\frac{\nu}{2}-1)!\)

\(\begin{align}
\frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\pi}\,\Gamma(\frac{\nu}{2})}&=\frac{(\nu-1)!}{\sqrt{\pi}\,2^{n-1}\,(\frac{\nu}{2}-1)!\,(\frac{\nu}{2}-1)!}\,\sqrt{\pi} \\
&=\frac{(\nu-1)!}{2^{n-1}\,((\frac{\nu}{2}-1)!)^2} \\
\end{align}\)

For \(\nu\) odd:

\(\Gamma(\tfrac{\nu+1}{2})=(\tfrac{\nu+1}{2}-1)!=\frac{\nu-1}{2}!\)

\(\begin{align}
\Gamma(\frac{\nu}{2})&=\frac{\nu-2}{2}\cdot\frac{\nu-4}{2}\cdots\frac{1}{2}\cdot\Gamma\left(\tfrac{1}{2}\right) \\
&=\frac{\nu-1}{\nu-1}\cdot\frac{\nu-2}{2}\cdot\frac{\nu-3}{\nu-3}\cdot\frac{\nu-4}{2}\cdots\frac{2}{2}\cdot\frac{1}{2}\cdot\Gamma\left(\tfrac{1}{2}\right) \\
&=\frac{\nu-1}{2\cdot\frac{\nu-1}{2}}\cdot\frac{\nu-2}{2}\cdot\frac{\nu-3}{2\cdot(\frac{\nu-1}{2}-1)}\cdot\frac{\nu-4}{2}\cdots\frac{2}{2\cdot1}\cdot\frac{1}{2}\cdot\Gamma\left(\tfrac{1}{2}\right)​ \\
&=\frac{(\nu-1)!}{2^{\nu-1}\,\frac{\nu-1}{2}!}\,\sqrt{\pi} \\
\end{align}\)

\(\begin{align}
\frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\pi}\,\Gamma(\frac{\nu}{2})}&=\frac{\frac{\nu-1}{2}!\,2^{\nu-1}\,\frac{\nu-1}{2}!}{\sqrt{\pi}\,(\nu-1)!\,\sqrt{\pi}} \\
&=\frac{2^{\nu-1}\,(\frac{\nu-1}{2}!)^2}{(\nu-1)!\,\pi} \\
\end{align}\)

Kind regards
Thomas
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RE: HP17bII+ Programming t-distribution - Thomas Klemm - 07-25-2015 10:12 AM



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