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- \begin{tikzpicture}[scale=1.9]
- \draw [white] node at (3,1.5){\textbf{\textcolor{yellow}{\Huge{Función de Densidad $\chi^2$ (pdf)}}}};
- \draw [white] node at (3,-0.5)
- {
- \begin{minipage}{13cm}\textcolor{white}{\huge{Sea $\mathsf{X}$ la variable aleatoria correspondiente a una distribución muestral $\chi^2$ con $\mathsf{\nu}$ grados de libertad ($\nu > 0$), entonces la función de densidad probabilística para $\mathsf{X}$ es:
- $$\mathsf{f(x) = \displaystyle{\frac{x^{\frac{\nu}{2}-1}e^{-\frac{x}{2}}}{2^{\nu/2}\ \Gamma(\frac{\nu}{2})},\ x\geq 0,\ x\in (0, \infty)}}$$}}\\
- \end{minipage};
- };
- \draw [white] node at (2.5,-2){\textbf{\textcolor{white}{\huge{$\nu$}: \Large{Número de grados de libertad} (\huge{$\nu > 0$}) }}};
- \draw [white] node at (2.5,-2.5){\textbf{\huge{\textcolor{white}{¡¡ $\mu\ =\ \nu$, $\sigma^2\ =\ 2\nu$ !!! }}}};
- \draw [white] node at (3,-3.5){\textbf{\textcolor{yellow}{\Huge{Función Acumulada $\chi^2$ (CDF)}}}};
- \draw [white] node at (3.2,-4.5){\textbf{\huge{$\mathsf{\displaystyle{P(X \leq x) = F(x)\ = \frac{1}{2^{\nu/2}\Gamma(\nu/2)}\int_0^x t^{\frac{\nu}{2}-1}e^{-t/2}\ dt }}$ }}};
- \draw [white] node at (3,-7)
- {
- \begin{minipage}{13cm}\textcolor{white}{\huge{Es común referirse a la integral en el lado derecho como \textcolor{green!90!white!80!blue!50}{$\mathsf{\gamma(\nu/2,\ x/2)}$}, en dónde \textcolor{green!90!white!80!blue!50}{$\gamma$} es la función Gama Incompleta:\\
- \textcolor{green!90!white!80!blue!50}{$\mathsf{\gamma(z, x)}$} = \textcolor{orange}{$\mathsf{\displaystyle{\int_0^x u^{z-1}e^{-u}du}}$}}}
- \end{minipage};
- };
- \end{tikzpicture}
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