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%\textcolor{white}{\textbf{Seguridad en Redes y Cifrado De Texto}}\\
%\textcolor{yellow}{Nimrod Rodríguez}\\
%\href{ENLACE WEB}{\underline{\textcolor{white}{enlace:} \textcolor{green}{Código \LaTeX} }}
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%\draw (6,1,-3)node[white, below=2cm] {\bf\large{Acercándose al \textit{O}rigen}};

%\draw (6,4,-3)node[white, below=2cm] {\bf\huge{$\mathsf{C_1 :}$ $\mathsf{u = x e^{i\pi}}$, $\ \infty > x \geq \delta$}};

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{\begin{minipage}{7cm}\textcolor{white}{\textbf{La aproximación parte de la relación binomial $\mathsf{\displaystyle{\mu\ =\ np}}$
y toma como objetivo la forma $\mathsf{\displaystyle{p\ =\ \frac{\mu}{n}}}$} \textbf{
Con esto en mano, consideramos la familia de funciones binomiales, con $\mathsf{\displaystyle{n \rightarrow \infty :}}$
$$\mathsf{\displaystyle{f_n(x)=\frac{n!}{x!(n-x)!}\left(\frac{\mu}{n}\right)^x\left(1-\frac{\mu}{n}\right)^n\left(1-\frac{\mu}{n}\right)^{-x} }}$$ Y para obtener más explícitamente el límite, reescribimos como:
$$\mathsf{\displaystyle{f_n(x) = \frac{\mu^x}{x!}\frac{n!}{n^x(n-x)!}\left(1-\frac{\mu}{n}\right)^n \left(1-\frac{\mu}{n}\right)^{-x}}}$$
Expresión que permite verificar el límite, como el producto de los límites de tres sucesiones como sigue:
$$\mathsf{\displaystyle{\lim_{n \to \infty}\frac{n!}{n^x (n-x)!} = 1}}$$
$$\mathsf{\displaystyle{\lim_{n \to \infty}\left(1 -\frac{\mu}{n}\right)^n = e^{-\mu}}}$$
$$\mathsf{\displaystyle{\lim_{n \to \infty}\left(1 -\frac{\mu}{n}\right)^{-x} = 1}}$$
Así que podemos escribir:
$$\mathsf{\displaystyle{\lim_{n\to\infty}f_n(x) = f^*(x) = \frac{\mu^x\ e^{-\mu}}{x!}}}$$
Expresión que es justamente la distribución de Poisson.}}
\end{minipage}};
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%\bibitem{Stinson2019} STINSON, Douglas R. \& PATERSON, Maura B. \underline{Cryptography
%Theory and Practice.}
%Fourth Edition CRC Press
%Taylor \& Francis Group
%6000 Broken Sound Parkway NW, Suite 300.
%Boca Raton, FL 33487-2742
%© 2019 by Taylor \& Francis Group, LLC
%\bibitem{tanen2003} TANENBAUM, Andrew S.\& WETHERALL, David J. \underline{Redes de Computadoras.} © Pearson-Prentice Hall, Educación. México 2003.


%\end{thebibliography}
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