Aproximación Binomial a Poisson

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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.}}
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