Poisson distribution mgf

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Poisson distribution MGF  

Poisson Distribution is derived from a binomial Distribution.

Notably, it is the limiting form of a binomial distribution under the following conditions;

1. Probability of success,p, in each trial is small. That is $p\underset{}{\to }0$
2. The number of trials n is large. $(n\underset{}{\to \infty )}$
3. $np=\lambda$ is finite and positive real number.

Thus the PDF will be;

$=\left\{\begin{array}{l}\frac{{\lambda }^x e^{-\lambda }}{x!} x=0,1,2,3...\\ 0, elsewhere\end{array}\right.$

$M_x\left(t\right)=E\left(e^{tx}\right)$

$=\frac{\sum _{} e^{tx}{\lambda }^x{e}^{-\lambda }}{x!}$

$=e^{-\lambda }\sum _{x=0}^{\infty }\frac{{\left(\lambda e^t\right)}^x}{x!}$

$=e^{-\lambda }[1+\lambda e^t+\frac{{\left(\lambda e^t\right)}^2}{2!}+\frac{{\left(\lambda e^t\right)}^3}{3!}+.......]$

$=e^{-\lambda }{e}^{\lambda e^t}$

$={e^{-\lambda +}}^{\lambda e^t}$

$=e^{\lambda e^t-\lambda }$

$=e^{\lambda (e^t-1)}$

The MGF of Poisson Distribution is $e^{\lambda (e^t-1)}$

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