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**Answer**

## 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;

- Probability of success,p, in each trial is small. That is $p\underset{}{\to}0$
- The number of trials n is large. $(n\underset{}{\to \infty )}$
- $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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