Lecture 4. D) Poisson
Poisson Distribution
A r.v. [math]X[/math] follows a Poisson distribution with parameter [math]\lambda\gt 0[/math], if [math]X[/math] is discrete with pmf
[math]f_{X}\left(x\right)=\begin{cases} \exp\left(-\lambda\right)\frac{\lambda^{x}}{x!}, & x\in\mathbb{N}_{0}\\ 0, & otherwise \end{cases}[/math]
The Poisson distribution characterizes a process with constant arrival rate, [math]\lambda[/math] (expressed as number of arrivals per unit of time).
Fun fact: [math]Bin\left(n,p\right)\simeq Pois\left(np\right)[/math] for [math]n[/math] large and [math]np[/math] small.
Mean
[math]E\left(X\right)=\sum_{x=0}^{^{\infty}}xf_{X}\left(x\right)=\sum_{x=0}^{^{\infty}}x\exp\left(-\lambda\right)\frac{\lambda^{x}}{x!}=\sum_{x=1}^{^{\infty}}\exp\left(-\lambda\right)\frac{\lambda^{x}}{\left(x-1\right)!}=\lambda\sum_{x=1}^{^{\infty}}\underset{f_{X}\left(\left.x-1\right|\lambda\right)}{\underbrace{\exp\left(-\lambda\right)\frac{\lambda^{x-1}}{\left(x-1\right)!}}}=\lambda\underset{=1}{\underbrace{\sum_{t=0}^{^{\infty}}f_{X}\left(\left.t\right|\lambda\right)}}=\lambda[/math].
Variance
[math]Var\left(X\right)=\lambda[/math] since [math]\underset{2nd\,factorial\,moment\,X}{\underbrace{E\left(X\left(X-1\right)\right)}}=\sum_{x=0}^{^{\infty}}x\left(x-1\right)\exp\left(-\lambda\right)\frac{\lambda^{x}}{x!}=\sum_{x=2}^{^{\infty}}\exp\left(-\lambda\right)\frac{\lambda^{x}}{\left(x-2\right)!}=\lambda^{2}\sum_{x=2}^{^{\infty}}\exp\left(-\lambda\right)\frac{\lambda^{x-2}}{x!}=\lambda^{2}[/math] and [math]Var\left(X\right)=E\left(X^{2}\right)-E\left(X\right)^{2}=E\left(X\left(X-1\right)\right)+E\left(X\right)-E\left(X\right)^{2}=\lambda[/math].
MGF
[math]M_{X}\left(t\right)=\exp\left(\lambda\left(\exp\left(t\right)-1\right)\right)[/math]
