Poisson Limit Theorem

Poisson Limit Theorem (Simplified) §

thm. Poisson Limit Theorem. With a random variable $X\sim \text{Binom}(n,p)$, as $n\to \infty ,p\to 0$, $X\approx \text{Poi}(np)$, since:

$X={\mathbb{I}}_1+\cdots {\mathbb{I}}_n$ where each is an indicator of success of the $i$-th event, and as ${\text{lim}}_{n\to \infty ,p\to 0}$ this defines the Poisson Distribution.

Poisson Scatter Theorem §

def. An experiment process which adheres to the following creteria is a Poisson Scatter process.

1. Number of hits are finite
2. No multiple hits on one point
3. Hits are homogenous and independent (any non-overlapping region’s hit number is independent.)

THM Poisson Scatter Theorem. In a poisson scatter process:

1. Number of hits over area R is a Poisson Random variable
2. The number of hits in each disjoint region is independent of each other (definiiton #3)
3. The rate of hits ($\lambda$) is proportional to its area

Poisson Addition Rule §

thm. If $X\sim \text{Poi}({\lambda }_X)$ and $Y\sim \text{Poi}({\lambda }_Y)$ and $X\perp Y$ then:

$$X+Y\sim \text{Poi}({\lambda }_X+{\lambda }_Y)$$

Info

Note also that the bionmial distribution as something like this too. If $X\sim \text{Bi}(n_X,p)$ and Y\sim \text{Bi}(n_Y,p) $$Y\sim \text{Bi}(\lambda_Y) and $X\perp Y$ then $X+Y\sim \text{Bi}(n_X+n_Y,p)

## Poisson Point Process def. Memory-less-ness (continuous). $X$ is memoryless iff:

\forall s,t>0,\ \ \mathbb{P}(X>s+t|X\geq s)=\mathbb{P}(X>t)

> [!info] Read it like this: > LHS: “probability of waiting $t$ **more** minues, given that you’ve **already** waited $s$ minutes.” RHS: “probability of just waiting $t$ minutes in total.” Geometric and **_Exponential_** distributions satisfy this property. See the following. def. **Poisson Point Process (PPP)**. PPP can be described in the following _three equivalent definitions_ in region $R$ with intensity $\lambda$ _(per unit)_: 1. Given a region $R_i$, let random variable $N(R_i)$ be defined as the number of events in the region. If..: - $\forall i\in R, \ \ N(I_i)\sim \text{Poi}(\lambda\cdot {|R_i|})$, where $|R_i|$ is the “size” of the region and $\lambda$ the intensity - $N(R_i)$ are all independent of each other - ⇒ …then the event occurs in a PPP 2. A process where the waiting time $W_i$ between two sequential events is distributed as an exponential distribution $\forall i\in R,\ \ W_i\sim \text{Exp}(\lambda)$ 3. Total region size $X$ for $r$ events is distributed over gamma $X\sim\Gamma(r,1/\lambda)$