# Poisson distribution TODO: summarize https://www.math.uchicago.edu/~may/VIGRE/VIGRE2010/REUPapers/Mcquighan.pdf --- Describes the distribution of events that occurs in a bounded period time (or space) e.g., you count the number of soldiers being kick to death by a horse along 200 years of cavalries unit. another example, you count the number of bombs fell on London per area. In both cases you would like to know if the events are random or not. A random model for those events is the Binomial distribution. The time or space is divided into $n$ Bernoulli's trials s.t. the event could occurred in probability $p$. This means that we split the bounded time interval to $n$ subinterval where we toss a coin with probability $p$. note that in this model the binomial expectation $\lambda$ is fixed while $p$ depends on the choice of $n$ i.e., $\lambda = pn$ . As you increase $n$ you get smaller $p$ . $Binom(p,n,k)=\binom{n}{k}p^n(1-p)^{n-k}=\frac{n(n-1)\cdots(n-k+1)}{k!}(\frac{\lambda}{n})^k(1-\frac{\lambda}{n})^{n-k}$ $ \begin{align} \lim_{n\rightarrow \infty}Binom(p,n,k) & =\lim_{n\rightarrow \infty}\frac{n^k+O(n^{k-1})}{k!}\frac{\lambda^k}{n^k}\bigg(\big(1+\frac{1}{(\frac{n}{-\lambda})}\big)^{\frac{n}{-\lambda}}\bigg)^{-\lambda}(1-\frac{\lambda}{n})^{-k}\\ &= \lim_{n\rightarrow \infty}\frac{n^k+O(n^{k-1})}{n^k}\frac{\lambda^k}{k!}\bigg(\big(1+\frac{1}{(\frac{n}{-\lambda})}\big)^{\frac{n}{-\lambda}}\bigg)^{-\lambda}(1-\frac{\lambda}{n})^{-k} \end{align} $ The left most and right most terms are 1 and $\lim_{n\rightarrow \infty}\bigg(\big(1+\frac{1}{(\frac{n}{-\lambda})}\big)^{\frac{n}{-\lambda}}\bigg) = e$ and therefore, $\lim_{n\rightarrow \infty}Binom(p,n,k)=Poisson(\lambda,k)=\frac{\lambda^k e^{-\lambda}}{k!}$ ## Examples - [[The Prussian cavalries (Poisson distribution)]] - [[Bombing London (Poisson distribution)]] Reference: [[@What is Random? Chance and Order in Mathematics and Life]] ## Created 2024-06-28 09:30