Poisson distribution

Motivation

Often used as a model for data we’re counting. This implies non-negative values and integers in a sequence, like $0,1,2,3,\dots$

A r.v. $X\sim \text{Pois}$ models the total number of “successes” (in this case, something that we count) as a sequence of Bernoulli trials, where we have a large amount of trials and the probability of “success” (being counted) is small. Some examples:

• Number of text messages during this class period
• Number of car accidents within the next hour
• Number of raisins in a loaf of raisin bread: here, a “trial” is a very small surface area of the bread, where we can either have 0 or 1 raisin (count, or no count).

Notice the similarities with the Binomial distribution. Often, when the number of trials is exceedingly large, calculating the PMF for $X\sim \text{Bin}$ takes more compute time (a lot of multiplications), and accuracy issues (multiplying very large numbers). So the Poisson is often used as a good approximation in this case.

PMF

Start with the PMF of the Binomial distribution, but take the number of trials to infinity, and $p$ to zero. Keep $np$ constant though (otherwise this would make no sense), so let $\lambda =np$. This is called the rate.

Calculate ${\lim }_{n\to \infty }{\lim }_{p\to 0}P(X=0)=(1-p{)}^n=(1-\frac{\lambda }{n}{)}^n$. Taking the limit, $P(X=0)\to e^{-\lambda }$.

For $k=1,2,3,\dots$, we calculate the ratio $\frac{P(X=k)}{P(X=k-1)}$, beginning with $k=1$, because we have the denominator $P(X=k-1)=P(X=0)$.

Taking the limits, we have $\frac{P(X=k)}{P(X=k-1)}\to \frac{\lambda }{k}$. From this, we can see $P(X=1)=e^{-\lambda }\lambda$, and generalizing this we have

$$P(X=k)=\frac{e^{-\lambda }{\lambda }^k}{k!}$$

There is no upper bound on the support $k$ for a r.v. $X$ that has the Poisson distribution. It takes one parameter, $\lambda$, so we can say $X\sim \text{Pois}(\lambda )$.

Properties of the distribution

If $X\sim \text{Bin}(n,p)$, then as we take the limits of $n$ and $p$ to infinity and 0 respectively, then the PMF of $X$ converges to the PMF of the Poisson with $\lambda =np$.

The sum of independent Poissons is also a Poisson: given $X\sim \text{Pois}({\lambda }_1)$ and $Y\sim \text{Pois}({\lambda }_2)$, if $X$ and $Y$ are independent, then $X+Y\sim \text{Pois}({\lambda }_1+{\lambda }_2)$.

If $X\sim \text{Pois}(\lambda )$, then both Expectation $\mathbb{E}(X)=\lambda$ and Variance $\text{Var}(X)=\lambda$.

Poisson paradigm (informal)

This distribution is a good approximation when we have $n$ Bernoulli trials that each have different success probabilities, and aren’t perfectly independent.

Let $A_1,A_2,\dots ,A_n$ be events where $p_i=P(A_i)$, where $n\gg p_i$, and $A_i$ is independent or weakly dependent. Let $I_i$ be an Indicator variable for $A_i$, and $X={\sum }_{i=1}^n{I}_i$. Then $X$ is approximately $\text{Pois}(\lambda )$, where $\lambda ={\sum }_{i=1}^n{p}_i$.