Poisson Distribution

Motivation

The Poisson distribution models the number of events occurring in a fixed interval (time, space, etc.) when events occur:

• Independently of each other
• At a constant average rate
• One at a time

Examples:

• Number of emails arriving in an hour
• Number of typos per page
• Number of earthquakes per year in a region
• Number of customers entering a shop per hour

Preview: The Constant $e$

The Poisson distribution uses the mathematical constant $e \approx 2.71828$.

Key identities: $e = \sum_{n=0}^{\infty} \frac{1}{n!}, \quad e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}$

$e^{-\lambda} = \lim_{n \to \infty} \left(1 - \frac{\lambda}{n}\right)^n$

Definition

$X \sim \text{Poisson}(\lambda)$ if:

$P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}, \quad k = 0, 1, 2, \ldots$

The parameter $\lambda > 0$ is both the mean and the variance.

Range

$X \in \{0, 1, 2, 3, \ldots\}$ (all non-negative integers)

Expectation

$E[X] = \sum_{k=0}^{\infty} k \cdot \frac{\lambda^k e^{-\lambda}}{k!} = e^{-\lambda} \sum_{k=1}^{\infty} \frac{\lambda^k}{(k-1)!} = e^{-\lambda} \cdot \lambda \sum_{k=1}^{\infty} \frac{\lambda^{k-1}}{(k-1)!} = e^{-\lambda} \cdot \lambda \cdot e^{\lambda} = \lambda$

Variance

By similar calculation: $\text{Var}(X) = \lambda$.

Not a coincidence: The mean and variance are equal for Poisson. This is a distinguishing feature.

Derivation from Binomial

Let $Y \sim \text{Bin}(n, p)$. If $n \to \infty$ and $p \to 0$ with $np = \lambda$ fixed:

$\text{Bin}(n, p) \to \text{Poisson}(\lambda)$

Intuition

Consider events occurring over time. Divide the interval into $n$ tiny subintervals:

• Each subinterval has probability $p = \lambda/n$ of containing an event
• Number of events in the whole interval is $\text{Bin}(n, p)$
• As $n \to \infty$, this approaches $\text{Poisson}(\lambda)$

Proof Sketch

$P(Y = k) = \binom{n}{k} p^k (1-p)^{n-k} = \frac{n(n-1)\cdots(n-k+1)}{k!} \left(\frac{\lambda}{n}\right)^k \left(1 - \frac{\lambda}{n}\right)^{n-k}$

As $n \to \infty$:

• $\frac{n(n-1)\cdots(n-k+1)}{n^k} \to 1$
• $(1 - \lambda/n)^{n-k} \to e^{-\lambda}$

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

Example: Call Centre

A call centre receives an average of 5 calls per minute. What is the probability of exactly 3 calls in a minute?

$X \sim \text{Poisson}(\lambda = 5)$

$P(X = 3) = \frac{5^3 e^{-5}}{3!} = \frac{125 \times 0.00674}{6} \approx 0.14$

What is the probability of at most 2 calls?

$P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2) = e^{-5}\left(1 + 5 + \frac{25}{2}\right) = 18.5 e^{-5} \approx 0.125$

Example: Rare Events

If events are rare (small $p$), but we have many trials (large $n$), use Poisson as an approximation to Binomial.

Earthquakes: Suppose there’s a 0.001 chance of a major earthquake each year. In 1000 years, expected number is $\lambda = 1$.

$P(\text{at least one in 1000 years}) = 1 - P(X = 0) = 1 - e^{-1} \approx 0.632$

Poisson Process

A Poisson process with rate $\lambda$ satisfies:

1. $N(0) = 0$ (no events at time 0)
2. Independent increments: counts in disjoint intervals are independent
3. $N(t) \sim \text{Poisson}(\lambda t)$: number of events in time interval of length $t$

The inter-arrival times are exponentially distributed with parameter $\lambda$.

Additivity

If $X \sim \text{Poisson}(\lambda_1)$ and $Y \sim \text{Poisson}(\lambda_2)$ are independent:

$X + Y \sim \text{Poisson}(\lambda_1 + \lambda_2)$

Intuition: Merging two independent streams of events at rates $\lambda_1$ and $\lambda_2$ gives rate $\lambda_1 + \lambda_2$.

Summary Table

Property	Value
PMF	$P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}$
Range	$\{0, 1, 2, \ldots\}$
Mean	$\lambda$
Variance	$\lambda$
Relation to binomial	Limit of $\text{Bin}(n, p)$ as $n \to \infty$, $np = \lambda$