1 Introduction
The Poisson distribution is a discrete probability distribution that models the number of events occurring in a fixed interval of time or space, when events occur independently at a constant average rate.
A random variable \(X\) follows a Poisson distribution with rate parameter \(\lambda > 0\) if
\[ X \sim \text{Poisson}(\lambda) \quad \text{or} \quad X \sim \mathcal{P}(\lambda). \]
2 Probability Mass Function
The probability mass function is
\[ \mathbb{P}(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}, \quad k \in \{0, 1, 2, \ldots\}. \]
Interpretation: The Poisson distribution arises as the limit of binomial distributions: when the number of trials \(n \to \infty\) and success probability \(p \to 0\) such that \(np = \lambda\) remains constant, \(\text{Binomial}(n,p) \xrightarrow{d} \text{Poisson}(\lambda)\).
3 Key Properties
The expectation and variance of a Poisson random variable \(X \sim \text{Poisson}(\lambda)\) are given by
\[ \mathbb{E}[X] = \lambda \quad \& \quad \operatorname{Var}(X) = \lambda. \]
Remarkable property: The mean and variance are equal, both equal to the rate parameter \(\lambda\).
Proof. We have that
\[ \mathbb{E}[X] = \sum_{k=0}^\infty k \frac{\lambda^k e^{-\lambda}}{k!} = \sum_{k=1}^\infty \frac{\lambda^k e^{-\lambda}}{(k-1)!} = \lambda e^{-\lambda} \sum_{j=0}^\infty \frac{\lambda^j}{j!} = \lambda. \]
Similarly, \(\mathbb{E}[X(X-1)] = \lambda^2\), so \(\operatorname{Var}(X) = \lambda^2 + \lambda - \lambda^2 = \lambda\).
4 Generating Functions
The Poisson distribution has several Generating Functions that are useful for deriving properties and relationships.
The probability generating function (PGF) of a Poisson random variable \(X \sim \text{Poisson}(\lambda)\) is given by
\[ G_X(s) = e^{\lambda(s-1)}. \]
Proof. We compute the PGF as
\[ \begin{align*} G_X(s) &= \mathbb{E}[s^X] = \sum_{k=0}^\infty s^k \frac{\lambda^k e^{-\lambda}}{k!} \\ & = e^{-\lambda} \sum_{k=0}^\infty \frac{(\lambda s)^k}{k!} = e^{-\lambda} e^{\lambda s} = e^{\lambda(s-1)}. \end{align*} as required. \]
The probabilities can be recovered from the PGF by taking derivatives:
\[ \mathbb{P}(X = k) = \frac{G_X^{(k)}(0)}{k!}. \]
The moment generating function (MGF) of a Poisson random variable \(X \sim \text{Poisson}(\lambda)\) is given by
\[ M_X(t) = e^{\lambda(e^t - 1)}. \]
Proof. The computation of the MGF is similar to the PGF
\[ \begin{align*} M_X(t) &= \mathbb{E}[e^{tX}] = \sum_{k=0}^\infty e^{tk} \frac{\lambda^k e^{-\lambda}}{k!} \\ & = e^{-\lambda} \sum_{k=0}^\infty \frac{(\lambda e^t)^k}{k!} = e^{-\lambda} e^{\lambda e^t} = e^{\lambda(e^t - 1)}. \end{align*} \]
as required.
The characteristic function of a Poisson random variable \(X \sim \text{Poisson}(\lambda)\) is given by
\[ \varphi_X(t) = e^{\lambda(e^{it} - 1)} \]
Proof. The characteristic function is computed using the same method as above.
5 Sums of Independent Poisson RVs
If \(X \sim \text{Poisson}(\lambda_1)\) and \(Y \sim \text{Poisson}(\lambda_2)\) are independent, then
\[ X + Y \sim \text{Poisson}(\lambda_1 + \lambda_2). \]
Proof. The MGF product gives \(e^{\lambda_1(e^t-1)} \cdot e^{\lambda_2(e^t-1)} = e^{(\lambda_1+\lambda_2)(e^t-1)}\).
6 Relationship to Other Distributions
- Binomial Distribution: Limit of \(\text{Binomial}(n, p)\) as \(n \to \infty\), \(p \to 0\) with \(np = \lambda\).
- Normal Distribution: For large \(\lambda\), \(\frac{X - \lambda}{\sqrt{\lambda}} \xrightarrow{d} \mathcal{N}(0,1)\) (Central Limit Theorem).
- Inter-arrival times: If events follow a poisson-process with rate \(\lambda\), the time between consecutive events follows an Exponential Distribution with parameter \(\text{Exponential}(\lambda)\).
7 Examples and Applications
- Number of emails received per hour with average rate 15 emails/hour: \(X \sim \text{Poisson}(15)\).
- Number of car accidents at an intersection in a day with average 3 accidents/day: \(X \sim \text{Poisson}(3)\).
- Number of mutations in a DNA sequence of fixed length.
- Modeling rare events in reliability and quality control.