Poisson Distribution

The Poisson distribution, named after French mathematician Siméon Denis Poisson, is a fundamental probability distribution that models the number of events occurring within a specified interval of time or space, given that these events happen independently and at a constant average rate.

Mathematical Foundation

The probability mass function of the Poisson distribution is given by:

P(X = k) = (λᵏe⁻λ)/k!

Where:

• λ (lambda) is the expected number of occurrences
• k is the number of occurrences
• e is Euler's number
• k! represents the factorial of k

Key Properties

1. Expected Value (Mean):

   • E(X) = λ
   • The mean equals the variance: Var(X) = λ

2. Characteristics:

   • Discrete probability distribution
   • Always non-negative
   • No upper bound
   • Memoryless property

Applications

The Poisson distribution finds widespread applications in various fields:

Science and Research

• Radioactive decay measurements
• Quantum mechanics detection
• Genetic mutations

Business and Operations

• Customer arrivals at a service point
• Queueing theory modeling
• Website traffic analysis
• Quality control

Natural Phenomena

• Number of storms per year
• Random process occurrences in fixed intervals
• Biological cell counts

Relationship to Other Distributions

The Poisson distribution has important connections to other probability distributions:

1. Exponential distribution - Models the time between Poisson events
2. Binomial distribution - Poisson is the limit of binomial as n→∞
3. Normal distribution - Approximates Poisson for large λ

Assumptions and Limitations

For accurate modeling, several key assumptions must be met:

1. Independence of events
2. Constant rate of occurrence
3. Non-overlapping events
4. Events cannot occur exactly simultaneously

Historical Context

Developed by Siméon Denis Poisson in 1838, the distribution was initially used to model legal cases of false convictions. Its applications have since expanded dramatically, making it one of the most important probability distributions in statistical analysis.

Computational Considerations

Modern statistical software packages include built-in functions for:

• Probability calculations
• Random number generation
• Parameter estimation
• Maximum likelihood estimation for Poisson parameters

The distribution's simple mathematical form makes it computationally efficient and widely implemented in statistical computing environments.