Understanding the Poisson Distribution

What Is "Poisson" in English?

The word "Poisson" is French and literally translates to "fish" in English. However, in the context of mathematics and statistics, "Poisson distribution" refers to a discrete probability distribution named after the French mathematician Siméon Denis Poisson.

Definition of Poisson Distribution

The Poisson distribution is a probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space, assuming these events happen with a known constant mean rate and independently of the time since the last event.

Probability Mass Function (PMF)

The formula for the Poisson probability mass function is:

Where:

• k = number of occurrences (k = 0, 1, 2, ...)
• λ (lambda) = average number of occurrences in the interval
• e ≈ 2.71828 (Euler's number)
• k! = factorial of k

Common Applications

• Modeling the number of phone calls received by a call center per hour
• Counting the number of cars passing through a toll booth in a minute
• Predicting the number of mutations in a DNA sequence
• Estimating website visits per minute
• Analyzing rare events in quality control

Key Properties

• Mean = λ
• Variance = λ
• Events are independent
• The probability of more than one event in an infinitesimally small interval is negligible