Statistics Poisson Distribution

What Is a Poisson Distribution?

The Poisson distribution counts how many times an event occurs in a fixed window — an hour, a page, a square meter — when events happen independently at a known average rate. It's the natural tool for questions like 'how many customers will arrive in the next hour?' or 'how many typos appear on a page?', where you know the average but not the exact count in advance.

The Poisson Formula

The probability of observing exactly k events is P(X = k) = (λ^k × e^(−λ)) / k!, where λ (lambda) is the average number of events per interval, and k is the specific count you're asking about. Unlike the binomial distribution, Poisson has no fixed 'number of trials' — only a rate.

import math

lam = 4  # average arrivals per hour
k = 6    # exactly 6 arrivals this hour

probability = (lam ** k) * math.exp(-lam) / math.factorial(k)
print(f"P(X = {k}) = {probability:.4f}")  # 0.1042

• Events occur independently of one another.
• The average rate, λ, stays constant across the interval.
• Two events essentially never occur at exactly the same instant.
• The count of events has no fixed upper limit — it can, in principle, be any non-negative integer.

When to Use Poisson vs Binomial

Poisson is often used as an approximation to the binomial distribution when n is large, p is small, and the product λ = n × p stays moderate — for example, modeling rare manufacturing defects across a huge production run. Where binomial needs both n and p, Poisson only needs the single rate λ, which makes it convenient when 'number of trials' isn't a meaningful concept, like typos per page.

import math

lam = 2  # average typos per page
k = 0    # a page with zero typos

p_zero = (lam ** k) * math.exp(-lam) / math.factorial(k)
print(f"P(no typos on a page) = {p_zero:.4f}")  # 0.1353

Poisson modeling shows up in call-center staffing (calls per minute), radioactive decay (particle emissions per second), website traffic (hits per second), and defect tracking (flaws per unit produced).