Poisson distribution

When to use the Poisson distribution

The Poisson distribution models the number of times a rare event occurs in a fixed interval of time, space or volume, such as accidents at a crossing per week, misprints per page or calls per minute. It applies when:

• events occur independently of one another,
• the average rate λ is constant over the interval, and
• two events cannot occur at exactly the same instant.

It is also the limiting case of the binomial distribution when n is large, p is small and np = λ stays moderate.

The formula

P(X = x) = e^(−λ) λˣ / x!, x = 0, 1, 2, … where λ > 0 is the mean number of occurrences.

Mean and variance

For the Poisson distribution, mean = variance = λ, and the standard deviation is √λ.

Examples

• λ = 2: P(X = 0) = e^(−2).
• λ = 3: P(X = 2) = e^(−3) × 9/2.
• A machine makes defective items with probability 0.01. In a batch of 200, λ = np = 2, so P(at most one defective) = e^(−2)(1 + 2) = 3e^(−2).
• If P(X = 1) = P(X = 2), then λe^(−λ) = λ²e^(−λ)/2, so λ = 2 and the standard deviation is √2.

Key points

• Parameter: λ (mean rate in the interval).
• Mean = variance = λ.
• P(X = 0) = e^(−λ).
• Binomial ≈ Poisson with λ = np when n is large and p is small.

Exam tips

• If the rate is given per hour and the question asks about 3 hours, multiply λ by 3.
• Equal mean and variance in data is a clue that a Poisson model fits.