Poisson distribution
The Poisson distribution describes rare events in a fixed time or area, such as requests per hour.
Formulas
- P(X = k) = e^(−λ) · λᵏ / k!
- E(X) = Var(X) = λ
- P(X ≥ 1) = 1 − e^(−λ)
Your turn
Three problems on this topic. Comma or point, fractions like 1/6 work too.
Task
A server fails on average 1.5 times a month (Poisson). What is P(at least one failure in a month)?
P(X ≥ 1) (4 decimal places)
Comma or point; fractions like 1/6 work too.
Worked example
On average 2.0 support requests arrive per hour (Poisson). What is P(exactly 1 requests in an hour)?
Worked solution
- 1.P(X = k) = e^(−λ) · λ^k / k!
- 2.= e^(−2.0) · 2.0^1 / 1! = 0.2707
Typical mistake: Forgetting k! in the denominator.
A server fails on average 2.0 times a month (Poisson). What is P(at least one failure in a month)?
Worked solution
- 1.Complement: P(X ≥ 1) = 1 − P(X = 0)
- 2.= 1 − e^(−2.0) = 0.8647
Typical mistake: Trying to add up every value from 1 instead of using the complement.
Frequently asked questions
When do I use Poisson instead of binomial?
For many trials with a very small probability, or when only the average rate λ is known.