Poisson distribution calculator
Type the mean rate λ and choose the probability you need. The answer is the exact Poisson probability, with the formula and the working.
- Mean and variance λ
- 4.5
Show the working
X ~ Po(4.5): events happen independently, at random, at a constant average rate of 4.5 per interval.
P(X = r) = e−4.5 × 4.5r ÷ r!
P(X = 3) = e−4.5 × 4.53 ÷ 3! = 0.169
For a Poisson distribution the mean and the variance are both λ = 4.5.
Answers are rounded to 3 significant figures, which is what IB papers ask for unless a question says otherwise.
How to do it by hand
- Use the Poisson distribution for events that happen independently and at random at a constant average rate λ per interval.
- Scale λ to the interval in the question first: 3 calls an hour is 1.5 calls in half an hour.
- P(X = r) = e^(−λ) λʳ ÷ r!. For “at most r”, add the probabilities from 0 to r; for “at least r”, use 1 − P(X ≤ r − 1).
Worked example
A help desk receives calls at random at an average rate of 3 per hour. Find the probability that it receives more than 4 calls in a 2-hour period.
Solution
X ~ Po(6): events happen independently, at random, at a constant average rate of 6 per interval.
P(X = r) = e−6 × 6r ÷ r!
There is no largest value, so use the complement: one minus the probabilities from 0 to 4.
P(X > 4) = 1 − P(X ≤ 4) = 1 − 0.285 = 0.715
For a Poisson distribution the mean and the variance are both λ = 6.
Answer: P(X > 4) = 0.715
Every number in this example is worked out by the same code as the tool above, and the code is tested against independent results from Python's SciPy library.
Where you need it
| AA SL | Not in the syllabus |
|---|---|
| AA HL | Not in the syllabus |
| AI SL | Not in the syllabus |
| AI HL | Poisson distribution, and sums of independent Poisson variables. |
On your calculator
In the exam you use your own calculator, so practise it too. Our guides give the exact keystrokes for the Casio fx-CG50, TI-Nspire CX II and TI-84 Plus CE, with a worked example to check against:
Common mistakes
- Not scaling λ to the time in the question: 3 per hour means λ = 6 for 2 hours.
- “More than 4” is P(X ≥ 5) = 1 − P(X ≤ 4), not 1 − P(X ≤ 5).
- Forgetting that the mean and the variance are both λ: if a question gives a mean and a variance that differ a lot, the Poisson model may not fit.
Notes and practice
Questions students ask
When is a Poisson distribution a good model?
When events occur singly, independently and at random, at a constant average rate. Counts of emails per hour or flaws per metre of cloth are typical examples.
Why are the mean and variance of a Poisson distribution equal?
It is a property of the distribution: E(X) = Var(X) = λ. Comparing the mean and the variance of some data is a quick check of whether a Poisson model is reasonable.
Can I add Poisson distributions?
Yes, if they are independent: X + Y ~ Po(λ + μ). That is why you can scale λ to a longer or shorter interval.