Poisson Distribution (AI HL)
The Poisson distribution models the number of times a rare, random event happens in a fixed interval of time or space - car crashes at a junction in an hour, typos on a page, flaws in a metre of cable. Unlike the binomial distribution it doesn't need a fixed number of trials, just a known average rate. This page covers the formula, the conditions that make it an appropriate model, and the mistakes that lose marks. It's part of the broader Probability & Distributions topic.
12 questions on this sub-topic.
The formula and its conditions
Covered under IB syllabus reference AHL4.17. The probability formula and the mean/variance result are both in the formula booklet, but you're expected to justify when the model applies.
Poisson probability
\(X \sim \text{Po}(\lambda) \Rightarrow P(X=x) = \dfrac{e^{-\lambda}\lambda^{x}}{x!}\)
\(\lambda\) is the mean number of occurrences in the interval you're working over - rescale it first if the question's interval differs from the one \(\lambda\) was quoted for.
Mean and variance
\(E(X) = \text{Var}(X) = \lambda\)
The defining feature of a Poisson variable: its mean and variance are always equal. This is also the usual check for whether a Poisson model is appropriate for a real dataset.
Need the full syllabus wording and formula-booklet reference table? See Probability & Distributions. GDC keystrokes for these problems are in the same page's GDC section.
Worked examples
Emails arrive at a help desk at a mean rate of 2.5 per hour, following a Poisson distribution.
(a) Find the probability of exactly 6 emails in a 2-hour period.
(b) Find the probability of at least 1 email in a 30-minute period.
Worked solution
(a) In 2 hours \(\lambda = 5\): \(P(X=6) = \dfrac{e^{-5}5^6}{6!}\) M1
\(\approx 0.146.\) A1
(b) In 30 min \(\lambda = 1.25\): \(P(X \ge 1)\) M1
\(= 1 - e^{-1.25}\) A1
\(\approx 0.713.\) A1
Calls arrive at \(\lambda = 3\) per hour (Poisson).
Find \(P(X=2)\) in one hour, to 3 significant figures.
Worked solution
\(P(X=r) = \dfrac{e^{-\lambda}\lambda^r}{r!}.\) M1
\(\lambda=3, r=2\): \(P(X=2) = \dfrac{9e^{-3}}{2}\approx 0.224.\) A1
Common mistakes
- Forgetting to rescale \(\lambda\). If \(\lambda\) is given per hour but the question asks about a 3-hour or 30-minute period, \(\lambda\) must be scaled proportionally before it goes into the formula - using the original hourly value is one of the most common errors.
- Confusing "exactly" with "at least" or "at most". \(P(X=x)\) uses the Poisson pdf directly, but "at least 1" needs \(1 - P(X=0)\) and "at most" needs a cumulative sum - reaching for the wrong GDC function flips the whole answer.
- Assuming a Poisson model without checking the conditions. The distribution only applies when events occur independently and at a uniform average rate; a quick check that the sample mean and variance are close is good practice before committing to it.
Ready to practise properly?
12 Poisson-distribution questions, marked instantly like the real exam.
Quick answers
What is the formula for the Poisson distribution?
If \(X \sim \text{Po}(\lambda)\), then \(P(X=x) = \dfrac{e^{-\lambda}\lambda^x}{x!}\), where \(\lambda\) is the mean number of occurrences in the interval.
How do I know when a Poisson distribution is an appropriate model?
Events must occur independently of each other and at a uniform average rate over the interval in question. A sample mean close to the sample variance also supports a Poisson model.