4 · IB Maths
Poisson distribution
Counts of rare events in a fixed interval, such as calls, typing slips or radioactive clicks. One parameter λ sets both the mean and the variance. Compare the bars with a normal curve.
Predict first
Start from the default settings (λ = 4.0). If you raise the mean rate λ from 4.0 to 16.0, what happens to the standard deviation σ?
For a Poisson distribution Var(X) = λ, so σ = √λ. Quadrupling λ doubles σ (from 2 to 4).
Check it in the model: Move the Mean rate λ slider from 4.0 to 16.0 and compare the Standard deviation σ readout.
E(X) = Var(X) = λ , σ = √λ
P(X ≤ k) = P(X = 0) + … + P(X = k)
normal: P(X ≤ k) ≈ Φ((k + ½ − λ) ÷ σ)
- P(X = k)
- P(X ≤ k)
- P(X ≥ k)
- Mean = variance = λ
- Standard deviation σ
- Normal approx. of P(X ≤ k)
Try this
- Set λ = 2.0 and k = 2. Read P(X = 2), then check it by hand from e⁻² × 2² ÷ 2! and also check P(X ≤ 2).
- Switch on the normal overlay and raise λ from 1 to 20. At roughly what λ does the curve follow the bar tops well, and what does the “Normal approx.” readout say about P(X ≤ k)?
- Tick “Shade P(X ≤ k)” with λ = 8 and k = 4. Read P(X ≤ 4) and P(X ≥ 4); explain why they add to more than 1.
Exam tip: a quick check for a Poisson model is that the mean and the variance of the data are about equal; then P(X = k) = e^(−λ) λᵏ ÷ k! for the interval you are given, rescaling λ if the interval changes.
More on 4: Venn diagrams and conditional probability · Discrete random variables · Continuous random variables · Bayes' theorem: medical test · Standard deviation and outliers · Binomial distribution · Markov chains · The normal distribution · Scatter and regression · Tree diagrams and Bayes’ theorem · Box plots and histograms