AHL 4.17 — Poisson distribution

The Poisson distribution models counts of rare events occurring independently at a constant average rate over a
fixed interval (time, area, volume, etc.). Its mean and variance are both equal to the rate parameter λ.
In AIHL problems you will: identify when Poisson is appropriate, compute probabilities using a GDC, and combine
independent Poisson processes by summing their λ parameters.

Term / concept Definition / short explanation
Poisson(λ) A discrete distribution for counts. P(X = k) = e-λ λk / k!, k = 0,1,2,…
Mean = λ, Variance = λ.
When appropriate Events occur independently, at a constant average rate, and counts per interval are small/rare relative to potential occurrences.
Sum of independent Poissons If X~Poisson(λ1) and Y~Poisson(λ2) independent then X+Y ~ Poisson(λ1+λ2).
Approximation use Poisson approximates Binomial(n,p) when n large, p small and np ≈ λ.

📌 1. Intuition & criteria for use

  • Independence: each event occurrence does not affect others.
  • Uniform rate: average rate λ is constant across the interval of interest.
  • Discreteness: we count occurrences (0,1,2,…).
  • Examples: calls to a switchboard per hour, number of typos per page, emergency admissions per night.

🔗 Connection

Telecommunications / call-centres: Poisson models are used to estimate arrival rates (calls per minute) and to help dimension staff.

📌 2. Probability formula & properties

For X ~ Poisson(λ):

P(X = k) = e-λ λk / k!    (k = 0,1,2,…)

  • Mean E(X) = λ
  • Var(X) = λ
  • Sum: If X~Poisson(λ1) and Y~Poisson(λ2) independent, X+Y~Poisson(λ1+λ2)

Example 1 (GDC): Probability of k events in an interval

Problem: A call centre receives on average 6 calls per hour (λ = 6). What is the probability of exactly 4 calls in one hour?

GDC steps (TI-84 / TI-83 style):

  1. Press 2nd then VARS to open DISTR.
  2. Choose PoissonPdf( or on some models poisspdf().
  3. Enter (6,4) where 6 is λ and 4 is k, i.e. PoissonPdf(6,4).
  4. Press ENTER. The calculator returns the probability ≈ 0.1339.

Answer: P(X = 4) ≈ 0.1339 (about 13.4%).

Example 2 (GDC): Cumulative probabilities & sum of independent Poissons

Problem A: With λ = 6, what is P(X ≤ 3)? (use GDC cumulative function)

GDC steps (TI-84 style):

  1. Open 2nd → VARS (DISTR).
  2. Choose PoissonCdf( (sometimes labelled poisscdf().
  3. Enter (6,0,3) (λ, lower k, upper k). Press ENTER.
  4. Calculator returns P(X ≤ 3) ≈ 0.1512.

Problem B: Two independent call streams have average rates 2 and 3 calls per hour. What is the probability that their combined total in an hour is exactly 4?

Solution idea: sum rates: λ_total = 2 + 3 = 5. Then compute P(Z = 4) with Z ~ Poisson(5).

GDC steps: use PoissonPdf(5,4) → result ≈ 0.1755.

🧠 Examiner Tip

  • Always state which λ you are using and why (context → interval chosen).
  • When combining independent Poisson processes, write λ_total = λ1 + λ2 before using your GDC.
  • Use cumulative functions (PoissonCdf) on the GDC for “at most” and “at least” type questions; remember P(X ≥ k) = 1 − P(X ≤ k−1).

📐 IA Spotlight

Collect counts (e.g., number of visitors per hour) across many intervals. Check the mean ≈ variance property and use goodness-of-fit (via simulation or technology) to justify a Poisson model. Discuss independence and changing rates as limitations.

📌 3. Worked AIHL-style questions (GDC shown)

Q1 (single λ): A website gets on average 0.2 error reports per hour. What is the probability of no errors in a 10-hour window?

Solution idea: λ per hour = 0.2, for 10 hours λ_total = 0.2 × 10 = 2. Use PoissonPdf(2,0).

GDC: PoissonPdf(2,0) → ≈ e-2 20/0! ≈ 0.1353.

Q2 (sum of Poissons): Two independent sensors record failures at mean rates 0.5/hour and 1.2/hour. What is P(total failures in 4 hours ≥ 3)?

Solution idea: λ1_4h = 0.5×4 = 2; λ2_4h = 1.2×4 = 4.8; λ_total = 6.8. Compute P(Z ≥ 3) = 1 − P(Z ≤ 2) with Z~Poisson(6.8).

GDC steps: PoissonCdf(6.8,0,2) → returns P(Z ≤ 2) ≈ value (say 0.015). Then answer = 1 − 0.015 ≈ 0.985.

Note: Always check independence and constant rate assumptions; if these fail, consider time-varying Poisson processes or alternative models.

🔗 Connections (integrated)

  • Traffic management: count of vehicles arriving at a junction per minute → helps design signal timing and predict congestion peaks.
  • Emergency admissions: Poisson can approximate arrivals to an ER in short time windows (discuss limits if outbreaks occur).
  • Typographical errors: modeling typos per page is a classic Poisson application in quality control.

📝 Paper strategy

  • Identify λ clearly from context (rate × interval length). Show the calculation for λ if you rescale the interval.
  • Use your GDC PoissonPdf or PoissonCdf functions; show the function you used and the parameters (e.g., PoissonPdf(5,4)).
  • When summing independent Poissons, state λ_total = λ1 + λ2 then use that λ in your GDC commands.