Normal and Poisson distributions (IB Math AI HL)

Key ideas

• Poisson conditions: events occur singly, at random, independently and at a constant mean rate. If the mean and variance of the data are close, Po(m) is a good model.
• Scale the mean with the interval: 3 per 10 minutes is 9 per 30 minutes. Independent Poisson variables add: Po(a) + Po(b) = Po(a + b).
• "More than 10" is P(X ≥ 11) = 1 − P(X ≤ 10) for a discrete count.
• For a normal variable, P(X = a) = 0, so "more than" and "at least" give the same probability.
• Combining: find a probability from one distribution, then use it as p in a binomial for several independent trials.

Worked example

Calls arrive at a help desk at random at a mean rate of 3 every 10 minutes. The lengths of calls are normally distributed with mean 4 minutes and standard deviation 1.2 minutes. Find (a) the probability of no calls in a 10-minute period, (b) the probability of more than 10 calls in 30 minutes, (c) the probability that a call lasts more than 6 minutes, and (d) the probability that at least one of 5 calls lasts more than 6 minutes.

1. (a) X ~ Po(3): P(X = 0) = e⁻³ ≈ 0.0498.
2. (b) In 30 minutes the mean is 9, so Y ~ Po(9): P(Y > 10) = 1 − P(Y ≤ 10) ≈ 0.294.
3. (c) L ~ N(4, 1.2²): P(L > 6) ≈ 0.0478.
4. (d) Number of long calls among 5 ~ B(5, 0.0478): P(at least one) = 1 − (1 − 0.0478)⁵ ≈ 0.217.

Answer: (a) 0.0498 (b) 0.294 (c) 0.0478 (d) 0.217

Exam-style HL AI questions on this topic

• Random Variables and Poisson questions — 47 questions with mark schemes, plus free worked examples
• All HL AI Statistics and Probability questions by topic

Key formulas for normal and Poisson distributions

Mean & variance of Poisson

E(X) = λ, Var(X) = λ

Poisson distribution

P(X = r) = (e^(−λ)λʳ)/(r!)

In the notation of the IB formula booklet. All HL AI formulas · see them used in worked examples

Do it on your GDC

A GDC is allowed on every AI paper. Keystrokes for the TI-84 Plus CE, TI-Nspire CX II and Casio fx-CG50, with a checked worked example:

• Normal distribution on your calculator
• Binomial and Poisson distributions on your calculator
• Try it: normal distribution calculator · inverse normal calculator · poisson distribution calculator (free, shows the working)

Maths IA ideas that use normal and poisson distributions

• Are NBA players getting taller?
• Where should I aim on a dartboard?
• Is the euro–dollar exchange rate a random walk?
• Will the pension pot last? Monte Carlo savings with random returns
• IA statistics, step by step: is my data normal?
• IA statistics, step by step: binomial and Poisson models
• All HL AI IA ideas

Related in Statistics and Probability

• Hypothesis testing — t-test and chi-squared
• Markov chains and transition matrices
• Chi-squared test and confidence intervals

Revise Normal and Poisson distributions for the exam

• HL AI revision notes — every unit summarised, with the key results to learn
• HL AI past papers — specimen papers and timed practice papers
• Formulas for Statistics and Probability — on the one-page HL AI formula sheet (free PDF), with the ones not in the IB booklet marked
• HL AI formula booklet by topic — every formula in the IB booklet's notation, with a free PDF
• IB Maths prediction papers — full papers in the IB format, with markschemes

Stretch yourself: original probability problems at Senior level (ages 16 to 18), with hints and full solutions: Problem S101 · Problem S29 · Problem S102 · Problem S30. All 20 →