Poisson questions test the value of λ, the terms in the expression, and the choice of approximation. The reports show a clear pattern: students know the formula, and the marks are lost in the interpretation of "more than" and in the choice of distribution. This guide collects what Cambridge examiners say students get wrong, based on recent reports.
Summarised in our own words from a reading of the examiner reports.
Mistakes that cost marks
- The wrong value of λ. One of the most common errors, with examples such as 1.9, 1.2 and 1.5 used in place of the correct mean, 2.5 used instead of 5.1 + 2.5 when two Poisson variables are combined, and 0.25 used instead of 5 × 0.25 when the time interval changes. In one question, the Poisson parameter was 1.5 when it should have been 4.5.
- Interpreting "more than". For "more than 3", writing 1 − P(0, 1, 2) when 1 − P(0, 1, 2, 3) was needed, or including P(4) in the expression. In another question, 1 − P(0, 1, 2) was calculated instead of 1 − P(0, 1). Some failed to subtract from 1 at all.
- Missing or extra terms. Leaving out one of the terms, such as P(Y = 2) or P(3). "More than 2 but not more than 5" was difficult, with P(3, 4) calculated instead of P(3, 4, 5).
- Combining two independent Poisson processes. Adding the two probability expressions instead of multiplying them.
- Wrong or no approximating distribution. Using a binomial or Normal distribution where a Poisson was required, or not using an approximating distribution at all. In some questions a Normal distribution was chosen when it was not valid.
- The continuity correction. Standardising without a continuity correction, with an incorrect one, or choosing the wrong probability area (an area greater than 0.5 where it should be less than 0.5). The correct approximating distribution, N(200, 200) in one question, was often used, and the continuity correction was the weak point.
- Justifying the approximation. Many candidates could not state the further condition required, and merely repeated conditions already given, with comments such as "the mean and variance need to be the same". Others wrote λ > 15 and did not say that λ = 25 > 15 in context. In a Poisson approximation to a binomial, saying n should be greater than 50 without saying that n was 200.
- Stating the distribution. Leaving out the parameter, giving the parameter without naming the Poisson distribution, or stating a different distribution.
- Equations. Sign errors from missing brackets, errors when using logarithms, and algebraic errors such as using 2μ² instead of (2μ)². Where the equation formed should be a quadratic, there were errors in rearranging it.
- Showing a given value. Not finding P(2) to 4 decimal places or better, and not showing that the given value and their calculated value were the same to 2 decimal places and different to 3.
- Accuracy. An answer given to two significant figures (0.044), or to 3 decimal places in place of 3 significant figures.
- Final integer. Reaching a value such as 6.411 and leaving it as the final answer, or drawing the wrong conclusion about the integer value of n.
- Following through. Some candidates did not use the distribution they had stated.
What examiners advise
- Write down the distribution with its parameter, such as Po(4.5), and check how λ scales with the time interval or number of items.
- Write out the probability in terms of P(X = r) for each value of X, and show the expression in full.
- Know and check the conditions for a valid approximating distribution, and use the numbers in the question.
- Draw a sketch to pick the correct probability area after a continuity correction.
- Keep working to at least 4 significant figures, and give the final answer to 3.
Habits that help
- Write "more than 3 means 4 or more" next to your expression, then write its complement.
- For several time intervals, change λ first and only then write the probability.
- When you approximate, say in a line which approximation you are using and which condition you have checked.
Questions students ask
How do I find the right value of λ?
Check how λ scales with the time interval or number of items, and change it first before writing the probability. The wrong λ is one of the most common errors, including 2.5 used instead of 5.1 + 2.5 when two Poisson variables are combined.
What does "more than 3" mean in a Poisson question?
It means 4 or more, so 1 − P(0, 1, 2, 3). Students wrote 1 − P(0, 1, 2) or included P(4). "More than 2 but not more than 5" is P(3, 4, 5), not P(3, 4).
What if I have two independent Poisson processes?
Multiply the probability expressions, not add them, when both must happen. Also combine means correctly when two Poisson variables are added.
How do I justify a Normal approximation to a Poisson?
State the condition using the numbers in the question, such as λ = 25 > 15. Many candidates merely repeated conditions already given, such as "the mean and variance need to be the same". Use a continuity correction and sketch to pick the right area.
How should I state the distribution?
With its name and parameter, such as Po(4.5). Reports record the parameter left out, or given without naming the Poisson distribution.
Practice this topic
Ready to try some? Work through past paper questions on this topic one at a time. Attempt each question fully first, and open the mark scheme only afterwards.