Cambridge International AS & A Level Mathematics 9709 (2026-2027) · Paper 6: Probability & Statistics 2 · official syllabus
POISSON DISTRIBUTION LAB
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Four short missions on interval means, count probabilities, rare-event binomial approximations and normal continuity corrections. Check the model before calculating.
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Mission 1 of 4
Match the interval to the rate
Calls follow a Poisson model at a constant average rate of 4 per 10 minutes. Set the mean count for a 30-minute interval. Assume the model remains appropriate over that interval.
For a Poisson count with parameter lambda, the variance is...
These blocks show equal exposures and expected counts, not guaranteed observations. Add their means for the 30-minute interval.
Your mean is 4. Scale 4 by the interval multiplier 3.
Study notes
Played the lab above? These notes are the reference version of what you just felt - use them to check exact formulas and the exam wording.
Use a Poisson model for a count over an interval 6.1
A Poisson variable X counts events in a stated interval of time, length, area or another exposure. Its parameter lambda is the mean count for that interval, not a probability. Suitable modelling assumptions include independent occurrence in disjoint intervals and a constant average rate, with negligible chance of more than one event in a sufficiently short interval.
Check the situation: clustering, changing rates or events that influence subsequent events may make a simple Poisson model unsuitable. An event being described as random is not by itself enough to justify the model.
Worked example: a constant average rate of 4 calls per 10 minutes gives mean 12 calls per 30 minutes and mean 2 calls per 5 minutes. Scale the interval and parameter together. Do not multiply a previously calculated probability by the time multiplier.
Calculate Poisson probabilities 6.1
For X distributed as Po(lambda), P(X=r)=exp(-lambda) lambda^r/r!, for r=0,1,2,... . There is no fixed upper count. Use 0!=1, so P(X=0)=exp(-lambda). Negative counts and non-integer counts have probability zero.
For Po(3), P(X=2)=exp(-3) times 9/2, approximately 0.2240 to 4 d.p. The exact form contains the exponential factor; 4.5 alone is only its coefficient, not a probability.
Cumulative events require a sum of relevant terms. For Po(3), P(X<=2)=exp(-3)(1+3+9/2)=(17/2)exp(-3), approximately 0.4232 to 4 d.p. P(X>=3)=1-P(X<=2), not 1-P(X<=3). Check endpoint inclusion carefully.
Mean and variance share the same parameter 6.1
For Po(lambda), E(X)=lambda and Var(X)=lambda. Standard deviation is sqrt(lambda), not lambda. The equal mean and variance are properties of this model, not a rule for every count distribution.
A Po(12) variable has mean 12, variance 12 and standard deviation sqrt(12), approximately 3.4641 to 4 d.p. Individual counts remain integers even though the standard deviation or an expectation in another example need not be an integer.
For calculations, it can be useful to generate successive probabilities from P(X=r+1)=lambda P(X=r)/(r+1), starting at P(X=0). This follows directly from the factorial formula. Do not treat successive count events as independent events and multiply their probabilities.
Approximate a rare binomial count 6.1
A binomial B(n,p) can be approximated by Po(np) when n is large and p small. The syllabus gives n>50 and np<5 as approximate conditions. State the conditions and match the mean np. These are guides to suitability, not claims of exact equality.
For B(200,0.01), n=200>50 and np=2<5, so Po(2) is suitable under the stated guide. The exact zero-count probability is 0.99^200, approximately 0.1340 to 4 d.p.; the Poisson approximation is exp(-2), approximately 0.1353 to 4 d.p.
Use the Poisson count probabilities directly for this discrete-to-discrete approximation; no half-unit continuity correction is needed. Keep the original experiment's binomial assumptions in view, including independent trials and constant p.
Approximate a large Poisson count by a normal 6.1
For large lambda, approximate Po(lambda) by N(lambda,lambda), whose standard deviation is sqrt(lambda). The syllabus gives lambda>15 as an approximate guide. A continuity correction is required because an integer count is being approximated by a continuous variable.
For X distributed as Po(25), approximate P(X<=30) by P(Y<30.5), with Y distributed as N(25,25). Then z=(30.5-25)/5=1.1 and the probability is approximately 0.8643 to 4 d.p. Show the corrected event before standardisation.
Translate X>=31 to Y>30.5; X<30 to Y<29.5; and X=30 to 29.5<Y<30.5. The equality case needs an interval, since a continuous normal variable has zero probability at one exact point.
Choose and check the calculation route 6.1
Name the original distribution and parameters before selecting an approximation. A rare-event binomial may suit a Poisson approximation; a large-parameter Poisson may suit a normal approximation. Do not apply a remembered approximation merely because a count is large in the observed data.
Check probabilities lie between 0 and 1 and that the event's inequality has been translated correctly. An at-least-one event is often easiest as 1-P(X=0). For Po(3), this is 1-exp(-3), approximately 0.9502 to 4 d.p.
Keep exact expressions during working and label approximation or rounding in the final result. Separate a rate, its interval-specific mean, a realised count and a probability: they answer different questions and have different units or meanings.
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