CAIE A-Level Mathematics 6.1 The Poisson Distribution Question Bank

Cambridge International AS & A Level Mathematics 9709 syllabus for exams in 2028, 2029 and 20302028–2030

Practise calculating Poisson probabilities, rescaling the mean across time or space and choosing Poisson or normal approximations with conditions and continuity corrections.

Exam points

rescale the Poisson mean in direct proportion to the required time, area or exposure
calculate exact, cumulative or complementary probabilities from Po(λ) with correct limits
select a Poisson-to-binomial or normal-to-Poisson approximation and justify its conditions

Covered syllabus points

6.1.1—Poisson probabilities 6.1.2—The fact that if X Po+6.1.3—Poisson distribution 6.1.4—Binomial distribution 6.1.5—Normal distribution

[Maximum number: 6]

The random variables X and Y have independent distributions $X \sim \operatorname{Po}(3)$ and $Y \sim \operatorname{Po}(2)$ respectively.

(a)

Find P(2<X<5).

[ 2 ]

(b)

The total of 100 random values of X and 150 random values of Y is denoted by T.

Use a suitable approximating distribution to find P(T<560).

[ 4 ]

[Maximum number: 8]

The number, X, of used computers donated to a charity has a constant average rate of 2.4 computers per week.

(a)

State a necessary condition for X to have a Poisson distribution.

Now assume that X has a Poisson distribution.

[ 1 ]

(b)

Calculate the probability that the number of computers donated during a 4-week period is more than 6 and less than 9 .

[ 3 ]

(c)

Use a suitable approximating distribution to calculate the probability that more than 50 computers are donated during a 20-week period.

[ 4 ]

(a)

The random variable W has a Poisson distribution.
State the relationship between E(W) and $\operatorname{Var}(W)$ .

[ 1 ]

(b)

The random variable X has the distribution B(n, p). Jyothi wishes to use a Poisson distribution as an approximate distribution for X.

Use the formulae for E(X) and $\operatorname{Var}(X)$ to explain why it is necessary for p to be close to 0 for this to be a reasonable approximation.

[ 1 ]

(c)

Given that Y has the distribution B(20000,0.00007), use a Poisson distribution to calculate an estimate of P(Y>2).

[ 3 ]

[Maximum number: 6]

The data produced by a certain data entry firm always include a small number of incorrect characters that occur at random. The proportion of incorrect characters is denoted by p, and experience has shown that p=0.0001. A particular data set from the firm contains 14500 characters, of which X characters are incorrect.

(a)

Use a suitable approximating distribution to find P(X<4).

The firm's management wishes to decrease the value of p by giving their employees some training. Their aim is that, for a data set containing 14500 characters, the value of P(X=0) for the new value of p should be double the value of P(X=0) when p=0.0001.

[ 3 ]

(b)

Use a suitable approximating distribution to find the new value of p.

[ 3 ]

[Maximum number: 10]

A sports fan produces a magazine each month.

(a)

On average 1 in 540 characters in the magazine is incorrect.

[ 4 ]

(i)

Use an appropriate approximating distribution to find the probability that, in a magazine containing 2430 characters, there are at least 4 incorrect characters.

[ 3 ]

(ii)

Justify your approximating distribution.

[ 1 ]

(b)

On average the number of copies, X, of the magazine sold per month is 123.4.

[ 6 ]

(i)

State one condition for X to have a Poisson distribution.
You are now given that X has a Poisson distribution.

[ 1 ]

(ii)

Use an appropriate approximating distribution to find the probability that in a randomly chosen month, more than 130 copies of the magazine are sold.

[ 5 ]

Exam points

Covered syllabus points

Question 1

Question 1(a)

Question 1(c)

Question 1

Question 1(a)

Question 1(b)

Question 1(c)

Question 2

Question 2(a)

Question 2(b)

Question 2(c)

Question 3

Question 3(a)

Question 3(b)

Question 4

Question 4(a)

Question 4(a)(i)

Question 4(a)(ii)

Question 4(b)

Question 4(b)(i)

Question 4(b)(ii)