Poisson Distribution (Cambridge International A-Level Mathematics): Exam Questions

A random variable X \sim \text{Po}(5).

Write down the integer values of $a$ and $b$ which make the following statements true.

(i) $\text{P} ( X \leq 7 ) = 1 - \text{P} ( X \geq a )$

(ii) $\text{P} ( X \geq 3 ) = 1 - \text{P} ( X \leq b )$

Find, giving your answers correct to 3 significant figures,

(i) $\text{P} ( X = 4 )$

(ii) $\text{P} ( X \leq 2 )$

(iii) $\text{P} ( X \geq 2 )$

A random variable X \sim \text{Po}(3).

Find the exact value, in terms of $\text{e}$, of

(i) $\text{P} ( X = 5 )$

(ii) $\text{P} ( X \leq 1 )$

(iii) $\text{P} ( X \geq 2 )$

The random variable X \sim \text{Po}(\lambda).

Find the value of $\text{P} ( X = 0 )$ in the case when

(i) $λ = 7$

(ii) $λ = 2 . 5$

(iii) $λ = 0 . 01$

Find the value of $\text{P} ( X = 1 )$ in the case when

(i) $λ = 7$

(ii) $λ = 2 . 5$

(iii) $λ = 0 . 01$

Given that $λ = k$, find a simplified expression in terms of $k$ for

(i) $\text{P} ( X = 0 )$

(ii) $\text{P} ( X = 1 )$

For each of the following random variables, find the mean and the standard deviation.

(i) X \sim \text{Po}(4)

(ii) X \sim \text{Po}(1)

(iii) X \sim \text{Po}(5.4)

The random variable X \sim \text{Po}(6.3).

(i) By taking out a common factor of $\text{e}^{-λ}$, factorise the expression

$\text{e}^{-λ} \frac{λ^{4}}{4!} + \text{e}^{-λ} \frac{λ^{5}}{5!} + \text{e}^{-λ} \frac{λ^{6}}{6!} + \text{e}^{-λ} \frac{λ^{7}}{7!} + \text{e}^{-λ} \frac{λ^{8}}{8!}$

(ii) Hence calculate $\text{P} ( 4 \leq X \leq 8 )$.

Calculate

(i) $\text{P} ( X \leq 3 )$

(ii) $\text{P} ( 5 \leq X \leq 7 )$

A random variable X \sim \text{Po}(6).

Find the exact value, in terms of $\text{e}$, of

(i) $\text{P} ( X = 3 )$

(ii) $\text{P} ( X = 4 )$

(iii) $\text{P} ( X = 5 )$

Use your answers to part (a) to find the exact value of

(i) $\frac{6}{4} \text{P} ( X = 3 )$

(ii) $\frac{6}{5} \text{P} ( X = 4 )$

Compare your answers to part (a) and part (b), then use the pattern to write down an expression, in terms of $k$, for

$\frac{6}{k+1} \text{P} ( X = k )$

where $k$ is an integer and $k \geq 0$.

Naomi, a teacher, receives emails at a constant average rate during the school day. Naomi models the number of emails she receives within a fixed period of time using a Poisson distribution, $\text{Po} ( λ )$.

Write down two assumptions that Naomi has made about the emails in order to use a Poisson model.

Naomi knows that on average she receives 10 emails in one hour.

Calculate the value of $λ$ in the case when Naomi is modelling the number of emails she receives in

(i) two hours

(ii) 15 minutes

(iii) one hour and 10 minutes.

The random variables R \sim \text{Po}(2), S \sim \text{Po}(3) and T \sim \text{Po}(4) are independent.

Write down the distribution of

(i) $R + S$

(ii) $S + T$

(i) Write down the distribution of $R + T$.

(ii) Find $\text{P} ( R + T = 5 )$.

The number of times that the internet connection at Amira's house disconnects during a day is denoted by $X$. On average, the internet disconnects 5 times each day.

State a suitable distribution for $X$, giving the value of any parameter, and state one assumption that is required for this model to be valid in this context.

Find the probability that, during a randomly chosen day, the internet at Amira's house disconnects

(i) exactly 4 times

(ii) at most 3 times

(iii) at least 2 times.

The numbers of pieces of ham and pieces of salami on a pizza from Pizza Palace have the independent distributions $\text{Po} ( 6 . 2 )$ and $\text{Po} ( 4 . 9 )$ respectively. Roberto orders a pizza from Pizza Palace.

State the distribution of the total number of pieces of meat on Roberto's pizza, giving the value of any parameter.

Find the probability that Roberto's pizza contains a total of exactly 10 pieces of meat.

Find the probability that Roberto's pizza contains more than 9 and fewer than 13 pieces of meat.

The number of times that Lucy goes to the cinema in a week has a Poisson distribution with mean 4.

Find the probability that Lucy goes to the cinema exactly 5 times in a randomly chosen week.

Find the probability that Lucy goes to the cinema not more than 4 times in a randomly chosen 2-week period.

Find the probability that Lucy goes to the cinema at least once on a randomly chosen day.

Customers enter Comic Stans, a comic book store, at random and independently at a constant average rate of 8 every 15 minutes.

Find the probability that exactly 3 customers enter the store during a randomly chosen 1-minute period.

Find the probability that at least 1 customer enters the store during a randomly chosen 15-second period.

Find the probability that at most 3 customers enter the store during a randomly chosen 10-minute period.

Find the variance of the number of customers entering the store during a 1-hour period.

Amber suggests that the number of times that she hiccups can be modelled by a Poisson distribution.

State two conditions that must be satisfied for a Poisson distribution to be a suitable model in this context.

The mean number of hiccups in a 30-second period is 2.9. Assume now that a Poisson distribution is a suitable model.

Find the probability that Amber hiccups

(i) exactly 3 times in a randomly chosen 30-second period

(ii) at least twice but not more than 5 times in a randomly chosen 15-second period

(iii) at least once in a randomly chosen 1-minute period.

The table below shows the data from a sample of 50 observations of a variable $x$.

(i) Calculate unbiased estimates for the mean and the variance.

(ii) State, with a reason, whether a Poisson distribution could be used to model the population’s data.

The table below shows the data from a sample of 100 observations of a variable $y$.

(i) Calculate unbiased estimates for the mean and the variance.

(ii) State, with a reason, whether a Poisson distribution could be used to model the population’s data.

Jim records the number of birds that fly past his window during each of 120 randomly chosen 10-minute periods. The results are summarised in the table.

Calculate unbiased estimates of the population mean and variance of the number of birds that fly past Jim's window in a 10-minute period.

Explain how your answers to part (a) suggest that a Poisson distribution may be a suitable model for the number of birds that fly past Jim's window in a 10-minute period.

Jim models the number of birds that fly past his window in a 10-minute period by the distribution $\text{Po} ( 1 )$.

Use Jim's model to find the probability that

(i) exactly 2 birds fly past Jim's window in a 30-minute period

(ii) fewer than 2 birds fly past Jim's window in a 1-minute period

(iii) at least 4 birds fly past Jim's window in a 1-hour period.

The number of times that Blue, a dog, barks is modelled by a Poisson distribution with mean 25 per hour.

State two assumptions that are required for the Poisson model to be valid in this context.

Find the probability that Blue barks

(i) exactly 5 times in a randomly chosen 10-minute period

(ii) at most 4 times in a randomly chosen 15-minute period

(iii) more than 47 times but not more than 51 times in a randomly chosen 2-hour period.

Roger is investigating historical data from World War II. In south London, an area of 144 km² was divided into 576 equal squares, and the number of times each square was hit by a flying bomb over a period of time was recorded. During this time, a total of 537 bombs landed in the area. The results are shown in the table below.

Roger suggests that a Poisson distribution with mean $\frac{537}{576}$ would be an appropriate model for the number of times a square was hit by a flying bomb.

(i) By finding probabilities using Roger's model, estimate the expected number of squares that would be hit, respectively, by 0, 1, 2, 3, 4 and 5 or more flying bombs.

(ii) By comparing the actual number of squares in the table with the expected numbers of squares from part (i), state whether Roger's model is appropriate.

The number of customer enquiries received per hour at an office is denoted by $X$. It is given that $X$ has a Poisson distribution with standard deviation 2.25.

Write down the value of $\text{E} ( X )$.

Find $\text{P} ( X = 4 )$.

Find $\text{P} ( 1 . 5 \leq X < 5 )$.

Find the probability that $X = 4$ given that $X > 0$.

Grace attends nursery 5 days a week. The number of tantrums that Grace has in a day at nursery is modelled by a Poisson distribution with variance 3.14.

Find the probability that Grace has exactly 17 tantrums during a randomly chosen week at nursery.

Find the probability that Grace has fewer than 4 tantrums in a randomly chosen 2-day period at nursery.

Given that Grace has fewer than 4 tantrums at nursery on a randomly chosen day, find the probability that she has no tantrums at nursery on that day.

The numbers of detentions issued per school day by Ms Ottway, a teacher, over a period of 150 days are summarised in the table.

Calculate unbiased estimates of the population mean and variance of the number of detentions issued per day, and explain how these estimates suggest that a Poisson distribution is an appropriate model.

Using a Poisson distribution with mean equal to your estimate from part (a), find the probability that Ms Ottway issues at least 5 detentions on a randomly chosen day.

Students at Ms Ottway's school attend school 5 days a week for 40 weeks in a year.

Estimate the number of weeks in a school year in which Ms Ottway issues fewer than 3 detentions.

$X$, $Y$ and $Z$ are random variables with X \sim \text{Po}(\alpha), Y \sim \text{Po}(\beta) and Z \sim \text{Po}(\gamma).

(i) Given that $\text{P} ( X = 3 ) = \text{P} ( X = 7 )$, find the value of $α$.

(ii) Given that $\text{P} ( Y = 0 ) = 0 . 253$, find the value of $β$.

(iii) Given that 5\text{Var}(Z) = \left(\text{E}(Z)\right)^{2} - 6, find the value of $γ$.

In a computer game, enemies appear at random and at a constant average rate of 2.5 per minute. You may assume that enemies appear independently of one another.

Find the probability that exactly 10 enemies appear during a randomly chosen 5-minute period.

Find the probability that at least 3 enemies appear during a randomly chosen 90-second period.

The probability that at least one enemy appears during a $k$-minute period is 0.999.

Find the value of $k$.

Matt receives messages from three friends: Danny, Jessica and Luke. On average, Matt receives 1.7 messages per hour from Danny, 1.5 messages per hour from Jessica and 1.1 messages per hour from Luke.

Stating a necessary assumption, find the probability that Matt receives fewer than 4 messages from Danny during a randomly chosen 2-hour period.

Stating a further assumption needed, find the probability that Matt receives exactly 6 messages in total during a randomly chosen 1-hour period.

Find the probability that Matt receives at least one message from any of the three friends during a randomly chosen 1-hour period.

Find the probability that Matt receives at least one message from each friend during a randomly chosen 1-hour period.

Phoebe has a faulty smoke detector which rings at an average rate of 5 times per day.

State two conditions necessary for a Poisson distribution to be a suitable model for the number of times that the smoke detector rings.

Assume now that a Poisson distribution is a suitable model.

(i) Find the probability that the smoke detector rings exactly 3 times on each day of a randomly chosen 4-day period.

(ii) Find the probability that the smoke detector rings exactly 12 times during a randomly chosen 4-day period.

(iii) Explain why the answers to part (b)(i) and part (b)(ii) are different.

Given that the smoke detector rings at least once during a randomly chosen 6-hour period, find the probability that it rings no more than 4 times during this period.

The random variables X \sim \text{Po}(\alpha), Y \sim \text{Po}(\beta) and Z \sim \text{Po}(\gamma).

(i) Given that $\text{P} ( X = 4 ) = \text{P} ( X = 2 ) + \text{P} ( X = 3 )$, find the value of $α$.

(ii) Given that $\text{P} ( Y \geq 1 ) = 0 . 314$, find the value of $β$.

(iii) Given that $\text{P} ( Z = k + 1 ) = \text{P} ( Z = k - 1 )$ where $k$ is a positive integer, find an expression for $γ$ in terms of $k$.

The number of mistakes made by a student, Priya, in a 20-minute revision period is modelled by the distribution $\text{Po} ( 1 . 2 )$. The number of mistakes made by a different student, Qays, in a 30-minute revision period is modelled by the distribution $\text{Po} ( 2 . 2 )$.

Stating a necessary assumption, find the probability that, in a randomly chosen 1-hour revision period, Priya makes exactly 2 mistakes and Qays makes exactly 1 mistake. Give your answer in the form $\frac{a}{b} \text{e}^{c}$, where $a$, $b$ and $c$ are integers.

Given that Priya and Qays make a total of exactly 9 mistakes in a randomly chosen 1-hour revision period, find the probability that Priya makes exactly 5 mistakes.

Given instead that Priya makes exactly 5 mistakes in a randomly chosen 1-hour revision period, find the probability that Priya and Qays make a total of exactly 9 mistakes in that period.

In the week before a test, Priya has ten 30-minute revision periods.

Estimate the number of these revision periods during which Priya makes at least one mistake.

Spelling mistakes in essays written by Gamu occur at random and at a constant average rate of 7 per 150 words. Gamu models the number of spelling mistakes using a Poisson distribution.

Find the largest number of words that Gamu can write so that the probability that she makes at least one spelling mistake does not exceed 0.75.

Gamu writes three short essays, containing 100 words, 200 words and 250 words. An essay is returned by her lecturer if more than 1% of its words contain spelling mistakes.

You should assume that each spelling mistake is in a different word, and that the numbers of spelling mistakes in the three essays are independent.

Find the probability that all three essays are returned.