5.02k Calculate Poisson probabilities

3 The numbers of CD players sold in a shop on three consecutive weekends were 7,6 and 2 . It may be assumed that sales of CD players occur randomly and that nobody buys more than one CD player at a time. The number of CD players sold on a randomly chosen weekend is denoted by \(X\).

1. How appropriate is the Poisson distribution as a model for \(X\) ? Now assume that a Poisson distribution with mean 5 is an appropriate model for \(X\).
2. Find
   1. \(\mathrm { P } ( X = 6 )\),
   2. \(\mathrm { P } ( X \geqslant 8 )\). The number of integrated sound systems sold in a weekend at the same shop can be assumed to have the distribution \(\operatorname { Po } ( 7.2 )\).
3. Find the probability that on a randomly chosen weekend the total number of CD players and integrated sound systems sold is between 10 and 15 inclusive.
4. State an assumption needed for your answer to part (c) to be valid.
5. Give a reason why the assumption in part (d) may not be valid in practice.

1. A multiple-choice test consists of 25 questions, each having 5 responses, only one of which is correct.

Each correct answer gains 4 marks but each incorrect answer loses 1 mark.
Sam answers all 25 questions by choosing at random one response for each question.
Let \(X\) be the number of correct answers that Sam achieves.

1. State the distribution of \(X\) Let \(M\) be the number of marks that Sam achieves.
2. 1. State the distribution of \(M\) in terms of \(X\)
   2. Hence, show clearly that the number of marks that Sam is expected to achieve is zero. In order to pass the test at least 30 marks are required.
3. Find the probability that Sam will pass the test. Past records show that when the test is done properly, the probability that a student answers the first question correctly is 0.5 A random sample of 50 students that did the test properly was taken.
   Given that the probability that more than \(n\) but at most 30 students answered the first question correctly was 0.9328 to 4 decimal places,
4. find the value of \(n\)

1 A company hires out narrowboats on a canal. It may be assumed that demands to hire a narrowboat occur independently and randomly at a constant mean rate of 25 per week. Using a suitable normal approximation, find

1. the probability that 15 or fewer narrowboats are hired out during a certain week,
2. the number of narrowboats that the company needs to have available for a week in order that the probability of running out of boats is 0.05 or less.

3

1. Given that \(X \sim \operatorname { Po } ( 5 )\), find \(\mathrm { P } ( X > 6 \mid X > 3 )\).
2. Given that \(Y \sim \operatorname { Po } ( \lambda )\) and \(\mathrm { P } ( Y \leqslant 1 ) = \frac { 1 } { 2 }\), show that \(\lambda\) satisfies the equation \(\lambda = \ln \{ 2 ( 1 + \lambda ) \}\).
3. Starting with a suitable approximation from the table of cumulative Poisson probabilities, use iteration to find \(\lambda\) correct to 3 decimal places.

3 The number of signal failures in a certain region of the railway network averages 10 every 3 weeks. Assume that signal failures occur independently, randomly and at constant mean rate.

1. Find the probability that
   1. there are between 7 and 12 (inclusive) signal failures in a three-week period,
   2. there are more than 4 signal failures in a one-week period.
   3. It has been calculated, using a suitable distributional approximation, that the probability of more than 62 signal failures in a period of \(n\) weeks is 0.0385 . Find the value of \(n\).

4

1. (a) Derive the moment generating function for a Poisson distribution with mean \(\lambda\).
   (b) The independent random variables \(X\) and \(Y\) are such that \(X \sim \operatorname { Po } ( \mu )\) and \(Y \sim \operatorname { Po } ( v )\). Use moment generating functions to show that \(( X + Y ) \sim \operatorname { Po } ( \mu + v )\).
2. The number of goals scored per match by Camford Academicals FC may be modelled by a Poisson distribution with mean 2. The number of goals scored against Camford during a match may be modelled by an independent Poisson distribution with mean \(k\). The probability that no goals are scored, by either side, in a match involving Camford is 0.045 . Find
   (a) the value of \(k\),
   (b) the probability that exactly 3 goals are scored against Camford in a match,
   (c) the probability that the total number of goals scored, in a match involving Camford, is between 2 and 5 inclusive.

2

1. The probability that a shopper obtains a parking space on the river embankment on any given Saturday morning is 0.2 . Using a suitable normal approximation, find the probability that, over a period of 100 Saturday mornings, the shopper finds a parking space at least 15 times. Justify the use of the normal approximation in this case.
2. The number of parking tickets that a traffic warden issues on the river embankment during the course of a week has a Poisson distribution with mean 36 . The probability that the traffic warden issues more than \(N\) parking tickets is less than 0.05 . Using a suitable normal approximation, find the least possible value of \(N\).

5 The number of calls to a car breakdown service during any one hour of the day is modelled by the distribution \(\operatorname { Po } ( 20 )\).

1. Find the probability that in a randomly chosen 12 -minute period there are at least 7 calls to the service.
2. Find the period of time, correct to the nearest second, for which the probability that no calls are made to the service is 0.6 .
3. Use a suitable approximation to find the probability that, in a randomly chosen 3-hour period, there are no more than 65 calls to the service.

2 Secret radio messages received under difficult conditions are subject to errors caused by random instantaneous breaks in transmission. The number of errors caused by breaks in transmission in a 10-minute period is denoted by \(B\).

1. State two conditions, other than randomness, needed for a Poisson distribution to be a suitable model for \(B\). Assume now that \(B \sim \mathrm { Po } ( 5 )\).
2. Calculate the probability that in a 15-minute period there are between 6 and 10 errors, inclusive, caused by random breaks in transmission. Secret radio messages are also subject to errors caused by mistakes made by the sender. The number of errors caused by mistakes made by the sender in a 10 -minute period, \(M\), has the independent distribution \(\operatorname { Po } ( 8 )\).
3. Calculate the period of time, in seconds, for which the probability that a message contains no errors of either sort is 0.6 .

10

1. \(X , Y\) and \(Z\) are independent random variables having Poisson distributions with means \(\lambda , \mu\) and \(\lambda + \mu\) respectively. Find \(\mathrm { P } ( X = 0\) and \(Y = 2 ) , \mathrm { P } ( X = 1\) and \(Y = 1 )\) and \(\mathrm { P } ( X = 2\) and \(Y = 0 )\). Hence verify that \(\mathrm { P } ( X + Y = 2 ) = \mathrm { P } ( Z = 2 )\).
2. In an office the male absence rate, i.e. the number of working days lost each month due to the absence of male employees, has a Poisson distribution with mean 4.5. In the same office the female absence rate has an independent Poisson distribution with mean 4.1. Calculate the probability that
   1. during a particular month both the male absence rate and the female absence rate are equal to 3,
   2. during a particular month the total of the male and female absence rates is equal to 6,
   3. during a particular month the male and female absence rates were each equal to 3 , given that the total of the male and female absence rates was equal to 6 .

In a study, samples of soil were collected during the summer. Soil samples of dimensions \(25 \mathrm {~cm} \times 25 \mathrm {~cm} \times 40 \mathrm {~cm}\) were collected for analysis. The study found that there were, on average, 11 earthworms per sample. a) Explain briefly the conditions under which a Poisson distribution could be used to model the number of earthworms per sample.
b) In July, pupils at a primary school are asked to dig a smaller hole, \(25 \mathrm {~cm} \times 25 \mathrm {~cm} \times 10 \mathrm {~cm}\), and to count the number of earthworms they find. Calculate the probability that the pupils find exactly 5 earthworms.
c) In the autumn, the average number of earthworms per sample is greater than in the summer. The probability that, in the autumn, there are fewer than 13 earthworms in a soil sample of dimensions \(25 \mathrm {~cm} \times 25 \mathrm {~cm} \times 40 \mathrm {~cm}\) is close to \(36 \%\). Find the mean number of earthworms, to the nearest whole number, per \(25 \mathrm {~cm} \times 25 \mathrm {~cm} \times 40 \mathrm {~cm}\) soil sample in the autumn. Jessica is studying the relationship between hip girth, \(h \mathrm {~cm}\), and thigh girth, \(t \mathrm {~cm}\), for American adults who are physically active. She takes a random sample of 11 people from a very large dataset which she has downloaded into a spreadsheet software package. The results are shown below. a) Jessica notes that, for the thigh girth data, the lower quartile is 49.5 and the upper quartile is \(64 \cdot 0\).
i) Show that 87.2 should be classified as an outlier for \(t\).
ii) Give a reason why Jessica might exclude the outlier.
iii) Give a reason why Jessica might include the outlier. Jessica decides to exclude the outlier and produces the following scatter diagram. \section*{Thigh girth versus Hip girth} \includegraphics[max width=\textwidth, alt={}, center]{77c62e6d-58e4-42d3-9982-5a8325e8e826-04_647_1250_1439_404}
b) Interpret, in context, the correlation in the data shown in the diagram. The equation of the regression line of \(t\) on \(h\) for this sample is $$t = 0.69 h - 11.26$$ c) Interpret the gradient of the regression line in this context.
d) Use your knowledge of large data sets and spreadsheet software packages to suggest a way in which Jessica could improve her investigation. A company, Run4Lyfe, sponsors an athletic event. The organisers of the event claim that \(70 \%\) of the participants know the name of the sponsoring company. Run4Lyfe is concerned that the proportion, \(p\), of participants knowing the name of the sponsoring company is less than \(70 \%\). They decide to survey 60 randomly selected participants to carry out a significance test.
a) State suitable hypotheses for carrying out the test.
b) i) Explain what is meant by the critical region for this test.
ii) Determine the critical region if the test is to be carried out at a significance level as close as possible to, but not exceeding, \(5 \%\).
iii) Given that 40 participants out of the 60 in the sample know the name of the company, complete the significance test.
c) State, with a reason, how you would advise Run4Lyfe with regards to sponsoring the event next year. The fertility rate for a country is the average number of children that are born to a woman over her lifetime. The graphs and table below show some data on the fertility rates for 197 countries in the years 1914 and 2014. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure}