5.02i Poisson distribution: random events model

8 A new light was installed on a certain footpath. A town councillor decided to use a hypothesis test to investigate whether the number of people using the path in the evening had increased. Before the light was installed, the mean number of people using the path during any 20 -minute period during the evening was 1.01. After the light was installed, the total number, \(n\), of people using the path during 3 randomly chosen 20 -minute periods during the evening was noted.

1. Given that the value of \(n\) was 6 , use a Poisson distribution to carry out the test at the \(5 \%\) significance level.
2. Later a similar test, at the \(5 \%\) significance level, was carried out using another 3 randomly chosen 20 -minute periods during the evening. Find the probability of a Type I error.
3. State what is meant by a Type I error in this context.
4. State, in context, what further information would be needed in order to find the probability of a Type II error. Do not carry out any further calculation.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

5 Sales of cell phones at a certain shop occur singly, randomly and independently.

1. State one further condition that must be satisfied for the number of sales in a certain time period to be well modelled by a Poisson distribution.
   The average number of sales per hour is 1.2 .
   Assume now that a Poisson distribution is a suitable model.
2. Find the probability that the number of sales during a randomly chosen 12 -hour period will be more than 12 and less than 16 .
3. Use a suitable approximating distribution to find the probability that the number of sales during a randomly chosen 1-month period (140 hours) will be less than 150 .

1 A random variable \(X\) has the distribution \(\mathrm { Po } ( 145 )\).

1. Use a suitable approximating distribution to calculate \(\mathrm { P } ( X \leqslant 150 )\).
2. Justify the use of your approximating distribution in this case.

5 The number of goals scored by a sports team in the first half of any match has the distribution \(X \sim \mathrm { Po }\) (3.1). The number of goals scored by the same team in the second half of any match has the distribution \(Y \sim \operatorname { Po } ( 2.4 )\). You may assume that the distributions of \(X\) and \(Y\) are independent.

1. Find \(\mathrm { P } ( X < 4 )\).
2. Find the probability that, in a randomly chosen match, the team scores at least 5 goals.
3. Given that the team scores a total of 5 goals in a randomly chosen match, find the probability that they score exactly 3 goals in the first half.

1 The random variable \(X\) has the distribution \(\mathrm { B } ( 4000,0.001 )\).

1. Use a suitable approximating distribution to find \(\mathrm { P } ( 2 \leqslant X < 5 )\).
2. Justify your approximating distribution in this case.

7 The independent random variables \(X\) and \(Y\) have the distributions \(\operatorname { Po } ( 1.9 )\) and \(\operatorname { Po } ( 2.2 )\) respectively.

1. Find \(\mathrm { P } ( X + Y < 4 )\). \includegraphics[max width=\textwidth, alt={}, center]{7c078a14-98f9-4292-ae76-a2642238176f-10_74_1581_406_322} \includegraphics[max width=\textwidth, alt={}, center]{7c078a14-98f9-4292-ae76-a2642238176f-10_75_1581_497_322}
2. Find the probability that \(X = 2\) given that \(X + Y < 4\). \includegraphics[max width=\textwidth, alt={}, center]{7c078a14-98f9-4292-ae76-a2642238176f-10_2715_35_144_2012}
3. A sample of 60 randomly chosen pairs of values of \(X\) and \(Y\) is taken,and the value of \(X + Y\) is calculated for each pair.The sample mean of these 60 values is found. Find the probability that the sample mean of \(X + Y\) is less than 4.0 .
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

1 The booklets produced by a certain publisher contain, on average, 1 incorrect letter per 30000 letters, and these errors occur randomly. A randomly chosen booklet from this publisher contains 12500 letters. Use a suitable approximating distribution to find the probability that this booklet contains at least 2 errors.

4 The number of accidents on a certain road has a Poisson distribution with mean 0.4 per 50-day period.

1. Find the probability that there will be fewer than 3 accidents during a year (365 days).
2. The probability that there will be no accidents during a period of \(n\) days is greater than 0.95 . Find the largest possible value of \(n\).

4 On average, 1 in 400 microchips made at a certain factory are faulty. The number of faulty microchips in a random sample of 1000 is denoted by \(X\).

1. State the distribution of \(X\), giving the values of any parameters.
2. State an approximating distribution for \(X\), giving the values of any parameters.
3. Use this approximating distribution to find each of the following.
   1. \(\mathrm { P } ( X = 4 )\).
   2. \(\mathrm { P } ( 2 \leqslant X \leqslant 4 )\).
4. Use a suitable approximating distribution to find the probability that, in a random sample of 700 microchips, there will be at least 1 faulty one.

7

1. Two ponds, \(A\) and \(B\), each contain a large number of fish. It is known that \(2.4 \%\) of fish in pond \(A\) are carp and \(1.8 \%\) of fish in pond \(B\) are carp. Random samples of 50 fish from pond \(A\) and 60 fish from pond \(B\) are selected. Use appropriate Poisson approximations to find the following probabilities.
   1. The samples contain at least 2 carp from pond \(A\) and at least 2 carp from pond \(B\).
   2. The samples contain at least 4 carp altogether.
2. The random variables \(X\) and \(Y\) have the distributions \(\operatorname { Po } ( \lambda )\) and \(\operatorname { Po } ( \mu )\) respectively. It is given that
   Find the value of \(k\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

4

1. The random variable \(W\) has the distribution \(\operatorname { Po } ( 1.5 )\). Find the probability that the sum of 3 independent values of \(W\) is greater than 2 .
2. The random variable \(X\) has the distribution \(\operatorname { Po } ( \lambda )\). Given that \(\mathrm { P } ( X = 0 ) = 0.523\), find the value of \(\lambda\) correct to 3 significant figures.
3. The random variable \(Y\) has the distribution \(\operatorname { Po } ( \mu )\), where \(\mu \neq 0\). Given that $$\mathrm { P } ( Y = 3 ) = 24 \times \mathrm { P } ( Y = 1 )$$ find \(\mu\).

6 People arrive at a checkout in a store at random, and at a constant mean rate of 0.7 per minute. Find the probability that

1. exactly 3 people arrive at the checkout during a 5 -minute period,
2. at least 30 people arrive at the checkout during a 1-hour period. People arrive independently at another checkout in the store at random, and at a constant mean rate of 0.5 per minute.
3. Find the probability that a total of more than 3 people arrive at this pair of checkouts during a 2-minute period.

7

1. A large number of spoons and forks made in a factory are inspected. It is found that \(1 \%\) of the spoons and \(1.5 \%\) of the forks are defective. A random sample of 140 items, consisting of 80 spoons and 60 forks, is chosen. Use the Poisson approximation to the binomial distribution to find the probability that the sample contains
   1. at least 1 defective spoon and at least 1 defective fork,
   2. fewer than 3 defective items.
2. The random variable \(X\) has the distribution \(\operatorname { Po } ( \lambda )\). It is given that $$\mathrm { P } ( X = 1 ) = p \quad \text { and } \quad \mathrm { P } ( X = 2 ) = 1.5 p$$ where \(p\) is a non-zero constant. Find the value of \(\lambda\) and hence find the value of \(p\).

4 At a certain company, computer faults occur randomly and at a constant mean rate. In the past this mean rate has been 2.1 per week. Following an update, the management wish to determine whether the mean rate has changed. During 20 randomly chosen weeks it is found that 54 computer faults occur. Use a suitable approximation to test at the \(5 \%\) significance level whether the mean rate has changed.

6 At a certain shop the demand for hair dryers has a Poisson distribution with mean 3.4 per week.

1. Find the probability that, in a randomly chosen two-week period, the demand is for exactly 5 hair dryers.
2. At the beginning of a week the shop has a certain number of hair dryers for sale. Find the probability that the shop has enough hair dryers to satisfy the demand for the week if
   1. they have 4 hair dryers in the shop,
   2. they have 5 hair dryers in the shop.
   3. Find the smallest number of hair dryers that the shop needs to have at the beginning of a week so that the probability of being able to satisfy the demand that week is at least 0.9 .

6 The number of sports injuries per month at a certain college has a Poisson distribution. In the past the mean has been 1.1 injuries per month. The principal recently introduced new safety guidelines and she decides to test, at the \(2 \%\) significance level, whether the mean number of sports injuries has been reduced. She notes the number of sports injuries during a 6-month period.

1. Find the critical region for the test and state the probability of a Type I error.
2. State what is meant by a Type I error in this context.
3. During the 6 -month period there are a total of 2 sports injuries. Carry out the test.
4. Assuming that the mean remains 1.1 , calculate the probability that there will be fewer than 30 sports injuries during a 36-month period.

2 Javier writes an article containing 52460 words. He plans to upload the article to his website, but he knows that this process sometimes introduces errors. He assumes that for each word in the uploaded version of his article, the probability that it contains an error is 0.00008 . The number of words containing an error is denoted by \(X\).

1. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\), giving your answers correct to three decimal places.
   Javier wants to use the Poisson distribution as an approximating distribution to calculate the probability that there will be fewer than 5 words containing an error in his uploaded article.
2. Explain how your answers to part (i) are consistent with the use of the Poisson distribution as an approximating distribution.
3. Use the Poisson distribution to calculate \(\mathrm { P } ( X < 5 )\).

6 Old televisions arrive randomly and independently at a recycling centre at an average rate of 1.2 per day.

1. Find the probability that exactly 2 televisions arrive in a 2-day period.
2. Use an appropriate approximating distribution to find the probability that at least 55 televisions arrive in a 50-day period.
   Independently of televisions, old computers arrive randomly and independently at the same recycling centre at an average rate of 4 per 7-day week.
3. Find the probability that the total number of televisions and computers that arrive at the recycling centre in a 3-day period is less than 4.

5

1. A random variable \(X\) has the distribution \(\operatorname { Po } ( 42 )\).
   1. Use an appropriate approximating distribution to find \(\mathrm { P } ( X \geqslant 40 )\).
   2. Justify your use of the approximating distribution.
   3. A random variable \(Y\) has the distribution \(\mathrm { B } ( 60,0.02 )\).
      (a) Use an appropriate approximating distribution to find \(\mathrm { P } ( Y > 2 )\).
      (b) Justify your use of the approximating distribution.

7 In the past the number of accidents per month on a certain road was modelled by a random variable with distribution \(\operatorname { Po } ( 0.47 )\). After the introduction of speed restrictions, the government wished to test, at the 5\% significance level, whether the mean number of accidents had decreased. They noted the number of accidents during the next 12 months. It is assumed that accidents occur randomly and that a Poisson model is still appropriate.

1. Given that the total number of accidents during the 12 months was 2 , carry out the test.
2. Explain what is meant by a Type II error in this context.
   It is given that the mean number of accidents per month is now in fact 0.05 .
3. Using another random sample of 12 months the same test is carried out again, with the same significance level. Find the probability of a Type II error.

4 The numbers, \(M\) and \(F\), of male and female students who leave a particular school each year to study engineering have means 3.1 and 0.8 respectively.

1. State, in context, one condition required for \(M\) to have a Poisson distribution.
   Assume that \(M\) and \(F\) can be modelled by independent Poisson distributions.
2. Find the probability that the total number of students who leave to study engineering in a particular year is more than 3 .
3. Given that the total number of students who leave to study engineering in a particular year is more than 3 , find the probability that no female students leave to study engineering in that year.

1 On average, 2 people in every 10000 in the UK have a particular gene. A random sample of 6000 people in the UK is chosen. The random variable \(X\) denotes the number of people in the sample who have the gene. Use an approximating distribution to calculate the probability that there will be more than 2 people in the sample who have the gene.

6 The number of injuries per month at a certain factory has a Poisson distribution. In the past the mean was 2.1 injuries per month. New safety procedures are put in place and the management wishes to use the next 3 months to test, at the \(2 \%\) significance level, whether there are now fewer injuries than before, on average.

1. Find the critical region for the test.
2. Find the probability of a Type I error.
3. During the next 3 months there are a total of 3 injuries. Carry out the test.
4. Assuming that the mean remains 2.1 , calculate an estimate of the probability that there will be fewer than 20 injuries during the next 12 months.

5 A random variable \(X\) has the distribution \(\operatorname { Po } ( 3.2 )\).

1. A random value of \(X\) is found.
   1. Find \(\mathrm { P } ( X \geqslant 3 )\).
   2. Find the probability that \(X = 3\) given that \(X \geqslant 3\).
   3. Random samples of 120 values of \(X\) are taken.
      (a) Describe fully the distribution of the sample mean.
      (b) Find the probability that the mean of a random sample of size 120 is less than 3.3.

7 At work Jerry receives emails randomly at a constant average rate of 15 emails per hour.

1. Find the probability that Jerry receives more than 2 emails during a 20 -minute period at work.
2. Jerry's working day is 8 hours long. Find the probability that Jerry receives fewer than 110 emails per day on each of 2 working days.
3. At work Jerry also receives texts randomly and independently at a constant average rate of 1 text every 10 minutes. Find the probability that the total number of emails and texts that Jerry receives during a 5 -minute period at work is more than 2 and less than 6 .