5.02n Sum of Poisson variables: is Poisson

6 Computer breakdowns occur randomly on average once every 48 hours of use.

1. Calculate the probability that there will be fewer than 4 breakdowns in 60 hours of use.
2. Find the probability that the number of breakdowns in one year (8760 hours) of use is more than 200.
3. Independently of the computer breaking down, the computer operator receives phone calls randomly on average twice in every 24 -hour period. Find the probability that the total number of phone calls and computer breakdowns in a 60-hour period is exactly 4 .

4 The random variable \(A\) has the distribution \(\operatorname { Po } ( 1.5 ) . A _ { 1 }\) and \(A _ { 2 }\) are independent values of \(A\).

1. Find \(\mathrm { P } \left( A _ { 1 } + A _ { 2 } < 2 \right)\).
2. Given that \(A _ { 1 } + A _ { 2 } < 2\), find \(\mathrm { P } \left( A _ { 1 } = 1 \right)\).
3. Give a reason why \(A _ { 1 } - A _ { 2 }\) cannot have a Poisson distribution.

4 The number, \(X\), of books received at a charity shop has a constant mean of 5.1 per day.

1. State, in context, one condition for \(X\) to be modelled by a Poisson distribution.
   Assume now that \(X\) can be modelled by a Poisson distribution.
2. Find the probability that exactly 10 books are received in a 3-day period.
3. Use a suitable approximating distribution to find the probability that more than 180 books are received in a 30-day period.
   The number of DVDs received at the same shop is modelled by an independent Poisson distribution with mean 2.5 per day.
4. Find the probability that the total number of books and DVDs that are received at the shop in 1 day is more than 3 .

1 A bus station has exactly four entrances. In the morning the numbers of passengers arriving at these entrances during a 10 -second period have the independent distributions \(\operatorname { Po } ( 0.4 ) , \operatorname { Po } ( 0.1 ) , \operatorname { Po } ( 0.2 )\) and \(\mathrm { Po } ( 0.5 )\). Find the probability that the total number of passengers arriving at the four entrances to the bus station during a randomly chosen 1 -minute period in the morning is more than 3 .

7 Every July, as part of a research project, Rita collects data about sightings of a particular kind of bird. Each day in July she notes whether she sees this kind of bird or not, and she records the number \(X\) of days on which she sees it. She models the distribution of \(X\) by \(\mathrm { B } ( 31 , p )\), where \(p\) is the probability of seeing this kind of bird on a randomly chosen day in July. Data from previous years suggests that \(p = 0.3\), but in 2022 Rita suspected that the value of \(p\) had been reduced. She decided to carry out a hypothesis test. In July 2022, she saw this kind of bird on 4 days.

1. Use the binomial distribution to test at the \(5 \%\) significance level whether Rita's suspicion is justified.
   In July 2023, she noted the value of \(X\) and carried out another test at the \(5 \%\) significance level using the same hypotheses.
2. Calculate the probability of a Type I error.
   Rita models the number of sightings, \(Y\), per year of a different, very rare, kind of bird by the distribution \(B ( 365,0.01 )\).
3. 1. Use a suitable approximating distribution to find \(\mathrm { P } ( Y = 4 )\).
   2. Justify your approximating distribution in this context.
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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.

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.

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.

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.

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.

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 .

6 The numbers of customers arriving at service desks \(A\) and \(B\) during a 10 -minute period have the independent distributions \(\operatorname { Po } ( 1.8 )\) and \(\operatorname { Po } ( 2.1 )\) respectively.

1. Find the probability that during a randomly chosen 15 -minute period more than 2 customers will arrive at \(\operatorname { desk } A\).
2. Find the probability that during a randomly chosen 5-minute period the total number of customers arriving at both desks is less than 4 . \includegraphics[max width=\textwidth, alt={}, center]{acd6f1c9-bbaf-40ca-b5cb-8466ddb9f596-08_2720_35_109_2012}
3. An inspector waits at desk \(B\). She wants to wait long enough to be \(90 \%\) certain of seeing at least one customer arrive at the desk. Find the minimum time for which she should wait, giving your answer correct to the nearest minute.

5 A machine puts sweets into bags at random. The numbers of lemon and orange sweets in a bag have the independent distributions \(\operatorname { Po } ( 3.7 )\) and \(\operatorname { Po } ( 2.6 )\) respectively. A bag of sweets is chosen at random.

1. Find the probability that the number of lemon sweets in the bag is more than 2 but not more than 5 .
2. Find the probability that the total number of lemon and orange sweets in the bag is less than 4 . \includegraphics[max width=\textwidth, alt={}, center]{9ac74d4c-f5e0-4c5d-ab25-5692dfb06f0b-06_2725_47_107_2002} \includegraphics[max width=\textwidth, alt={}, center]{9ac74d4c-f5e0-4c5d-ab25-5692dfb06f0b-07_2716_29_107_22} 10 bags of sweets are chosen at random.
3. Use approximating distributions to find the probability that the total number of lemon sweets in the 10 bags is less than the total number of orange sweets in the 10 bags.

6 At a petrol station cars arrive independently and at random times at constant average rates of 8 cars per hour travelling east and 5 cars per hour travelling west.

1. Find the probability that, in a quarter-hour period,
   1. one or more cars travelling east and one or more cars travelling west will arrive,
   2. a total of 2 or more cars will arrive.
   3. Find the approximate probability that, in a 12 -hour period, a total of more than 175 cars will arrive.

7 A clinic deals only with flu vaccinations. The number of patients arriving every 15 minutes is modelled by the random variable \(X\) with distribution \(\operatorname { Po } ( 4.2 )\).

1. State two assumptions required for the Poisson model to be valid.
2. Find the probability that
   1. at least 1 patient will arrive in a 15-minute period,
   2. fewer than 4 patients will arrive in a 10-minute period.
   3. The clinic is open for 20 hours each week. At the beginning of one week the clinic has enough vaccine for 370 patients. Use a suitable approximation to find the probability that this will not be enough vaccine for that week.

1 A hotel kitchen has two dish-washing machines. The numbers of breakdowns per year by the two machines have independent Poisson distributions with means 0.7 and 1.0 . Find the probability that the total number of breakdowns by the two machines during the next two years will be less than 3 .

4 Bacteria of a certain type are randomly distributed in the water in two ponds, \(A\) and \(B\). The average numbers of bacteria per \(\mathrm { cm } ^ { 3 }\) in \(A\) and \(B\) are 0.32 and 0.45 respectively.

1. Samples of \(8 \mathrm {~cm} ^ { 3 }\) of water from \(A\) and \(12 \mathrm {~cm} ^ { 3 }\) of water from \(B\) are taken at random. Find the probability that the total number of bacteria in these samples is at least 3 .
2. Find the probability that in a random sample of \(155 \mathrm {~cm} ^ { 3 }\) of water from \(A\), the number of bacteria is less than 35 .

4 The number of lions seen per day during a standard safari has the distribution \(\operatorname { Po } ( 0.8 )\). The number of lions seen per day during an off-road safari has the distribution \(\operatorname { Po } ( 2.7 )\). The two distributions are independent.

1. Susan goes on a standard safari for one day. Find the probability that she sees at least 2 lions.
2. Deena goes on a standard safari for 3 days and then on an off-road safari for 2 days. Find the probability that she sees a total of fewer than 5 lions.
3. Khaled goes on a standard safari for \(n\) days, where \(n\) is an integer. He wants to ensure that his chance of not seeing any lions is less than \(10 \%\). Find the smallest possible value of \(n\).

6 The number of cases of asthma per month at a clinic has a Poisson distribution. In the past the mean has been 5.3 cases per month. A new treatment is introduced. In order to test at the \(5 \%\) significance level whether the mean has decreased, the number of cases in a randomly chosen month is noted.

1. Find the critical region for the test and, given that the number of cases is 2 , carry out the test.
2. Explain the meaning of a Type I error in this context and state the probability of a Type I error.
3. At another clinic the mean number of cases of asthma per month has the independent distribution \(\mathrm { Po } ( 13.1 )\). Assuming that the mean for the first clinic is still 5.3, use a suitable approximating distribution to estimate the probability that the total number of cases in the two clinics in a particular month is more than 20.

7 The number of absences by girls from a certain class on any day is modelled by a random variable with distribution \(\operatorname { Po } ( 0.2 )\). The number of absences by boys from the same class on any day is modelled by an independent random variable with distribution \(\operatorname { Po } ( 0.3 )\).

1. Find the probability that, during a randomly chosen 2-day period, the total number of absences is less than 3 .
2. Find the probability that, during a randomly chosen 5-day period, the number of absences by boys is more than 3.
3. The teacher claims that, during the football season, there are more absences by boys than usual. In order to test this claim at the 5\% significance level, he notes the number of absences by boys during a randomly chosen 5-day period during the football season.
   1. State what is meant by a Type I error in this context.
   2. State appropriate null and alternative hypotheses and find the probability of a Type I error.
   3. In fact there were 4 absences by boys during this period. Test the teacher's claim at the 5\% significance level.
      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 numbers of alpha, beta and gamma particles emitted per minute by a certain piece of rock have independent distributions \(\operatorname { Po } ( 0.2 ) , \operatorname { Po } ( 0.3 )\) and \(\operatorname { Po } ( 0.6 )\) respectively. Find the probability that the total number of particles emitted during a 4 -minute period is less than 4.

5

1. The random variable \(X\) has the distribution \(\operatorname { Po } ( 2.3 )\).
   1. Find \(\mathrm { P } ( 2 \leqslant X \leqslant 4 )\).
   2. Find the probability that the sum of two independent values of \(X\) is greater than 2 .
   3. The random variable \(S\) is the sum of 50 independent values of \(X\). Use a suitable approximating distribution to find \(\mathrm { P } ( S \leqslant 110 )\).
2. The random variable \(Y\) has the distribution \(\mathrm { Po } ( \lambda )\). Given that \(\mathrm { P } ( Y = 3 ) = \mathrm { P } ( Y = 5 )\), find \(\lambda\).

7 All the seats on a certain daily flight are always sold. The number of passengers who have bought seats but fail to arrive for this flight on a particular day is modelled by the distribution \(\mathbf { B } ( 320,0.005 )\).

1. Explain what the number 320 represents in this context.
2. The total number of passengers who have bought seats but fail to arrive for this flight on 2 randomly chosen days is denoted by \(X\). Use a suitable approximating distribution to find \(\mathrm { P } ( 2 < X < 6 )\).
3. Justify the use of your approximating distribution.
   After some changes, the airline wishes to test whether the mean number of passengers per day who fail to arrive for this flight has decreased.
4. During 5 randomly chosen days, a total of 2 passengers failed to arrive. Carry out the test at the 2.5\% significance level.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.