Sum of Poisson processes

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 .

2 The random variable \(A\) has a Poisson distribution with mean 2 The random variable \(B\) has a Poisson distribution with standard deviation 4 The random variables \(A\) and \(B\) are independent.
State the distribution of \(A + B\) Circle your answer.
[0pt] [1 mark]
Po(4)
Po(6)
Po(8)
Po(18)

1 A car repair firm receives call-outs both as a result of breakdowns and also as a result of accidents. On weekdays (Monday to Friday), call-outs resulting from breakdowns occur at random, at an average rate of 6 per 5 -day week; call-outs resulting from accidents occur at random, at an average rate of 2 per 5 -day week. The two types of call-out occur independently of each other. Find the probability that the total number of call-outs received by the firm on one randomly chosen weekday is more than 3 .

2. On a stretch of motorway accidents occur at a rate of 0.9 per month.

1. Show that the probability of no accidents in the next month is 0.407 , to 3 significant figures. Find the probability of
2. exactly 2 accidents in the next 6 month period,
3. no accidents in exactly 2 of the next 4 months.

2 The number of telephone calls per day, \(X\), received by Candice may be modelled by a Poisson distribution with mean 3.5. The number of e-mails per day, \(Y\), received by Candice may be modelled by a Poisson distribution with mean 6.0.

1. For any particular day, find:
   1. \(\mathrm { P } ( X = 3 )\);
   2. \(\quad \mathrm { P } ( Y \geqslant 5 )\).
2. 1. Write down the distribution of \(T\), the total number of telephone calls and e-mails per day received by Candice.
   2. Determine \(\mathrm { P } ( 7 \leqslant T \leqslant 10 )\).
   3. Hence calculate the probability that, on each of three consecutive days, Candice will receive a total of at least 7 but at most 10 telephone calls and e-mails.
      (2 marks)

2. A company produces chocolate chip biscuits. The number of chocolate chips per biscuit has a Poisson distribution with mean 8

1. Find the probability that one of these biscuits, selected at random, does not contain 8 chocolate chips. A small packet contains 4 of these biscuits, selected at random.
2. Find the probability that each biscuit in the packet contains at least 8 chocolate chips. A large packet contains 9 of these biscuits, selected at random.
3. Use a suitable approximation to find the probability that there are more than 75 chocolate chips in the packet. A shop sells packets of biscuits, randomly, at a rate of 1.5 packets per hour. Following an advertising campaign, 11 packets are sold in 4 hours.
4. Test, at the \(5 \%\) level of significance, whether or not there is evidence that the rate of sales of packets of biscuits has increased. State your hypotheses clearly.

2 People arrive randomly and independently at a supermarket checkout at an average rate of 2 people every 3 minutes.

1. Find the probability that exactly 4 people arrive in a 5 -minute period. At another checkout in the same supermarket, people arrive randomly and independently at an average rate of 1 person each minute.
2. Find the probability that a total of fewer than 3 people arrive at the two checkouts in a 3 -minute period.

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.

7 Men arrive at a clinic independently and at random, at a constant mean rate of 0.2 per minute. Women arrive at the same clinic independently and at random, at a constant mean rate of 0.3 per minute.

1. Find the probability that at least 2 men and at least 3 women arrive at the clinic during a 5 -minute period.
2. Find the probability that fewer than 36 people arrive at the clinic during a 1-hour period.

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.

1. At a particular junction on a train line, signal failures are known to occur randomly at a rate of 1 every 4 days.
   1. Find the probability that there are no signal failures on a randomly selected day.
   2. Find the probability that there is at least 1 signal failure on each of the next 3 days.
   3. Find the probability that in a randomly selected 7 -day week, there are exactly 5 days with no signal failures.
   Repair works are carried out on the line. After these repair works, the number, \(f\), of signal failures in a 32-day period is recorded. A test is carried out, at the \(5 \%\) level of significance, to determine whether or not there has been a decrease in the rate of signal failures following the repair works.
2. State the hypotheses for this test.
3. Find the largest value of \(f\) for which the null hypothesis should be rejected.

4 In summer, wasps' nests occur randomly in the south of England at an average rate of 3 nests for every 500 houses.

1. Find the probability that two villages in the south of England, with 600 houses and 700 houses, have a total of exactly 3 wasps' nests.
2. Use a suitable approximation to estimate the probability of there being fewer than 369 wasps' nests in a town with 64000 houses.

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

1. The number of inhabitants of a village who are selected for jury service in the course of a 10-year period is a random variable with the distribution \(\operatorname { Po } ( 4.2 )\).
   1. Find the probability that in the course of a 10-year period, at least 7 inhabitants are selected for jury service.
   2. Find the probability that in 1 year, exactly 2 inhabitants are selected for jury service.
   3. Explain why the number of inhabitants of the village who contract influenza in 1 year can probably not be well modelled by a Poisson distribution.

4 Goals scored by Femchester United occur at random with a constant average of 1.2 goals per match. Goals scored against Femchester United occur independently and at random with a constant average of 0.9 goals per match.

1. Find the probability that in a randomly chosen match involving Femchester,
   1. a total of 3 goals are scored,
   2. a total of 3 goals are scored and Femchester wins. The manager promises the Femchester players a bonus if they score at least 35 goals in the next 25 matches.
   3. Find the probability that the players receive the bonus.

5 Two guesthouses, the Albion and the Blighty, have 8 and 6 rooms respectively. The demand for rooms at the Albion has a Poisson distribution with mean 6.5 and the demand for rooms at the Blighty has an independent Poisson distribution with mean 5.5. The owners have agreed that if their guesthouse is full, they will re-direct guests to the other.

1. Find the probability that, on any particular night, the two guesthouses together do not have enough rooms to meet demand.
2. The Albion charges \(\pounds 60\) per room per night, and the Blighty \(\pounds 80\). Find the probability, that on a particular night, the total income of the two guesthouses is exactly \(\pounds 400\).
3. If \(A\) is the number of rooms demanded at the Albion each night, and \(B\) the number of rooms demanded at the Blighty each night, find the mean and variance of the variable \(C = 60 A + 80 B\). State whether \(C\) has a Poisson distribution, giving a reason for your answer.