5.02n Sum of Poisson variables: is Poisson

1 It is known that, on average, 1 in 300 flowers of a certain kind are white. A random sample of 200 flowers of this kind is selected.

1. Use an appropriate approximating distribution to find the probability that more than 1 flower in the sample is white.
2. Justify the approximating distribution used in part (a).
   The probability that a randomly chosen flower of another kind is white is 0.02 . A random sample of 150 of these flowers is selected.
3. Use an appropriate approximating distribution to find the probability that the total number of white flowers in the two samples is less than 4 .

7 In the past the number of cars sold per day at a showroom has been modelled by a random variable with distribution \(\operatorname { Po } ( 0.7 )\). Following an advertising campaign, it is hoped that the mean number of sales per day will increase. In order to test at the \(10 \%\) significance level whether this is the case, the total number of sales during the first 5 days after the campaign is noted. You should assume that a Poisson model is still appropriate.

1. Given that the total number of cars sold during the 5 days is 5 , carry out the test.
   The number of cars sold per day at another showroom has the independent distribution \(\operatorname { Po } ( 0.6 )\). Assume that the distribution for the first showroom is still \(\operatorname { Po } ( 0.7 )\).
2. Find the probability that the total number of cars sold in the two showrooms during 3 days is exactly 2 .

1

1. 1. A random variable \(X\) has the distribution \(\mathrm { B } ( 2540,0.001 )\). Use the Poisson approximation to the binomial distribution to find \(\mathrm { P } ( X > 1 )\).
   2. Explain why the Poisson approximation is appropriate in this case.
2. Two independent random variables, \(S\) and \(T\), have distributions \(\operatorname { Po } ( 2.1 )\) and \(\operatorname { Po } ( 3.5 )\) respectively. Find the mean and standard deviation of \(S + T\).

4 Small drops of two liquids, \(A\) and \(B\), are randomly and independently distributed in the air. The average numbers of drops of \(A\) and \(B\) per cubic centimetre of air are 0.25 and 0.36 respectively.

1. A sample of \(10 \mathrm {~cm} ^ { 3 }\) of air is taken at random. Find the probability that the total number of drops of \(A\) and \(B\) in this sample is at least 4 .
2. A sample of \(100 \mathrm {~cm} ^ { 3 }\) of air is taken at random. Use an approximating distribution to find the probability that the total number of drops of \(A\) and \(B\) in this sample is less than 60 .

7 The independent random variables \(X\) and \(Y\) have the distributions \(\operatorname { Po } ( 2.1 )\) and \(\operatorname { Po } ( 3.5 )\) respectively.

1. Find \(\mathrm { P } ( X + Y = 3 )\).
2. Given that \(X + Y = 3\), find \(\mathrm { P } ( X = 2 )\).
3. A random sample of 100 values of \(X\) is taken. Find the probability that the sample mean is more than 2.2.
   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 Failures of two computers occur at random and independently. On average the first computer fails 1.2 times per year and the second computer fails 2.3 times per year. Find the probability that the total number of failures by the two computers in a 6-month period is more than 1 and less than 4 .

5 The number of eagles seen per hour in a certain location has the distribution \(\operatorname { Po } ( 1.8 )\). The number of vultures seen per hour in the same location has the independent distribution \(\operatorname { Po } ( 2.6 )\).

1. Find the probability that, in a randomly chosen hour, at least 2 eagles are seen.
2. Find the probability that, in a randomly chosen half-hour period, the total number of eagles and vultures seen is less than 5 .
   Alex wants to be at least \(99 \%\) certain of seeing at least 1 eagle.
3. Find the minimum time for which she should watch for eagles.

2 A certain machine makes matches. One match in 10000 on average is defective. Using a suitable approximation, calculate the probability that a random sample of 45000 matches will include 2,3 or 4 defective matches.

1 The number of radioactive particles emitted per second by a certain metal is random and has mean 1.7. The radioactive metal is placed next to an object which independently emits particles at random such that the mean number of particles emitted per second is 0.6 . Find the probability that the total number of particles emitted in the next 3 seconds is 6, 7 or 8 .

5 Of people who wear contact lenses, 1 in 1500 on average have laser treatment for short sight.

1. Use a suitable approximation to find the probability that, of a random sample of 2700 contact lens wearers, more than 2 people have laser treatment.
2. In a random sample of \(n\) contact lens wearers the probability that no one has laser treatment is less than 0.01 . Find the least possible value of \(n\).

3 Flies stick to wet paint at random points. The average number of flies is 2 per square metre. A wall with area \(22 \mathrm {~m} ^ { 2 }\) is painted with a new type of paint which the manufacturer claims is fly-repellent. It is found that 27 flies stick to this wall. Use a suitable approximation to test the manufacturer's claim at the \(1 \%\) significance level. Take the null hypothesis to be \(\mu = 44\), where \(\mu\) is the population mean.

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.

6 The random variable \(X\) denotes the number of worms on a one metre length of a country path after heavy rain. It is given that \(X\) has a Poisson distribution.

1. For one particular path, the probability that \(X = 2\) is three times the probability that \(X = 4\). Find the probability that there are more than 3 worms on a 3.5 metre length of this path.
2. For another path the mean of \(X\) is 1.3.
   1. On this path the probability that there is at least 1 worm on a length of \(k\) metres is 0.96 . Find \(k\).
   2. Find the probability that there are more than 1250 worms on a one kilometre length of this path.

6 In their football matches, Rovers score goals independently and at random times. Their average rate of scoring is 2.3 goals per match.

1. State the expected number of goals that Rovers will score in the first half of a match.
2. Find the probability that Rovers will not score any goals in the first half of a match but will score one or more goals in the second half of the match.
3. Football matches last for 90 minutes. In a particular match, Rovers score one goal in the first 30 minutes. Find the probability that they will score at least one further goal in the remaining 60 minutes. Independently of the number of goals scored by Rovers, the number of goals scored per football match by United has a Poisson distribution with mean 1.8.
4. Find the probability that a total of at least 3 goals will be scored in a particular match when Rovers play United.

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.

7 A random variable \(X\) has the distribution \(\operatorname { Po } ( 1.6 )\).

1. The random variable \(R\) is the sum of three independent values of \(X\). Find \(\mathrm { P } ( R < 4 )\).
2. The random variable \(S\) is the sum of \(n\) independent values of \(X\). It is given that $$\mathrm { P } ( S = 4 ) = \frac { 16 } { 3 } \times \mathrm { P } ( S = 2 )$$ Find \(n\).
3. The random variable \(T\) is the sum of 40 independent values of \(X\). Find \(\mathrm { P } ( T > 75 )\).

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.

2 The probability that a randomly chosen plant of a certain kind has a particular defect is 0.01 . A random sample of 150 plants is taken.

1. Use an appropriate approximating distribution to find the probability that at least 1 plant has the defect. Justify your approximating distribution. The probability that a randomly chosen plant of another kind has the defect is 0.02 . A random sample of 100 of these plants is taken.
2. Use an appropriate approximating distribution to find the probability that the total number of plants with the defect in the two samples together is more than 3 and less than 7 .

1 Failures of two computers occur at random and independently. On average the first computer fails 1.2 times per year and the second computer fails 2.3 times per year. Find the probability that the total number of failures by the two computers in a 6-month period is more than 1 and 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.

2 It is given that on average one car in forty is yellow. Using a suitable approximation, find the probability that, in a random sample of 130 cars, exactly 4 are yellow.

1 A roller-coaster ride has a safety system to detect faults on the track.

1. State conditions for a Poisson distribution to be a suitable model for the number of faults occurring on a randomly selected day. Faults are detected at an average rate of 0.15 per day. You may assume that a Poisson distribution is a suitable model.
2. Find the probability that on a randomly chosen day there are
   (A) no faults,
   (B) at least 2 faults.
3. Find the probability that, in a randomly chosen period of 30 days, there are at most 3 faults. There is also a separate safety system to detect faults on the roller-coaster train itself. Faults are detected by this system at an average rate of 0.05 per day, independently of the faults detected on the track. You may assume that a Poisson distribution is also suitable for modelling the number of faults detected on the train.
4. State the distribution of the total number of faults detected by the two systems in a period of 10 days. Find the probability that a total of 5 faults is detected in a period of 10 days.
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5. The roller-coaster is operational for 200 days each year. Use a suitable approximating distribution to find the probability that a total of at least 50 faults is detected in 200 days. [5]

4 A multi-storey car park has two entrances and one exit. During a morning period the numbers of cars using the two entrances are independent Poisson variables with means 2.3 and 3.2 per minute. The number leaving is an independent Poisson variable with mean 1.8 per minute. For a randomly chosen 10-minute period the total number of cars that enter and the number of cars that leave are denoted by the random variables \(X\) and \(Y\) respectively.

1. Use a suitable approximation to calculate \(\mathrm { P } ( X \geqslant 40 )\).
2. Calculate \(\mathrm { E } ( X - Y )\) and \(\operatorname { Var } ( X - Y )\).
3. State, giving a reason, whether \(X - Y\) has a Poisson distribution.

1 The numbers of \(\alpha\)-particles emitted per minute from two types of source, \(A\) and \(B\), have the distributions \(\operatorname { Po } ( 1.5 )\) and \(\operatorname { Po } ( 2 )\) respectively. The total number of \(\alpha\)-particles emitted over a period of 2 minutes from three sources of type \(A\) and two sources of type \(B\), all of which are independent, is denoted by \(X\). Calculate \(\mathrm { P } ( X = 27 )\).

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 .