5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!

1 The random variable \(X\) has the distribution \(\operatorname { Po } ( 5 )\).

1. Find \(\mathrm { P } ( X = 2 )\).
   It is given that \(\mathrm { P } ( X = n ) = \mathrm { P } ( X = n + 1 )\).
2. Write down an equation in \(n\).
3. Hence or otherwise find the value of \(n\).

6 The battery in Sue's phone runs out at random moments. Over a long period, she has found that the battery runs out, on average, 3.3 times in a 30-day period.

1. Find the probability that the battery runs out fewer than 3 times in a 25-day period.
2. (a) Use an approximating distribution to find the probability that the battery runs out more than 50 times in a year ( 365 days).
   (b) Justify the approximating distribution used in part (ii)(a).
3. Independently of her phone battery, Sue's computer battery also runs out at random moments. On average, it runs out twice in a 15-day period. Find the probability that the total number of times that her phone battery and her computer battery run out in a 10-day period is at least 4 .

4 At a doctors' surgery, the number of missed appointments per day has a Poisson distribution. In the past the mean number of missed appointments per day has been 0.9 . Following some publicity, the manager carries out a hypothesis test to determine whether this mean has decreased. If there are fewer than 3 missed appointments in a randomly chosen 5-day period, she will conclude that the mean has decreased.

1. Find the probability of a Type I error.
2. State what is meant by a Type I error in this context.
3. Find the probability of a Type II error if the mean number of missed appointments per day is 0.2 .

7 The number of planes arriving at an airport every hour during daytime is modelled by the random variable \(X\) with distribution \(\operatorname { Po } ( 5.2 )\).

1. State two assumptions required for the Poisson model to be valid in this context.
2. (a) Find the probability that the number of planes arriving in a 15 -minute period is greater than 1 and less than 4,
   (b) Find the probability that more than 3 planes will arrive in a 40-minute period.
3. The airport has enough staff to deal with a maximum of 60 planes landing during a 10-hour day. Use a suitable approximation to find the probability that, on a randomly chosen 10-hour day, staff will be able to deal with all the planes that land.

3 In a certain lottery, on average 1 in every 10000 tickets is a prize-winning ticket. An agent sells 6000 tickets.

1. Use a suitable approximating distribution to find the probability that at least 3 of the tickets sold by the agent are prize-winning tickets.
2. Justify the use of your approximating distribution in this context.

5 A teacher models the numbers of girls and boys who arrive late for her class on any day by the independent random variables \(G \sim \operatorname { Po } ( 0.10 )\) and \(B \sim \operatorname { Po } ( 0.15 )\) respectively.

1. Find the probability that during a randomly chosen 2-day period no girls arrive late.
2. Find the probability that during a randomly chosen 5-day period the total number of students who arrive late is less than 3 .
3. It is given that the values of \(\mathrm { P } ( G = r )\) and \(\mathrm { P } ( B = r )\) for \(r \geqslant 3\) are very small and can be ignored. Find the probability that on a randomly chosen day more girls arrive late than boys.
   Following a timetable change the teacher claims that on average more students arrive late than before the change. During a randomly chosen 5-day period a total of 4 students are late.
4. Test the teacher's claim at the \(5 \%\) significance level.

5 The number of absences per week by workers at a factory has the distribution \(\operatorname { Po } ( 2.1 )\).

1. Find the standard deviation of the number of absences per week.
2. Find the probability that the number of absences in a 2-week period is at least 2 .
3. Find the probability that the number of absences in a 3-week period is more than 4 and less than 8 .
   Following a change in working conditions, the management wished to test whether the mean number of absences has decreased. They found that, in a randomly chosen 3-week period, there were exactly 2 absences.
4. Carry out the test at the \(10 \%\) significance level.
5. State, with a reason, which of the errors, Type I or Type II, might have been made in carrying out the test in part (d).

5 Customers arrive at a shop at a constant average rate of 2.3 per minute.

1. State another condition for the number of customers arriving per minute to have a Poisson distribution.
   It is now given that the number of customers arriving per minute has the distribution \(\mathrm { Po } ( 2.3 )\).
2. Find the probability that exactly 3 customers arrive during a 1 -minute period.
3. Find the probability that more than 3 customers arrive during a 2 -minute period.
4. Five 1-minute periods are chosen at random. Find the probability that no customers arrive during exactly 2 of these 5 periods.

1 The random variable \(X\) has the distribution \(\operatorname { Po } ( 2.3 )\). Find \(\mathrm { P } ( 2 \leq X < 5 )\).

6 The number of accidents per month, \(X\), at a factory has a Poisson distribution. In the past the mean has been 1.1 accidents per month. Some new machinery is introduced and the management wish to test whether the mean has increased. They note the number of accidents in a randomly chosen month and carry out a hypothesis test at the 1\% significance level.

1. Show that the critical region for the test is \(X \geqslant 5\). Given that the number of accidents is 6 , carry out the test.
   Later they carry out a similar test, also at the \(1 \%\) significance level.
2. Explain the meaning of a Type I error in this context and state the probability of a Type I error.
3. Given that the mean is now 7.0 , find the probability of a Type II error.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

2 Cars arrive at a filling station randomly and at a constant average rate of 2.4 cars per minute.

1. Calculate the probability that fewer than 4 cars arrive in a 2 -minute period.
2. Use a suitable approximating distribution to calculate the probability that at least 140 cars arrive in a 1-hour period.

5

1. The random variable \(X\) has the distribution \(\mathrm { B } ( 300,0.01 )\). Use a Poisson approximation to find \(\mathrm { P } ( 2 < X < 6 )\).
2. The random variable \(Y\) has the distribution \(\mathrm { Po } ( \lambda )\), and \(\mathrm { P } ( Y = 0 ) = \mathrm { P } ( Y = 2 )\). Find \(\lambda\).
3. The random variable \(Z\) has the distribution \(\mathrm { Po } ( 5.2 )\) and it is given that \(\mathrm { P } ( Z = n ) < \mathrm { P } ( Z = n + 1 )\).
   1. Write down an inequality in \(n\).
   2. Hence or otherwise find the largest possible value of \(n\).

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 On average, 1 in 2500 adults has a certain medical condition.

1. Use a suitable approximation to find the probability that, in a random sample of 4000 people, more than 3 have this condition.
2. In a random sample of \(n\) people, where \(n\) is large, the probability that none has the condition is less than 0.05 . Find the smallest possible value of \(n\).

6 A shopkeeper sells electric fans. The demand for fans follows a Poisson distribution with mean 3.2 per week.

1. Find the probability that the demand is exactly 2 fans in any one week.
2. The shopkeeper has 4 fans in his shop at the beginning of a week. Find the probability that this will not be enough to satisfy the demand for fans in that week.
3. Given instead that he has \(n\) fans in his shop at the beginning of a week, find, by trial and error, the least value of \(n\) for which the probability of his not being able to satisfy the demand for fans in that week is less than 0.05 .

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.

1 The random variable \(X\) has the distribution \(\operatorname { Po } ( 1.3 )\). The random variable \(Y\) is defined by \(Y = 2 X\).

1. Find the mean and variance of \(Y\).
2. Give a reason why the variable \(Y\) does not have a Poisson distribution.

6 Customers arrive at an enquiry desk at a constant average rate of 1 every 5 minutes.

1. State one condition for the number of customers arriving in a given period to be modelled by a Poisson distribution. Assume now that a Poisson distribution is a suitable model.
2. Find the probability that exactly 5 customers will arrive during a randomly chosen 30 -minute period.
3. Find the probability that fewer than 3 customers will arrive during a randomly chosen 12-minute period.
4. Find an estimate of the probability that fewer than 30 customers will arrive during a randomly chosen 2-hour 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 \(\mathrm { P } ( T > 75 )\).

7 The number of workers, \(X\), absent from a factory on a particular day has the distribution \(\mathrm { B } ( 80,0.01 )\).

1. Explain why it is appropriate to use a Poisson distribution as an approximating distribution for \(X\).
2. Use the Poisson distribution to find the probability that the number of workers absent during 12 randomly chosen days is more than 2 and less than 6 . Following a change in working conditions, the management wishes to test whether the mean number of workers absent per day has decreased.
3. During 10 randomly chosen days, there were a total of 2 workers absent. Use the Poisson distribution to carry out the test at the \(2 \%\) significance level.

4 The number of radioactive particles emitted per 150-minute period by some material has a Poisson distribution with mean 0.7.

1. Find the probability that at most 2 particles will be emitted during a randomly chosen 10 -hour period.
2. Find, in minutes, the longest time period for which the probability that no particles are emitted is at least 0.99 .

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.

6 The number of accidents on a certain road has a Poisson distribution with mean 3.1 per 12-week period.

1. Find the probability that there will be exactly 4 accidents during an 18-week period. Following the building of a new junction on this road, an officer wishes to determine whether the number of accidents per week has decreased. He chooses 15 weeks at random and notes the number of accidents. If there are fewer than 3 accidents altogether he will conclude that the number of accidents per week has decreased. He assumes that a Poisson distribution still applies.
2. Find the probability of a Type I error.
3. Given that the mean number of accidents per week is now 0.1 , find the probability of a Type II error.
4. Given that there were 2 accidents during the 15 weeks, explain why it is impossible for the officer to make a Type II error.