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

6 The number of calls received at a small call centre has a Poisson distribution with mean 2.4 calls per 5 -minute period. Find the probability of

1. exactly 4 calls in an 8 -minute period,
2. at least 3 calls in a 3-minute period. The number of calls received at a large call centre has a Poisson distribution with mean 41 calls per 5-minute period.
3. Use an approximating distribution to find the probability that the number of calls received in a 5 -minute period is between 41 and 59 inclusive.

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 \(n\).

3 Particles are emitted randomly from a radioactive substance at a constant average rate of 3.6 per minute. Find the probability that

1. more than 3 particles are emitted during a 20 -second period,
2. more than 240 particles are emitted during a 1-hour period.

1 The random variable \(X\) has the distribution \(\operatorname { Po } ( 3.5 )\). Find \(\mathrm { P } ( X < 3 )\).

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.

5 The number of system failures per month in a large network is a random variable with the distribution \(\operatorname { Po } ( \lambda )\). A significance test of the null hypothesis \(\mathrm { H } _ { 0 } : \lambda = 2.5\) is carried out by counting \(R\), the number of system failures in a period of 6 months. The result of the test is that \(\mathrm { H } _ { 0 }\) is rejected if \(R > 23\) but is not rejected if \(R \leqslant 23\).

1. State the alternative hypothesis.
2. Find the significance level of the test.
3. Given that \(\mathrm { P } ( R > 23 ) < 0.1\), use tables to find the largest possible actual value of \(\lambda\). You should show the values of any relevant probabilities.

4 A zoologist investigates the number of snakes found in a given region of land. The zoologist intends to use a Poisson distribution to model the number of snakes.
[0pt]

1. One condition for a Poisson distribution to be valid is that snakes must occur at constant average rate. State another condition needed for a Poisson distribution to be valid. [1] Assume now that the number of snakes found in 1 acre of a region can be modelled by the distribution Po(4).
   [0pt]
2. Find the probability that, in 1 acre of the region, at least 6 snakes are found. [2]
   [0pt]
3. Find the probability that, in 0.77 acres of the region, the number of snakes found is either 2 or 3. [4]

8 The random variable \(W\) has the distribution \(\operatorname { Po } ( \lambda )\). A significance test is carried out of the null hypothesis \(\mathrm { H } _ { 0 } : \lambda = 3.60\), against the alternative hypothesis \(\mathrm { H } _ { 1 } : \lambda < 3.60\). The test is based on a single observation of \(W\). The critical region is \(W = 0\).
[0pt]

1. Find the significance level of the test. [2]
2. It is known that, when \(\boldsymbol { \lambda } = \boldsymbol { \lambda } _ { \mathbf { 0 } }\), the probability that the test results in a Type II error is \(\mathbf { 0 . 8 }\). Find the value of \(\lambda _ { 0 }\). [4] \section*{END OF QUESTION PAPER}

6 On average a motorway police force records one car that has run out of petrol every two days.

1. (a) Using a Poisson distribution, calculate the probability that, in one randomly chosen day, the police force records exactly two cars that have run out of petrol.
   (b) Using a Poisson distribution and a suitable approximation to the binomial distribution, calculate the probability that, in one year of 365 days, there are fewer than 205 days on which the police force records no cars that have run out of petrol.
2. State an assumption needed for the Poisson distribution to be appropriate in part (i), and explain why this assumption is unlikely to be valid.

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.
   [0pt]
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]

2 A large hotel has 90 bedrooms. Sometimes a guest makes a booking for a room, but then does not arrive. This is called a 'no-show'. On average \(10 \%\) of bookings are no-shows. The hotel manager accepts up to 94 bookings before saying that the hotel is full. If at least 4 of these bookings are no-shows then there will be enough rooms for all of the guests. 94 bookings have been made for each night in August. You should assume that all bookings are independent.

1. State the distribution of the number of no-shows on one night in August.
2. State the conditions under which the use of a Poisson distribution is appropriate as an approximation to a binomial distribution.
3. Use a Poisson approximating distribution to find the probability that, on one night in August,
   (A) there are exactly 4 no-shows,
   (B) there are enough rooms for all of the guests who do arrive.
4. Find the probability that, on all of the 31 nights in August, there are enough rooms for all of the guests who arrive.
5. (A) In August there are \(31 \times 94 = 2914\) bookings altogether. State the exact distribution of the total number of no-shows during August.
   (B) Use a suitable approximating distribution to find the probability that there are at most 300 no-shows altogether during August.

1 A student is collecting data on traffic arriving at a motorway service station during weekday lunchtimes. The random variable \(X\) denotes the number of cars arriving in a randomly chosen period of ten seconds.

1. State two assumptions necessary if a Poisson distribution is to provide a suitable model for the distribution of \(X\). Comment briefly on whether these assumptions are likely to be valid. The student counts the number of arrivals, \(x\), in each of 100 ten-second periods. The data are shown in the table below.

   \(x\)	0	1	2	3	4	5	\(> 5\)
   Frequency, \(f\)	18	39	20	12	8	3	0

2. Show that the sample mean is 1.62 and calculate the sample variance.
3. Do your calculations in part (ii) support the suggestion that a Poisson distribution is a suitable model for the distribution of \(X\) ? Explain your answer. For the remainder of this question you should assume that \(X\) may be modelled by a Poisson distribution with mean 1.62 .
4. Find \(\mathrm { P } ( X = 2 )\). Comment on your answer in relation to the data in the table.
5. Find the probability that at least ten cars arrive in a period of 50 seconds during weekday lunchtimes.
6. Use a suitable approximating distribution to find the probability that no more than 550 cars arrive in a randomly chosen period of one hour during weekday lunchtimes.

1 A low-cost airline charges for breakfasts on its early morning flights. On average, \(10 \%\) of passengers order breakfast.

1. Find the probability that, out of 8 randomly selected passengers, exactly 1 orders breakfast.
2. Use a suitable Poisson approximating distribution to find the probability that the number of breakfasts ordered by 30 randomly selected passengers is
   (A) exactly 6,
   (B) at least 8 .
3. State the conditions under which the use of a Poisson distribution is appropriate as an approximation to a binomial distribution.
4. The aircraft carries 120 passengers and the flight is always full. Find the mean \(\mu\) and variance \(\sigma ^ { 2 }\) of a Normal approximating distribution suitable for modelling the total number of passengers on the flight who order breakfast.
5. Use your Normal approximating distribution to calculate the probability that more than 15 breakfasts are ordered on a particular flight.
6. The airline wishes to be at least \(99 \%\) certain that the plane will have sufficient breakfasts for all passengers who order them. Find the minimum number of breakfasts which should be carried on each flight.

3 The number of calls received at an office per 5 minutes is modelled by a Poisson distribution with mean 3.2.

1. Find the probability of
   (A) exactly one call in a 5 -minute period,
   (B) at least 6 calls in a 5 -minute period.
2. Find the probability of
   (A) exactly one call in a 1 -minute period,
   (B) exactly one call in each of five successive 1-minute periods.
3. Use a suitable approximating distribution to find the probability of at most 45 calls in a period of 1 hour. Two assumptions required for a Poisson distribution to be a suitable model are that calls arrive

2 A public water supply contains bacteria. Each day an analyst checks the water quality by counting the number of bacteria in a random sample of 5 ml of water. Throughout this question, you should assume that the bacteria occur randomly at a mean rate of 0.37 bacteria per 5 ml of water.

1. Use a Poisson distribution to
   (A) find the probability that a 5 ml sample contains exactly 2 bacteria,
   (B) show that the probability that a 5 ml sample contains more than 2 bacteria is 0.0064 .
2. The month of September has 30 days. Find the probability that during September there is at most one day when a 5 ml sample contains more than 2 bacteria. The daily 5 ml sample is the first stage of the quality control process. The remainder of the process is as follows.

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.

5 A music store sells both upright and grand pianos. Grand pianos are sold at random times and at a constant average weekly rate \(\lambda\). The probability that in one week no grand pianos are sold is 0.45 .

1. Show that \(\lambda = 0.80\), correct to 2 decimal places. Upright pianos are sold, independently, at random times and at a constant average weekly rate \(\mu\). During a period of 100 weeks the store sold 180 upright pianos.
2. Calculate the probability that the total number of pianos sold in a randomly chosen week will exceed 3.
3. Calculate the probability that over a period of 3 weeks the store sells a total of 6 pianos during the first week and a total of 4 pianos during the next fortnight.

2 The random variable \(X\) has the Poisson distribution with parameter \(\lambda\).

1. Show that the probability generating function of \(X\) is \(\mathrm { G } ( t ) = \mathrm { e } ^ { \lambda ( t - 1 ) }\).
2. Hence obtain the mean \(\mu\) and variance \(\sigma ^ { 2 }\) of \(X\).
3. Write down the mean and variance of the random variable \(Z = \frac { X - \mu } { \sigma }\).
4. Write down the moment generating function of \(X\). State the linear transformation result for moment generating functions and use it to show that the moment generating function of \(Z\) is $$\mathrm { M } _ { Z } ( \theta ) = \mathrm { e } ^ { \mathrm { f } ( \theta ) } \quad \text { where } \mathrm { f } ( \theta ) = \lambda \left( \mathrm { e } ^ { \theta / \sqrt { \lambda } } - \frac { \theta } { \sqrt { \lambda } } - 1 \right)$$
5. Show that the limit of \(\mathrm { M } _ { Z } ( \theta )\) as \(\lambda \rightarrow \infty\) is \(\mathrm { e } ^ { \theta ^ { 2 } / 2 }\).
6. Explain briefly why this implies that the distribution of \(Z\) tends to \(\mathrm { N } ( 0,1 )\) as \(\lambda \rightarrow \infty\). What does this imply about the distribution of \(X\) as \(\lambda \rightarrow \infty\) ?

8 In a large city the number of traffic lights that fail in one day of 24 hours is denoted by \(Y\). It may be assumed that failures occur randomly.

1. Explain what the statement "failures occur randomly" means.
2. State, in context, two different conditions that must be satisfied if \(Y\) is to be modelled by a Poisson distribution, and for each condition explain whether you think it is likely to be met in this context.
3. For this part you may assume that \(Y\) is well modelled by the distribution \(\operatorname { Po } ( \lambda )\). It is given that \(\mathrm { P } ( Y = 7 ) = \mathrm { P } ( Y = 8 )\). Use an algebraic method to calculate the value of \(\lambda\) and hence calculate the corresponding value of \(\mathrm { P } ( Y = 7 )\).

2 A class investigated the number of dead rabbits found along a particular stretch of road.

1. The class agrees that dead rabbits occur randomly along the road. Explain what this statement means.
2. State, in this context, an assumption needed for the number of dead rabbits in a fixed length of road to be modelled by a Poisson distribution, and explain what your statement means. Assume now that the number of dead rabbits in a fixed length of road can be well modelled by a Poisson distribution with mean 1 per 600 m of road.
3. Use an appropriate formula, showing your working, to find the probability that in a road of length 1650 m there are exactly 3 dead rabbits.

7 A large railway network suffers points failures at an average rate of 1 every 3 days. Assume that the number of points failures can be modelled by a Poisson distribution. The network employs a new firm of engineers. After the new engineers have become established, it is found that in a randomly chosen period of 15 days there are 2 instances of points failures.

1. Test, at the \(5 \%\) significance level, whether there is evidence that the mean number of points failures has been reduced.
2. A new test is carried out over a period of 150 days. Use a suitable approximation to find the greatest number of points failures there could be in 150 days that would lead to a \(5 \%\) significance test concluding that the average number of points failures had been reduced.

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.

1 A newspaper article consists of 800 words. For each word, the probability that it is misprinted is 0.005 , independently of all other words. Use a suitable approximation to find the probability that the total number of misprinted words in the article is no more than 6 . Give a reason to justify your approximation.

3 The number of incidents of radio interference per hour experienced by a certain listener is modelled by a random variable with distribution \(\operatorname { Po } ( 0.42 )\).

1. Find the probability that the number of incidents of interference in one randomly chosen hour is
   1. 0 ,
   2. exactly 1 .
   3. Find the probability that the number of incidents in a randomly chosen 5-hour period is greater than 3.
   4. One hundred hours of listening are monitored and the numbers of 1 -hour periods in which 0,1 , \(2 , \ldots\) incidents of interference are experienced are noted. A bar chart is drawn to represent the results. Without any further calculations, sketch the shape that you would expect for the bar chart. (There is no need to use an exact numerical scale on the frequency axis.)