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

5 It is proposed to model the number of people per hour calling a car breakdown service between the times 0900 and 2100 by a Poisson distribution.

1. Explain why a Poisson distribution may be appropriate for this situation. People call the car breakdown service at an average rate of 20 per hour, and a Poisson distribution may be assumed to be a suitable model.
2. Find the probability that exactly 8 people call in any half hour.
3. By using a suitable approximation, find the probability that exactly 250 people call in the 12 hours between 0900 and 2100.

5 When a guitar is played regularly, a string breaks on average once every 15 months. Broken strings occur at random times and independently of each other.

1. Show that the mean number of broken strings in a 5 -year period is 4 . A guitar is fitted with a new type of string which, it is claimed, breaks less frequently. The number of broken strings of the new type was noted after a period of 5 years.
2. The mean number of broken strings of the new type in a 5 -year period is denoted by \(\lambda\). Find the rejection region for a test at the \(10 \%\) significance level when the null hypothesis \(\lambda = 4\) is tested against the alternative hypothesis \(\lambda < 4\).
3. Hence calculate the probability of making a Type I error. The number of broken guitar strings of the new type, in a 5 -year period, was in fact 1 .
4. State, with a reason, whether there is evidence at the \(10 \%\) significance level that guitar strings of the new type break less frequently.

6 People arrive randomly and independently at the elevator in a block of flats at an average rate of 4 people every 5 minutes.

1. Find the probability that exactly two people arrive in a 1-minute period.
2. Find the probability that nobody arrives in a 15 -second period.
3. The probability that at least one person arrives in the next \(t\) minutes is 0.9 . Find the value of \(t\).

3 Major avalanches can be regarded as randomly occurring events. They occur at a uniform average rate of 8 per year.

1. Find the probability that more than 3 major avalanches occur in a 3-month period.
2. Find the probability that any two separate 4 -month periods have a total of 7 major avalanches.
3. Find the probability that a total of fewer than 137 major avalanches occur in a 20 -year period.

6 In restaurant \(A\) an average of 2.2\% of tablecloths are stained and, independently, in restaurant \(B\) an average of 5.8\% of tablecloths are stained.

1. Random samples of 55 tablecloths are taken from each restaurant. Use a suitable Poisson approximation to find the probability that a total of more than 2 tablecloths are stained.
2. Random samples of \(n\) tablecloths are taken from each restaurant. The probability that at least one tablecloth is stained is greater than 0.99 . Find the least possible value of \(n\).

7 A hospital patient's white blood cell count has a Poisson distribution. Before undergoing treatment the patient had a mean white blood cell count of 5.2. After the treatment a random measurement of the patient's white blood cell count is made, and is used to test at the \(10 \%\) significance level whether the mean white blood cell count has decreased.

1. State what is meant by a Type I error in the context of the question, and find the probability that the test results in a Type I error.
2. Given that the measured value of the white blood cell count after the treatment is 2 , carry out the test.
3. Find the probability of a Type II error if the mean white blood cell count after the treatment is actually 4.1.

4 At a power plant, the number of breakdowns per year has a Poisson distribution. In the past the mean number of breakdowns per year has been 4.8. Following some repairs, the management carry out a hypothesis test at the 5\% significance level to determine whether this mean has decreased. If there is at most 1 breakdown in the following year, they will conclude that the mean has decreased.

1. State what is meant by a Type I error in this context.
2. Find the probability of a Type I error.
3. Find the probability of a Type II error if the mean is now 0.9 breakdowns per year.

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 On average, 1 in 2500 people have a particular gene.

1. Use a suitable approximation to find the probability that, in a random sample of 10000 people, more than 3 people have this gene.
2. The probability that, in a random sample of \(n\) people, none of them has the gene is less than 0.01 . Find the smallest possible value of \(n\).

1 The number of new enquiries per day at an office has a Poisson distribution. In the past the mean has been 3 . Following a change of staff, the manager wishes to test, at the \(5 \%\) significance level, whether the mean has increased.

1. State the null and alternative hypotheses for this test. The manager notes the number, \(N\), of new enquiries during a certain 6 -day period. She finds that \(N = 25\) and then, assuming that the null hypothesis is true, she calculates that \(\mathrm { P } ( N \geqslant 25 ) = 0.0683\).
2. What conclusion should she draw?

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 \(n\).

5 The probability that a new car of a certain type has faulty brakes is 0.008 . A random sample of 520 new cars of this type is chosen, and the number, \(X\), having faulty brakes is noted.

1. Describe fully the distribution of \(X\) and describe also a suitable approximating distribution. Justify this approximating distribution.
2. Use your approximating distribution to find
   1. \(\mathrm { P } ( X > 3 )\),
   2. the smallest value of \(n\) such that \(\mathrm { P } ( X = n ) > \mathrm { P } ( X = n + 1 )\).

1 It is known that \(1.2 \%\) of rods made by a certain machine are bent. The random variable \(X\) denotes the number of bent rods in a random sample of 400 rods.

1. State the distribution of \(X\).
2. State, with a reason, a suitable approximate distribution for \(X\).
3. Use your approximate distribution to find the probability that the sample will include more than 2 bent rods.

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.

4 The independent random variables \(X\) and \(Y\) have the distributions \(\operatorname { Po } ( 2 )\) and \(\operatorname { Po } ( 3 )\) respectively.

1. Given that \(X + Y = 5\), find the probability that \(X = 1\) and \(Y = 4\).
2. Given that \(\mathrm { P } ( X = r ) = \frac { 2 } { 3 } \mathrm { P } ( X = 0 )\), show that \(3 \times 2 ^ { r - 1 } = r\) ! and verify that \(r = 4\) satisfies this equation.

6 Calls arrive at a helpdesk randomly and at a constant average rate of 1.4 calls per hour. Calculate the probability that there will be

1. more than 3 calls in \(2 \frac { 1 } { 2 }\) hours,
2. fewer than 1000 calls in four weeks ( 672 hours).

8

1. The following tables show the probability distributions for the random variables \(V\) and \(W\).

   \(v\)	- 1	0	1	\(> 1\)
   \(\mathrm { P } ( V = v )\)	0.368	0.368	0.184	0.080

   \(w\)	0	0.5	1	\(> 1\)
   \(\mathrm { P } ( W = w )\)	0.368	0.368	0.184	0.080

   For each of the variables \(V\) and \(W\) state how you can tell from its probability distribution that it does NOT have a Poisson distribution.
2. The random variable \(X\) has the distribution \(\operatorname { Po } ( \lambda )\). It is given that $$\mathrm { P } ( X = 0 ) = p \quad \text { and } \quad \mathrm { P } ( X = 1 ) = 2.5 p$$ where \(p\) is a constant.
   1. Show that \(\lambda = 2.5\).
   2. Find \(\mathrm { P } ( X \geqslant 3 )\).
   3. The random variable \(Y\) has the distribution \(\operatorname { Po } ( \mu )\), where \(\mu > 30\). Using a suitable approximating distribution, it is found that \(\mathrm { P } ( Y > 40 ) = 0.5793\) correct to 4 decimal places. Find \(\mu\).

7 A Lost Property office is open 7 days a week. It may be assumed that items are handed in to the office randomly, singly and independently.

1. State another condition for the number of items handed in to have a Poisson distribution. It is now given that the number of items handed in per week has the distribution \(\operatorname { Po } ( 4.0 )\).
2. Find the probability that exactly 2 items are handed in on a particular day.
3. Find the probability that at least 4 items are handed in during a 10-day period.
4. Find the probability that, during a certain week, 5 items are handed in altogether, but no items are handed in on the first day of the week.

6 A publishing firm has found that errors in the first draft of a new book occur at random and that, on average, there is 1 error in every 3 pages of a first draft. Find the probability that in a particular first draft there are

1. exactly 2 errors in 10 pages,
2. at least 3 errors in 6 pages,
3. fewer than 50 errors in 200 pages.

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.

6 Accidents on a particular road occur at a constant average rate of 1 every 4.8 weeks.

1. State, in context, one condition for the number of accidents 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 4 accidents will occur during a randomly chosen 12-week period.
3. Find the probability that more than 3 accidents will occur during a randomly chosen 10 -week period.
4. Use a suitable approximating distribution to find the probability that fewer than 30 accidents will occur during a randomly chosen 2 -year period ( \(104 \frac { 2 } { 7 }\) weeks).

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 \(\lambda\).