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

2 John works from home. The number of business letters, \(X\), that he receives on a weekday may be modelled by a Poisson distribution with mean 5.0. The number of private letters, \(Y\), that he receives on a weekday may be modelled by a Poisson distribution with mean 1.5.

1. Find, for a given weekday:
   1. \(\mathrm { P } ( X < 4 )\);
   2. \(\quad \mathrm { P } ( Y = 4 )\).
2. 1. Assuming that \(X\) and \(Y\) are independent random variables, determine the probability that, on a given weekday, John receives a total of more than 5 business and private letters.
   2. Hence calculate the probability that John receives a total of more than 5 business and private letters on at least 7 out of 8 given weekdays.
3. The numbers of letters received by John's neighbour, Brenda, on 10 consecutive weekdays are $$\begin{array} { l l l l l l l l l l } 15 & 8 & 14 & 7 & 6 & 8 & 2 & 8 & 9 & 3 \end{array}$$
   1. Calculate the mean and the variance of these data.
   2. State, giving a reason based on your answers to part (c)(i), whether or not a Poisson distribution might provide a suitable model for the number of letters received by Brenda on a weekday.

1 In a study of urban foxes it is found that on average there are 2 foxes in every 3 acres.

1. Use a Poisson distribution to find the probability that, at a given moment,
   1. in a randomly chosen area of 3 acres there are at least 4 foxes,
   2. in a randomly chosen area of 1 acre there are exactly 2 foxes.
   3. Explain briefly why a Poisson distribution might not be a suitable model.

4 DVD players are tested after manufacture. The probability that a randomly chosen DVD player is defective is 0.02 . The number of defective players in a random sample of size 80 is denoted by \(R\).

1. Use an appropriate approximation to find \(\mathrm { P } ( R \geqslant 2 )\).
2. Find the smallest value of \(r\) for which \(\mathrm { P } ( R \geqslant r ) < 0.01\).

5 The number of letters per week received at home by Rosa may be modelled by a Poisson distribution with parameter 12.25.

1. Using a normal approximation, estimate the probability that, during a 4 -week period, Rosa receives at home at least 42 letters but at most 54 letters.
2. Rosa also receives letters at work. During a 16-week period, she receives at work a total of 248 letters.
   1. Assuming that the number of letters received at work by Rosa may also be modelled by a Poisson distribution, calculate a \(98 \%\) confidence interval for the average number of letters per week received at work by Rosa.
   2. Hence comment on Rosa's belief that she receives, on average, fewer letters at home than at work.

7 In a town, the total number, \(R\), of houses sold during a week by estate agents may be modelled by a Poisson distribution with a mean of 13 . A new housing development is completed in the town. During the first week in which houses on this development are offered for sale by the developer, the estate agents sell a total of 10 houses.

1. Using the \(10 \%\) level of significance, investigate whether the offer for sale of houses by the developer has resulted in a reduction in the mean value of \(R\).
2. Determine, for your test in part (a), the critical region for \(R\).
3. Assuming that the offer for sale of houses on the new housing development has reduced the mean value of \(R\) to 6.5, determine, for a test at the 10\% level of significance, the probability of a Type II error.
   (4 marks)

2 The discrete random variable \(Y\) has a Poisson distribution with mean 3 Find the value of \(\mathrm { P } ( Y > 1 )\) to three significant figures.
Circle your answer. \(0.149 \quad 0.199 \quad 0.801 \quad 0.950\)

7 Over a period of time it has been shown that the mean number of vehicles passing a service station on a motorway is 50 per minute. After a new motorway junction was built nearby, Xander observed that 30 vehicles passed the service station in one minute. 7

1. Xander claims that the construction of the new motorway junction has reduced the mean number of vehicles passing the service station per minute. Investigate Xander's claim, using a suitable test at the \(1 \%\) level of significance.
   7
2. For your test carried out in part (a) state, in context, the meaning of a Type 1 error. 7
3. Explain why the model used in part (a) might be invalid.

6 A company owns two machines, \(A\) and \(B\), which make toys. Both machines run continuously and independently. Machine \(A\) makes an average of 2 errors per hour.
6

1. Using a Poisson model, find the probability that the machine makes exactly 5 errors in 4 hours, giving your answer to three significant figures. 6
2. Machine \(B\) makes an average of 5 errors per hour. Both machines are switched on and run for 1 hour. The company finds the probability that the total number of errors made by machines \(A\) and \(B\) in 1 hour is greater than 8 . If the probability is greater than 0.4 , a new machine will be purchased.
   Using a Poisson model, determine whether or not the toy company will purchase a new machine.
   6
3. After investigation, the standard deviation of errors made by machine \(A\) is found to be 0.5 errors per hour and the standard deviation of errors made by machine \(B\) is also found to be 0.5 errors per hour. Explain whether or not the use of Poisson models in parts (a) and (b) is appropriate.

6 An insurance company models the number of motor claims received in 1 day using a Poisson distribution with mean 65 6

1. Find the probability that the company receives at most 60 motor claims in 1 day. Give your answer to three decimal places. 6
2. The company receives motor claims using a telephone line which is open 24 hours a day. Find the probability that the company receives exactly 2 motor claims in 1 hour. Give your answer to three decimal places.
   6
3. The company models the number of property claims received in 1 day using a Poisson distribution with mean 23 Assume that the number of property claims received is independent of the number of motor claims received. 6 (c) (i) Find the standard deviation of the variable that represents the total number of motor claims and property claims received in 1 day. Give your answer to three significant figures.
   6 (c) (ii) Find the probability that the company receives a total of more than 90 motor claims and property claims in 1 day. Give your answer to three significant figures.

7 Over a period of time, it has been shown that the mean number of customers entering a small store is 6 per hour. The store runs a promotion, selling many products at lower prices. 7

1. Luke randomly selects an hour during the promotion and counts 11 customers entering the store. He claims that the promotion has changed the mean number of customers per hour entering the store. Investigate Luke's claim, using the \(5 \%\) level of significance.
   7
2. Luke randomly selects another hour and carries out the same investigation as in part (a). Find the probability of a Type I error, giving your answer to four decimal places.
   Fully justify your answer.
   7
3. When observing the store, Luke notices that some customers enter the store together as a group. Explain why the model used in parts (a) and (b) might not be valid.
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256 2 The random variable \(T\) has an exponential distribution with mean 2 Find \(\mathrm { P } ( T \leq 1.4 )\) Circle your answer. \(\mathrm { e } ^ { - 2.8 }\) \(\mathrm { e } ^ { - 0.7 }\) \(1 - e ^ { - 0.7 }\) \(1 - \mathrm { e } ^ { - 2.8 }\) The continuous random variable \(Y\) has cumulative distribution function $$\mathrm { F } ( y ) = \left\{ \begin{array} { l r } 0 & y < 2 \\ - \frac { 1 } { 9 } y ^ { 2 } + \frac { 10 } { 9 } y - \frac { 16 } { 9 } & 2 \leq y < 5 \\ 1 & y \geq 5 \end{array} \right.$$ Find the median of \(Y\) Circle your answer. 2 \(\frac { 10 - 3 \sqrt { 2 } } { 2 }\) \(\frac { 7 } { 2 }\) \(\frac { 10 + 3 \sqrt { 2 } } { 2 }\) Turn over for the next question 4 Research has shown that the mean number of volcanic eruptions on Earth each day is 20 Sandra records 162 volcanic eruptions during a period of one week. Sandra claims that there has been an increase in the mean number of volcanic eruptions per week. Test Sandra's claim at the \(5 \%\) level of significance.
5 The continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} \frac { 1 } { 6 } e ^ { \frac { x } { 3 } } & 0 \leq x \leq \ln 27 \\ 0 & \text { otherwise } \end{cases}$$ Show that the mean of \(X\) is \(\frac { 3 } { 2 } ( \ln 27 - 2 )\) 6 Over time it has been accepted that the mean retirement age for professional baseball players is 29.5 years old. Imran claims that the mean retirement age is no longer 29.5 years old.
He takes a random sample of 5 recently retired professional baseball players and records their retirement ages, \(x\). The results are $$\sum x = 152.1 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 7.81$$ 6

1. State an assumption that you should make about the distribution of the retirement ages to investigate Imran's claim. 6
2. Investigate Imran's claim, using the 10\% level of significance.

16
256 2 The random variable \(T\) has an exponential distribution with mean 2 Find \(\mathrm { P } ( T \leq 1.4 )\) Circle your answer. \(\mathrm { e } ^ { - 2.8 }\) \(\mathrm { e } ^ { - 0.7 }\) \(1 - e ^ { - 0.7 }\) \(1 - \mathrm { e } ^ { - 2.8 }\) The continuous random variable \(Y\) has cumulative distribution function $$\mathrm { F } ( y ) = \left\{ \begin{array} { l r } 0 & y < 2 \\ - \frac { 1 } { 9 } y ^ { 2 } + \frac { 10 } { 9 } y - \frac { 16 } { 9 } & 2 \leq y < 5 \\ 1 & y \geq 5 \end{array} \right.$$ Find the median of \(Y\) Circle your answer. 2 \(\frac { 10 - 3 \sqrt { 2 } } { 2 }\) \(\frac { 7 } { 2 }\) \(\frac { 10 + 3 \sqrt { 2 } } { 2 }\) Turn over for the next question 4 Research has shown that the mean number of volcanic eruptions on Earth each day is 20 Sandra records 162 volcanic eruptions during a period of one week. Sandra claims that there has been an increase in the mean number of volcanic eruptions per week. Test Sandra's claim at the \(5 \%\) level of significance.
5 The continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} \frac { 1 } { 6 } e ^ { \frac { x } { 3 } } & 0 \leq x \leq \ln 27 \\ 0 & \text { otherwise } \end{cases}$$ Show that the mean of \(X\) is \(\frac { 3 } { 2 } ( \ln 27 - 2 )\) 6 Over time it has been accepted that the mean retirement age for professional baseball players is 29.5 years old. Imran claims that the mean retirement age is no longer 29.5 years old.
He takes a random sample of 5 recently retired professional baseball players and records their retirement ages, \(x\). The results are $$\sum x = 152.1 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 7.81$$ 6

1. State an assumption that you should make about the distribution of the retirement ages to investigate Imran's claim. 6
2. Investigate Imran's claim, using the 10\% level of significance.

8 Natural background radiation consists of various particles, including neutrons. A detector is used to count the number of neutrons per second at a particular location.

1. State the conditions required for a Poisson distribution to be a suitable model for the number of neutrons detected per second. The number of neutrons detected per second due to background radiation only is modelled by a Poisson distribution with mean 1.1.
2. Find the probability that the detector detects
   (A) no neutrons in a randomly chosen second,
   (B) at least 60 neutrons in a randomly chosen period of 1 minute. A neutron source is switched on. It emits neutrons which should all be contained in a protective casing. The detector is used to check whether any neutrons have not been contained; these are known as stray neutrons. If the detector detects more than 8 neutrons in a period of 1 second, an alarm will be triggered in case this high reading is due to stray neutrons.
3. Suppose that there are no stray neutrons and so the neutrons detected are all due to the background radiation. Find the expected number of times the alarm is triggered in 1000 randomly chosen periods of 1 second.
4. Suppose instead that stray neutrons are being produced at a rate of 3.4 per second in addition to the natural background radiation. Find the probability that at least one alarm will be triggered in 10 randomly chosen periods of 1 second. You should assume that all stray neutrons produced are detected.

2 In the manufacture of fibre optical cable (FOC), flaws occur randomly. Whether any point on a cable is flawed is independent of whether any other point is flawed. The number of flaws in 100 m of FOC of standard diameter is denoted by \(X\).

1. State a further assumption needed for \(X\) to be well modelled by a Poisson distribution. Assume now that \(X\) can be well modelled by the distribution \(\operatorname { Po } ( 0.7 )\).
2. Find the probability that in 300 m of FOC of standard diameter there are exactly 3 flaws. The number of flaws in 100 m of FOC of a larger diameter has the distribution \(\mathrm { Po } ( 1.6 )\).
3. Find the probability that in 200 m of FOC of standard diameter and 100 m of FOC of the larger diameter the total number of flaws is at least 4 . Judith believes that mathematical ability and chess-playing ability are related. She asks 20 randomly chosen chess players, with known British Chess Federation (BCF) ratings \(X\), to take a mathematics aptitude test, with scores \(Y\). The results are summarised as follows. $$n = 20 , \Sigma x = 3600 , \Sigma x ^ { 2 } = 660500 , \Sigma y = 1440 , \Sigma y ^ { 2 } = 105280 , \Sigma x y = 260990$$
   1. Calculate the value of Pearson's product-moment correlation coefficient \(r\).
   2. State an assumption needed to be able to carry out a significance test on the value of \(r\).
   3. Assume now that the assumption in part (b) is valid. Test at the \(5 \%\) significance level whether there is evidence that chess players with higher BCF ratings are better at mathematics.
   4. There are two different grading systems for chess players, the BCF system and the international ELO system. The two sets of ratings are related by $$\text { ELO rating } = 8 \times \text { BCF rating } + 650$$ Magnus says that the experiment should have used ELO ratings instead of BCF ratings. Comment on Magnus's suggestion.
   5. Calculate the value of Pearson's product-moment correlation coefficient \(r\).
   6. State an assumption needed to be able to carry out a significance test on the value of \(r\).
   7. Assume now that the assumption in part (b) is valid. Test at the \(5 \%\) significance level whether there is evidence that chess players with higher BCF ratings are better at mathematics.
   8. There are two different grading systems for chess players, the BCF system and the international ELO system. The two sets of ratings are related by $$\mathrm { ELO } \text { rating } = 8 \times \mathrm { BCF } \text { rating } + 650 .$$ Magnus says that the experiment should have used ELO ratings instead of BCF ratings. Comment on Magnus's suggestion. An environmentalist measures the mean concentration, \(c\) milligrams per litre, of a particular chemical in a group of rivers, and the mean mass, \(m\) pounds, of fish of a certain species found in those rivers. The results are given in the table.

      Question			Answer	Marks	AO	Guidance
      1	(a)		\(\begin{aligned}	0.25 + 0.36 + x + x ^ { 2 } = 1
      	x ^ { 2 } + x - 0.39 = 0
      	x = 0.3 \text { (or } - 1.3 \text { ) }
      	x \text { cannot be negative }
      	\mathrm { E } ( W ) = 2.23
      	\mathrm { E } \left( W ^ { 2 } \right) = \Sigma w ^ { 2 } \mathrm { p } ( w ) \quad [ = 5.83 ]
      	\text { Subtract } [ \mathrm { E } ( W ) ] ^ { 2 } \text { to get } \mathbf { 0 . 8 5 7 1 } \end{aligned}\)	\(\begin{gathered} \text { M1 }
      \text { A1 }
      \text { A1 }
      \text { B1ft }
      \text { B1 }
      \text { M1 }
      \text { A1 }
      { [ 7 ] } \end{gathered}\)	3.1a 1.1b 1.1b 2.3 1.1b 1.1 2.1	Equation using \(\Sigma p = 1\) Correct simplified quadratic Correctly obtain \(x = 0.3\) Explicitly reject other solution 2.23 or exact equivalent only Use \(\Sigma w ^ { 2 } \mathrm { p } ( w )\) Correctly obtain given answer, www	Can be implied Method needed ft on their quadratic Allow for \(\mathrm { E } ( W ) ^ { 2 } = 4.9729\) Need 2.23 or 4.9729 and 5.83 or full numerical \(\Sigma w ^ { 2 } \mathrm { p } ( w )\)
      1	(b)		\(9 \times 0.8571 = 7.7139\)	B1 [1]	1.1b	Allow 7.71 or 7.714
      2	(a)		Flaws must occur at constant average rate (uniform rate)	B1 [1]	1.2	Context (e.g. "flaws") needed Extra answers, e.g. "singly": B0	Not "constant rate" or "average constant rate".
      2	(b)		\(\operatorname { Po(2.1)~or~ } e ^ { - \lambda } \frac { \lambda ^ { 3 } } { 3 ! }\)	M1 A1 [2]	1.1 1.1b	Po(2.1) stated or implied, or formula with \(\lambda = 2.1\) stated Awrt 0.189
      2	(c)		Po(3) \(1 - \mathrm { P } ( \leq 3 )\)	M1 M1 A1 [3]	1.1 1.1 1.1b	\(\operatorname { Po } ( 2 \times 0.7 + 1.6 )\) stated or implied Allow \(1 - \mathrm { P } ( \leq 4 ) = 0.1847\), or from wrong \(\lambda\) Awrt 0.353	Or all combinations \(\leq 3\) \(1 -\) above, not just \(= 3\)

      Question			Answer	Marks	AO	Guidance
      3	(a)		0.4(00)	B2 [2]	1.1 1.1b	SC: if B0, give SC B1 for two of \(S _ { x x } = 12500 , S _ { y y } = 1600 , S _ { x y } = 1790\) and \(S _ { x y } / \sqrt { } \left( S _ { x x } S _ { y y } \right)\)	Also allow SC B1 for equivalent methods using Covariance \	SDs
      3	(b)		Data needs to have a bivariate normal distribution	B1 [1]	1.2	Needs "bivariate normal" or clear equivalent. Not just "both normally distributed"	Allow "scatter diagram forms ellipse"
      3	(c)		\(\mathrm { H } _ { 0 }\) : higher maths scores are not associated with higher BCF grading; \(\mathrm { H } _ { 1 }\) : positively associated CV 0.3783 \(0.400 > 0.3783\) so reject \(\mathrm { H } _ { 0 }\) Significant evidence that higher maths scores are associated with higher BCF grading	B1 B1 M1ft A1ft [4]	2.5 1.1b 2.2b 3.5a	Needs context and clearly onetailed \(O R \rho\) used and defined Not "evidence that ..." Allow 0.378 Reject/do not reject \(\mathrm { H } _ { 0 }\) Contextualised, not too definite Needn't say "positive" if \(\mathrm { H } _ { 1 } \mathrm { OK }\) SC 2-tail: B0; 0.4438, or 0.3783 B1; then M1A0	\(\mathrm { H } _ { 0 } : \rho = 0 , \mathrm { H } _ { 1 } : \rho > 0\) where \(\rho\) is population pmcc (not \(r\) ) FT on their \(r\), but not CV Not "scores are associated ...". FT on their \(r\) only
      3	(d)		It makes no difference as this is a linear transformation	B1 [1]	2.2a	Need both "unchanged" oe and reason, need "linear" or exact equivalent	"oe" includes "their 0.4"
      4	(a)		Neither	B1 [1]	2.5	OE	Not "neither is independent of the other"
      4	(b)		\(c = 2.848 - 0.1567 m\)	B1 B1 B1 [3]	1.1 1.1 1.1	Correct \(a\), awrt 2.85 Correct \(b\), awrt 0.157 Letters correct from correct method (If both wrongly rounded, e.g. \(c = 2.84 - 0.156 m\), give B2)	\(\mathrm { SC } : m\) on \(c\) : \(m = 15.65 - 4.832 c\) : B2 \(y = 15.65 - 4.832 x\) : B1 \(c = 15.65 - 4.832 m : \mathrm { B } 1\) If B0B0, give B1 for correct letters from valid working

      Question		Answer	Marks	AO	Guidance
      4	(c)	\(a\) unchanged, \(b\) multiplied by 2.2 (allow " \(a\) unchanged, \(b\) increases", etc)	B1 [1]	2.2a	oe, e.g. \(c = 2.848 - 0.345 m\); \(m = 7.114 - 2.196 c\)	SC: \(m\) on \(c\) in (b): Both divided by 2.2 B1
      4	(d)	Draw approximate line of best fit Draw at least one vertical from line to point Say that "Best fit" line minimises the sum of squares of these distances	M1 M1 A1 [3]	1.1 2.4 2.4	Needs M2 and "minimises" and "sums of squares" oe	SC: Horizontal(s): full marks (indept of (b))

1 On any day, the number of orders received in one randomly chosen hour by an online supplier can be modelled by the distribution \(\mathrm { Po } ( 120 )\).

1. Find the probability that at least 28 orders are received in a randomly chosen 10 -minute period.
2. Find the probability that in a randomly chosen 10-minute period on one day and a randomly chosen 10-minute period on the next day a total of at least 56 orders are received.
3. State a necessary assumption for the validity of your calculation in part (b).

3 The numbers of CD players sold in a shop on three consecutive weekends were 7,6 and 2 . It may be assumed that sales of CD players occur randomly and that nobody buys more than one CD player at a time. The number of CD players sold on a randomly chosen weekend is denoted by \(X\).

1. How appropriate is the Poisson distribution as a model for \(X\) ? Now assume that a Poisson distribution with mean 5 is an appropriate model for \(X\).
2. Find
   1. \(\mathrm { P } ( X = 6 )\),
   2. \(\mathrm { P } ( X \geqslant 8 )\). The number of integrated sound systems sold in a weekend at the same shop can be assumed to have the distribution \(\operatorname { Po } ( 7.2 )\).
3. Find the probability that on a randomly chosen weekend the total number of CD players and integrated sound systems sold is between 10 and 15 inclusive.
4. State an assumption needed for your answer to part (c) to be valid.
5. Give a reason why the assumption in part (d) may not be valid in practice.

9

1. Two independent discrete random variables \(X\) and \(Y\) follow Poisson distributions with means \(\lambda\) and \(\mu\) respectively. Prove that the discrete random variable \(Z = X + Y\) follows a Poisson distribution with mean \(\lambda + \mu\). A garage has a white limousine and a green limousine for hire. Demands to hire the white limousine occur at a constant mean rate of 3 per week and demands to hire the green limousine occur at a constant mean rate of 2 per week. Demands for hire are received independently and randomly.
2. Calculate the probability that in a period of two weeks
   1. no demands for hire are received, giving your answer to 3 significant figures,
   2. seven demands for hire are received.
   3. Find the least value of \(n\) such that the probability of at least \(n\) demands for hire in a period of three weeks is less than 0.005 .

2 The discrete random variable \(X\) has a Poisson distribution with mean 12.25.

1. Calculate \(\mathrm { P } ( X \leqslant 5 )\).
2. Calculate an approximate value for \(\mathrm { P } ( X \leqslant 5 )\) using a normal approximation to the Poisson distribution.
3. Comment, giving a reason, on the accuracy of using a normal approximation to the Poisson distribution in this case.

4

1. The random variable \(X\) has the distribution \(\operatorname { Po } ( \lambda )\). Prove that the probability generating function, \(\mathrm { G } _ { X } ( t )\), is given by $$\mathrm { G } _ { X } ( t ) = \mathrm { e } ^ { \lambda ( t - 1 ) } .$$
2. The independent random variables \(X\) and \(Y\) have distributions \(\operatorname { Po } ( \lambda )\) and \(\operatorname { Po } ( \mu )\) respectively. Use probability generating functions to show that the distribution of \(X + Y\) is \(\operatorname { Po } ( \lambda + \mu )\).
3. Given that \(X \sim \operatorname { Po } ( 1.5 )\) and \(Y \sim \operatorname { Po } ( 2.5 )\), find \(\mathrm { P } ( X \leqslant 2 \mid X + Y = 4 )\).

1 A company hires out narrowboats on a canal. It may be assumed that demands to hire a narrowboat occur independently and randomly at a constant mean rate of 25 per week. Using a suitable normal approximation, find

1. the probability that 15 or fewer narrowboats are hired out during a certain week,
2. the number of narrowboats that the company needs to have available for a week in order that the probability of running out of boats is 0.05 or less.

3

1. Given that \(X \sim \operatorname { Po } ( 5 )\), find \(\mathrm { P } ( X > 6 \mid X > 3 )\).
2. Given that \(Y \sim \operatorname { Po } ( \lambda )\) and \(\mathrm { P } ( Y \leqslant 1 ) = \frac { 1 } { 2 }\), show that \(\lambda\) satisfies the equation \(\lambda = \ln \{ 2 ( 1 + \lambda ) \}\).
3. Starting with a suitable approximation from the table of cumulative Poisson probabilities, use iteration to find \(\lambda\) correct to 3 decimal places.

3 The number of signal failures in a certain region of the railway network averages 10 every 3 weeks. Assume that signal failures occur independently, randomly and at constant mean rate.

1. Find the probability that
   1. there are between 7 and 12 (inclusive) signal failures in a three-week period,
   2. there are more than 4 signal failures in a one-week period.
   3. It has been calculated, using a suitable distributional approximation, that the probability of more than 62 signal failures in a period of \(n\) weeks is 0.0385 . Find the value of \(n\).

4

1. (a) Derive the moment generating function for a Poisson distribution with mean \(\lambda\).
   (b) The independent random variables \(X\) and \(Y\) are such that \(X \sim \operatorname { Po } ( \mu )\) and \(Y \sim \operatorname { Po } ( v )\). Use moment generating functions to show that \(( X + Y ) \sim \operatorname { Po } ( \mu + v )\).
2. The number of goals scored per match by Camford Academicals FC may be modelled by a Poisson distribution with mean 2. The number of goals scored against Camford during a match may be modelled by an independent Poisson distribution with mean \(k\). The probability that no goals are scored, by either side, in a match involving Camford is 0.045 . Find
   (a) the value of \(k\),
   (b) the probability that exactly 3 goals are scored against Camford in a match,
   (c) the probability that the total number of goals scored, in a match involving Camford, is between 2 and 5 inclusive.

2 Secret radio messages received under difficult conditions are subject to errors caused by random instantaneous breaks in transmission. The number of errors caused by breaks in transmission in a 10-minute period is denoted by \(B\).

1. State two conditions, other than randomness, needed for a Poisson distribution to be a suitable model for \(B\). Assume now that \(B \sim \mathrm { Po } ( 5 )\).
2. Calculate the probability that in a 15-minute period there are between 6 and 10 errors, inclusive, caused by random breaks in transmission. Secret radio messages are also subject to errors caused by mistakes made by the sender. The number of errors caused by mistakes made by the sender in a 10 -minute period, \(M\), has the independent distribution \(\operatorname { Po } ( 8 )\).
3. Calculate the period of time, in seconds, for which the probability that a message contains no errors of either sort is 0.6 .

10

1. \(X , Y\) and \(Z\) are independent random variables having Poisson distributions with means \(\lambda , \mu\) and \(\lambda + \mu\) respectively. Find \(\mathrm { P } ( X = 0\) and \(Y = 2 ) , \mathrm { P } ( X = 1\) and \(Y = 1 )\) and \(\mathrm { P } ( X = 2\) and \(Y = 0 )\). Hence verify that \(\mathrm { P } ( X + Y = 2 ) = \mathrm { P } ( Z = 2 )\).
2. In an office the male absence rate, i.e. the number of working days lost each month due to the absence of male employees, has a Poisson distribution with mean 4.5. In the same office the female absence rate has an independent Poisson distribution with mean 4.1. Calculate the probability that
   1. during a particular month both the male absence rate and the female absence rate are equal to 3,
   2. during a particular month the total of the male and female absence rates is equal to 6,
   3. during a particular month the male and female absence rates were each equal to 3 , given that the total of the male and female absence rates was equal to 6 .