Two independent Poisson sums

4 The number, \(X\), of books received at a charity shop has a constant mean of 5.1 per day.

1. State, in context, one condition for \(X\) to be modelled by a Poisson distribution.
   Assume now that \(X\) can be modelled by a Poisson distribution.
2. Find the probability that exactly 10 books are received in a 3-day period.
3. Use a suitable approximating distribution to find the probability that more than 180 books are received in a 30-day period.
   The number of DVDs received at the same shop is modelled by an independent Poisson distribution with mean 2.5 per day.
4. Find the probability that the total number of books and DVDs that are received at the shop in 1 day is more than 3 .

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 .

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 radioactive source is decaying at a mean rate of 3.4 counts per 5 seconds.

1. State conditions for a Poisson distribution to be a suitable model for the rate of decay of the source. You may assume that a Poisson distribution with a mean rate of 3.4 counts per 5 seconds is a suitable model.
2. State the variance of this Poisson distribution.
3. Find the probability of
   (A) exactly 3 counts in a 5 -second period,
   (B) at least 3 counts in a 5 -second period.
4. Find the probability of exactly 40 counts in a period of 60 seconds.
5. Use a suitable approximating distribution to find the probability of at least 40 counts in a period of 60 seconds.
6. The background radiation rate also, independently, follows a Poisson distribution and produces a mean count of 1.4 per 5 seconds. Find the probability that the radiation source together with the background radiation give a total count of at least 8 in a 5 -second period.

4 The manager of a car breakdown service uses the distribution \(\operatorname { Po } ( 2.7 )\) to model the number of punctures, \(R\), in a 24-hour period in a given rural area. The manager knows that, for this model to be valid, punctures must occur randomly and independently of one another.

1. State a further assumption needed for the Poisson model to be valid.
2. State the value of the standard deviation of \(R\).
3. Use the model to calculate the probability that, in a randomly chosen period of 168 hours, at least 22 punctures occur. The manager uses the distribution \(\operatorname { Po } ( 0.8 )\) to model the number of flat batteries in a 24 -hour period in the same rural area, and he assumes that instances of flat batteries are independent of punctures. A day begins and ends at midnight, and a "bad" day is a day on which there are more than 6 instances, in total, of punctures and flat batteries.
4. Assume first that both the manager's models are correct. Calculate the probability that a randomly chosen day is a "bad" day.
5. It is found that 12 of the next 100 days are "bad" days. Comment on whether this casts doubt on the validity of the manager's models.

5 The number of goals scored by the home team in a randomly chosen hockey match is denoted by \(X\).

1. In order for \(X\) to be modelled by a Poisson distribution it is assumed that goals scored are random events. State two other conditions needed for \(X\) to be modelled by a Poisson distribution in this context. Assume now that \(X\) can be modelled by the distribution \(\operatorname { Po } ( 1.9 )\).
2. (a) Write down an expression for \(\mathrm { P } ( X = r )\).
   (b) Hence find \(\mathrm { P } ( X = 3 )\).
3. Assume also that the number of goals scored by the away team in a randomly chosen hockey match has an independent Poisson distribution with mean \(\lambda\) between 1.31 and 1.32. Find an estimate for the probability that more than 3 goals are scored altogether in a randomly chosen match.

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.

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 \(X\)? [2]

Now assume that a Poisson distribution with mean 5 is an appropriate model for \(X\).

1. Find
   1. P\((X = 6)\), [2]
   2. P\((X \geqslant 8)\). [2]

The number of integrated sound systems sold in a weekend at the same shop can be assumed to have the distribution Po(7.2).

1. 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. [3]
2. State an assumption needed for your answer to part (c) to be valid. [1]
3. Give a reason why the assumption in part (d) may not be valid in practice. [1]

The number of goals scored by the home team in a randomly chosen hockey match is denoted by \(X\).

1. In order for \(X\) to be modelled by a Poisson distribution it is assumed that goals scored are random events. State two other conditions needed for \(X\) to be modelled by a Poisson distribution in this context. [2]

Assume now that \(X\) can be modelled by the distribution Po\((1.9)\).

1. 1. Write down an expression for P\((X = r)\). [1]
   2. Hence find P\((X = 3)\). [1]
2. Assume also that the number of goals scored by the away team in a randomly chosen hockey match has an independent Poisson distribution with mean \(\lambda\) between 1.31 and 1.32. Find an estimate for the probability that more than 3 goals are scored altogether in a randomly chosen match. [4]