Poisson parameter from given probability

5

1. The random variable \(X\) has the distribution \(\operatorname { Po } ( \lambda )\).
   1. State the values that \(X\) can take.
      It is given that \(\mathrm { P } ( X = 1 ) = 3 \times \mathrm { P } ( X = 0 )\).
   2. Find \(\lambda\).
   3. Find \(\mathrm { P } ( 4 \leqslant X \leqslant 6 )\).
2. The random variable \(Y\) has the distribution \(\operatorname { Po } ( \mu )\) where \(\mu\) is large. Using a suitable approximating distribution, it is found that \(\mathrm { P } ( Y < 46 ) = 0.0668\), correct to 4 decimal places. Find \(\mu\).

4

1. The random variable \(W\) has the distribution \(\operatorname { Po } ( 1.5 )\). Find the probability that the sum of 3 independent values of \(W\) is greater than 2 .
2. The random variable \(X\) has the distribution \(\operatorname { Po } ( \lambda )\). Given that \(\mathrm { P } ( X = 0 ) = 0.523\), find the value of \(\lambda\) correct to 3 significant figures.
3. The random variable \(Y\) has the distribution \(\operatorname { Po } ( \mu )\), where \(\mu \neq 0\). Given that $$\mathrm { P } ( Y = 3 ) = 24 \times \mathrm { P } ( Y = 1 )$$ find \(\mu\).

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 \(\mu\).

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 \(\lambda\).

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 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.

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 \(\mathrm { P } ( T > 75 )\).

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 \(\mathrm { P } ( Y = 7 )\).

1. During morning hours, employees arrive randomly at an office drinks dispenser at a rate of 2 every 10 minutes.

The number of employees arriving at the drinks dispenser is assumed to follow a Poisson distribution.

1. Find the probability that fewer than 5 employees arrive at the drinks dispenser during a 10-minute period one morning. During a 30 -minute period one morning, the probability that \(n\) employees arrive at the drinks dispenser is the same as the probability that \(n + 1\) employees arrive at the drinks dispenser.
2. Find the value of \(n\) During a 45-minute period one morning, the probability that between \(c\) and 12, inclusive, employees arrive at the drinks dispenser is 0.8546
3. Find the value of \(C\)
4. Find the probability that exactly 2 employees arrive at the drinks dispenser in exactly 4 of the 6 non-overlapping 10-minute intervals between 10 am and 11am one morning.

5 Biological cell membranes have receptor molecules which perform various functions. It is known that the number of receptor molecules of a particular type can be modelled by a Poisson distribution with mean 6 per area of 1 square unit.

1. 1. Determine the probability that there are at least 10 of these receptor molecules in an area of 1 square unit.
   2. Determine the probability that there are fewer than 50 of these receptor molecules in an area of 10 square units.
2. A scientist is looking at areas of 1 square unit of cell membrane in order to find one which has at least 10 receptor molecules. Find the probability that she has to look at more than 20 to find such an area. It is known that the number of receptor molecules of another type in an area of 1 square unit can be modelled by the random variable \(X\) which has a Poisson distribution with mean \(\mu\). It is given that \(\mathrm { E } \left( X ^ { 2 } \right) = 12\).
3. Determine \(\mathrm { P } ( X < 5 )\).

2 The number of cars arriving per minute to queue at a drive-through fast-food restaurant is modelled by the random variable \(X\). The standard deviation of \(X\) is 0.6 . You should assume that arrivals are random and independent and occur at a constant average rate.

1. Find the mean of \(X\).
2. 1. Calculate \(\mathrm { P } ( X = 1 )\).
   2. Calculate \(\mathrm { P } ( X > 1 )\).
3. Find the probability that fewer than 5 cars arrive in a randomly chosen 20 -minute period.

3. The number of claims made to the home insurance department of an insurance company follows a Poisson distribution with mean 4 per day.

1. Find the probability that more than 11 claims are made in a 2 -day period. The number of claims made in a day to the pet insurance department of the same company follows a Poisson distribution with parameter \(\lambda\). An insurance company worker notices that the probability of two claims being made in a day is three times the probability of four claims being made in a day.
2. Determine the value of \(\lambda\). The car insurance department models the length of time between claims for drivers aged 17 to 21 as an exponential distribution with mean 10 months. Rachel is 17 years old and has just passed her test. Her father says he will give her the car that they share if she does not make a claim in the first 12 months.
3. What is the probability that her father gives her the car?

6 The discrete random variable \(R\) has the distribution \(\operatorname { Po } ( \lambda )\).
Use an algebraic method to find the range of values of \(\lambda\) for which the single most likely value of \(R\) is 7. [7]

In a study, samples of soil were collected during the summer. Soil samples of dimensions \(25 \mathrm {~cm} \times 25 \mathrm {~cm} \times 40 \mathrm {~cm}\) were collected for analysis. The study found that there were, on average, 11 earthworms per sample. a) Explain briefly the conditions under which a Poisson distribution could be used to model the number of earthworms per sample.
b) In July, pupils at a primary school are asked to dig a smaller hole, \(25 \mathrm {~cm} \times 25 \mathrm {~cm} \times 10 \mathrm {~cm}\), and to count the number of earthworms they find. Calculate the probability that the pupils find exactly 5 earthworms.
c) In the autumn, the average number of earthworms per sample is greater than in the summer. The probability that, in the autumn, there are fewer than 13 earthworms in a soil sample of dimensions \(25 \mathrm {~cm} \times 25 \mathrm {~cm} \times 40 \mathrm {~cm}\) is close to \(36 \%\). Find the mean number of earthworms, to the nearest whole number, per \(25 \mathrm {~cm} \times 25 \mathrm {~cm} \times 40 \mathrm {~cm}\) soil sample in the autumn. Jessica is studying the relationship between hip girth, \(h \mathrm {~cm}\), and thigh girth, \(t \mathrm {~cm}\), for American adults who are physically active. She takes a random sample of 11 people from a very large dataset which she has downloaded into a spreadsheet software package. The results are shown below. a) Jessica notes that, for the thigh girth data, the lower quartile is 49.5 and the upper quartile is \(64 \cdot 0\).
i) Show that 87.2 should be classified as an outlier for \(t\).
ii) Give a reason why Jessica might exclude the outlier.
iii) Give a reason why Jessica might include the outlier. Jessica decides to exclude the outlier and produces the following scatter diagram. \section*{Thigh girth versus Hip girth} \includegraphics[max width=\textwidth, alt={}, center]{77c62e6d-58e4-42d3-9982-5a8325e8e826-04_647_1250_1439_404}
b) Interpret, in context, the correlation in the data shown in the diagram. The equation of the regression line of \(t\) on \(h\) for this sample is $$t = 0.69 h - 11.26$$ c) Interpret the gradient of the regression line in this context.
d) Use your knowledge of large data sets and spreadsheet software packages to suggest a way in which Jessica could improve her investigation. A company, Run4Lyfe, sponsors an athletic event. The organisers of the event claim that \(70 \%\) of the participants know the name of the sponsoring company. Run4Lyfe is concerned that the proportion, \(p\), of participants knowing the name of the sponsoring company is less than \(70 \%\). They decide to survey 60 randomly selected participants to carry out a significance test.
a) State suitable hypotheses for carrying out the test.
b) i) Explain what is meant by the critical region for this test.
ii) Determine the critical region if the test is to be carried out at a significance level as close as possible to, but not exceeding, \(5 \%\).
iii) Given that 40 participants out of the 60 in the sample know the name of the company, complete the significance test.
c) State, with a reason, how you would advise Run4Lyfe with regards to sponsoring the event next year. The fertility rate for a country is the average number of children that are born to a woman over her lifetime. The graphs and table below show some data on the fertility rates for 197 countries in the years 1914 and 2014. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure}