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 \(\mu\).

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.
   (a) On this path the probability that there is at least 1 worm on a length of \(k\) metres is 0.96 . Find \(k\).
   (b) 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 \(\mathrm { P } ( T > 75 )\).

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.

4 It is given that \(Y \sim \operatorname { Po } ( \lambda )\), where \(\lambda \neq 0\), and that \(\mathrm { P } ( Y = 4 ) = \mathrm { P } ( Y = 5 )\). Write down an equation for \(\lambda\). Hence find the value of \(\lambda\) and the corresponding value of \(\mathrm { P } ( Y = 5 )\).
\(555 \%\) of the pupils in a large school are girls. A member of the student council claims that the probability that a girl rather than a boy becomes Head Student is greater than 0.55 . As evidence for his claim he says that 6 of the last 8 Head Students have been girls.

1. Use an exact binomial distribution to test the claim at the \(10 \%\) significance level.
2. A statistics teacher says that considering only the last 8 Head Students may not be satisfactory. Explain what needs to be assumed about the data for the test to be valid.

5. The number of eruptions of a volcano in a 10 year period is modelled by a Poisson distribution with mean 1

1. Find the probability that this volcano erupts at least once in each of 2 randomly selected 10 year periods.
2. Find the probability that this volcano does not erupt in a randomly selected 20 year period. The probability that this volcano erupts exactly 4 times in a randomly selected \(w\) year period is 0.0443 to 3 significant figures.
3. Use the tables to find the value of \(w\) A scientist claims that the mean number of eruptions of this volcano in a 10 year period is more than 1 She selects a 100 year period at random in order to test her claim.
4. State the null hypothesis for this test.
5. Determine the critical region for the test at the \(5 \%\) level of significance.
   

8. A cloth manufacturer knows that faults occur randomly in the production process at a rate of 2 every 15 metres.

1. Find the probability of exactly 4 faults in a 15 metre length of cloth.
2. Find the probability of more than 10 faults in 60 metres of cloth. A retailer buys a large amount of this cloth and sells it in pieces of length \(x\) metres. He chooses \(x\) so that the probability of no faults in a piece is 0.80
3. Write down an equation for \(x\) and show that \(x = 1.7\) to 2 significant figures. The retailer sells 1200 of these pieces of cloth. He makes a profit of 60p on each piece of cloth that does not contain a fault but a loss of \(\pounds 1.50\) on any pieces that do contain faults.
4. Find the retailer's expected profit.

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?

1. A machine produces cloth. Faults occur randomly in the cloth at a rate of 0.4 per square metre.

The machine is used to produce tablecloths, each of area \(A\) square metres. One of these tablecloths is taken at random. The probability that this tablecloth has no faults is 0.0907

1. Find the value of \(A\) The tablecloths are sold in packets of 20
   A randomly selected packet is taken.
2. Find the probability that more than 1 of the tablecloths in this packet has no faults. A hotel places an order for 100 tablecloths each of area \(A\) square metres.
   The random variable \(X\) represents the number of these tablecloths that have no faults.
3. Find
   1. \(\mathrm { E } ( X )\)
   2. \(\operatorname { Var } ( X )\)
4. Use a Poisson approximation to estimate \(\mathrm { P } ( X = 10 )\) It is claimed that a new machine produces cloth with a rate of faults that is less than 0.4 per square metre. A piece of cloth produced by this new machine is taken at random.
   The piece of cloth has area 30 square metres and is found to have 6 faults.
5. Stating your hypotheses clearly, use a suitable test to assess the claim made for the new machine. Use a \(5 \%\) level of significance.
6. Write down the \(p\)-value for the test used in part (e).

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]