5.02c Linear coding: effects on mean and variance

2 A large hotel has 90 bedrooms. Sometimes a guest makes a booking for a room, but then does not arrive. This is called a 'no-show'. On average \(10 \%\) of bookings are no-shows. The hotel manager accepts up to 94 bookings before saying that the hotel is full. If at least 4 of these bookings are no-shows then there will be enough rooms for all of the guests. 94 bookings have been made for each night in August. You should assume that all bookings are independent.

1. State the distribution of the number of no-shows on one night in August.
2. State the conditions under which the use of a Poisson distribution is appropriate as an approximation to a binomial distribution.
3. Use a Poisson approximating distribution to find the probability that, on one night in August,
   (A) there are exactly 4 no-shows,
   (B) there are enough rooms for all of the guests who do arrive.
4. Find the probability that, on all of the 31 nights in August, there are enough rooms for all of the guests who arrive.
5. (A) In August there are \(31 \times 94 = 2914\) bookings altogether. State the exact distribution of the total number of no-shows during August.
   (B) Use a suitable approximating distribution to find the probability that there are at most 300 no-shows altogether during August.

5 The discrete random variable \(X\) has moment generating function \(\frac { 1 } { 4 } \mathrm { e } ^ { 2 t } + a \mathrm { e } ^ { 3 t } + b \mathrm { e } ^ { 4 t }\), where \(a\) and \(b\) are constants. It is given that \(\mathrm { E } ( X ) = 3 \frac { 3 } { 8 }\).

1. Show that \(a = \frac { 1 } { 8 }\), and find the value of \(b\).
2. Find \(\operatorname { Var } ( X )\).
3. State the possible values of \(X\).

4 The discrete random variable \(X\) has moment generating function \(\left( \frac { 1 } { 4 } + \frac { 3 } { 4 } \mathrm { e } ^ { t } \right) ^ { 3 }\).

1. Find \(\mathrm { E } ( X )\).
2. Find \(\mathrm { P } ( X = 2 )\).
3. Show that \(X\) can be expressed as a sum of 3 independent observations of a random variable \(Y\). Obtain the probability distribution of \(Y\), and the variance of \(Y\).

1. The random variable \(X\) has probability function

$$\mathrm { P } ( X = x ) = \frac { x ^ { 2 } } { k } \quad x = 1,2,3,4$$

1. Show that \(k = 30\)
2. Find \(\mathrm { P } ( X \neq 4 )\)
3. Find the exact value of \(\mathrm { E } ( X )\)
4. Find the exact value of \(\operatorname { Var } ( X )\) Given that \(Y = 3 X - 1\)
5. find \(\mathrm { E } \left( Y ^ { 2 } \right)\)

1. The discrete random variable \(X\) has the probability distribution given in the table below.

1. Write down the value of \(\mathrm { F } ( 5 )\)
2. Find \(\mathrm { E } ( X )\)
3. Find \(\operatorname { Var } ( X )\) The random variable \(Y = 7 - 2 X\)
4. Find
   1. \(\mathrm { E } ( Y )\)
   2. \(\operatorname { Var } ( Y )\)
   3. \(\mathrm { P } ( Y > X )\) \includegraphics[max width=\textwidth, alt={}, center]{70137e9a-0a6b-48b5-8dd4-c436cb063351-03_2261_47_313_37}

1. A scientist measured the salinity of water, \(x \mathrm {~g} / \mathrm { kg }\), and recorded the temperature at which the water froze, \(y ^ { \circ } \mathrm { C }\), for 12 different water samples. The summary statistics are listed below.

$$\begin{gathered} \sum x = 504 \quad \sum y = - 27 \quad \sum x ^ { 2 } = 22842 \quad \sum y ^ { 2 } = 62.98 \\ \sum x y = - 1190.7 \quad \mathrm {~S} _ { x x } = 1674 \quad \mathrm {~S} _ { y y } = 2.23 \end{gathered}$$

1. Find the mean and variance of the recorded temperatures.
   (3) Priya believes that the higher the salinity of water, the higher the temperature at which the water freezes.
2. 1. Calculate the product moment correlation coefficient between \(x\) and \(y\)
   2. State, with a reason, whether or not this value supports Priya's belief.
3. Find the least squares regression line of \(y\) on \(x\) in the form \(y = a + b x\) Give the value of \(a\) and the value of \(b\) to 3 significant figures.
4. Estimate the temperature at which water freezes when the salinity is \(32 \mathrm {~g} / \mathrm { kg }\) The coding \(w = 1.8 y + 32\) is used to convert the recorded temperatures from \({ } ^ { \circ } \mathrm { C }\) to \({ } ^ { \circ } \mathrm { F }\)
5. Find an equation of the least squares regression line of \(w\) on \(x\) in the form \(w = c + d x\)
6. Find
   1. the variance of the recorded temperatures when converted to \({ } ^ { \circ } \mathrm { F }\)
   2. the product moment correlation coefficient between \(w\) and \(x\) \href{http://PhysicsAndMathsTutor.com}{PhysicsAndMathsTutor.com}

1. In a game, the number of points scored by a player in the first round is given by the random variable \(X\) with probability distribution

Find

1. \(\mathrm { E } ( X )\)
2. \(\operatorname { Var } ( X )\)
3. \(\operatorname { Var } ( 3 - 2 X )\) The number of points scored by a player in the second round is given by the random variable \(Y\) and is independent of the number of points scored in the first round. The random variable \(Y\) has probability function $$\mathrm { P } ( Y = y ) = \frac { 1 } { 4 } \quad \text { for } y = 5,6,7,8$$
4. Write down the value of \(\mathrm { E } ( Y )\)
5. Find \(\mathrm { P } ( X = Y )\)
6. Find the probability that the number of points scored by a player in the first round is greater than the number of points scored by the player in the second round.

4. The discrete random variable \(X\) has probability distribution

1. Find \(\mathrm { E } ( X )\) in terms of \(a\) and \(b\) For this probability distribution, \(\operatorname { Var } ( X ) = \mathrm { E } \left( X ^ { 2 } \right)\)
2. 1. Write down the value of \(\mathrm { E } ( X )\)
   2. Find the value of \(a\) and the value of \(b\)
3. Find \(\operatorname { Var } ( 1 - 3 X )\) Given that \(Y = 1 - X\), find
4. 1. \(\mathrm { P } ( Y < 0 )\)
   2. the largest possible value of \(k\) such that \(\mathrm { P } ( Y < k ) = 0.2\)

5. Franca is the manager of an accountancy firm. She is investigating the relationship between the salary, \(\pounds x\), and the length of commute, \(y\) minutes, for employees at the firm. She collected this information from 9 randomly selected employees. The salary of each employee was then coded using \(w = \frac { x - 20000 } { 1000 }\) The table shows the values of \(w\) and \(y\) for the 9 employees. (You may use \(\sum w = 81 \quad \sum y = 405 \quad \sum w y = 2490 \quad S _ { w w } = 660 \quad S _ { y y } = 2500\) )

1. Calculate the salary of the employee with \(w = - 2\)
2. Show that, to 3 significant figures, the value of the product moment correlation coefficient between \(w\) and \(y\) is - 0.899
3. State, giving a reason, the value of the product moment correlation coefficient between \(x\) and \(y\) The least squares regression line of \(y\) on \(w\) is \(y = 60.75 - 1.75 w\)
4. Find the equation of the least squares regression line of \(y\) on \(x\) giving your answer in the form \(y = a + b x\)
5. Estimate the length of commute for an employee with a salary of \(\pounds 21000\) Franca uses the regression line to estimate the length of commute for employees with salaries between \(\pounds 25000\) and \(\pounds 40000\)
6. State, giving a reason, whether or not these estimates are reliable.

2. The discrete random variable \(X\) has the following probability distribution.

1. Find \(\mathrm { E } ( X )\). Given that \(\mathrm { E } \left( X ^ { 2 } \right) = 4.54\)
2. find the value of \(a\) and the value of \(b\). The random variable \(Y = 3 - 2 X\)
3. Find \(\operatorname { Var } ( Y )\).

1. The average minimum monthly temperature, \(x\) degrees Fahrenheit ( \({ } ^ { \circ } \mathrm { F }\) ), and the average maximum monthly temperature, \(y\) degrees Fahrenheit ( \({ } ^ { \circ } \mathrm { F }\) ), in Kolkata were recorded for 12 months.

Some of the summary statistics are given below. $$\sum x = 862 \quad \sum x ^ { 2 } = 62802 \quad \mathrm {~S} _ { y y } = 413.67 \quad S _ { x y } = 512.67 \quad n = 12$$

1. 1. Calculate the mean of the 12 values of the average minimum
      monthly temperature.
   2. Show that the standard deviation of the 12 values of the average minimum monthly temperature is \(8.57 ^ { \circ } \mathrm { F }\) to 3 significant figures.
2. Calculate the product moment correlation coefficient between \(x\) and \(y\) For comparative purposes with a UK city, it was necessary to convert the temperatures from degrees Fahrenheit ( \({ } ^ { \circ } \mathrm { F }\) ) to degrees Celsius ( \({ } ^ { \circ } \mathrm { C }\) ). The formula used was $$c = \frac { 5 } { 9 } ( f - 32 )$$ where \(f\) is the temperature in \({ } ^ { \circ } \mathrm { F }\) and \(c\) is the temperature in \({ } ^ { \circ } \mathrm { C }\)
3. Use this formula and the values from part (a) to calculate, in \({ } ^ { \circ } \mathrm { C }\), the mean and the standard deviation of the 12 values of the average minimum monthly temperature in Kolkata.
   Give your answers to 3 significant figures. Given that
   • \(u\) is the equivalent temperature in \({ } ^ { \circ } \mathrm { C }\) of \(x\)
   • \(\quad v\) is the equivalent temperature in \({ } ^ { \circ } \mathrm { C }\) of \(y\)
   • state, giving a reason, the product moment correlation coefficient between \(u\) and \(v\)

8 The random variable \(S\) has the distribution \(\mathrm { B } ( 14 , p )\). A significance test is carried out of the null hypothesis \(\mathrm { H } _ { 0 } : p = 0.3\) against the alternative hypothesis \(\mathrm { H } _ { 1 } : p > 0.3\). The critical region for the test is \(S \geqslant 8\).

1. Find the significance level of the test, correct to 3 significant figures.
2. It is given that, on each occasion that the test is carried out, the true value of \(p\) is equally likely to be \(0.3,0.5\) or 0.7 , independently of any other test. Four independent tests are carried out. Find the probability that at least one of the tests results in a Type II error.

3 Erika is a birdwatcher. The probability that she will see a woodpecker on any given day is \(\frac { 1 } { 8 }\). It is assumed that this probability is unaffected by whether she has seen a woodpecker on any other day.

1. Calculate the probability that Erika first sees a woodpecker
   1. on the third day,
   2. after the third day.
   3. Find the expectation of the number of days up to and including the first day on which she sees a woodpecker.
   4. Calculate the probability that she sees a woodpecker on exactly 2 days in the first 15 days.

7 At a factory that makes crockery the quality control department has found that \(10 \%\) of plates have minor faults. These are classed as 'seconds'. Plates are stored in batches of 12. The number of seconds in a batch is denoted by \(X\).

1. State an appropriate distribution with which to model \(X\). Give the value(s) of any parameter(s) and state any assumptions required for the model to be valid. Assume now that your model is valid.
2. Find
   1. \(\mathrm { P } ( X = 3 )\),
   2. \(\mathrm { P } ( X \geqslant 1 )\).
   3. A random sample of 4 batches is selected. Find the probability that the number of these batches that contain at least 1 second is fewer than 3 .

3 A random variable \(X\) has the distribution \(\mathrm { B } ( 13,0.12 )\).

1. Find \(\mathrm { P } ( X < 2 )\). Two independent values of \(X\) are found.
2. Find the probability that exactly one of these values is equal to 2 .

6 The diagrams illustrate all or part of the probability distributions of the discrete random variables \(V , W , X , Y\) and \(Z\). \includegraphics[max width=\textwidth, alt={}, center]{56ca7462-d061-48d3-bc5f-274d925e4e34-4_419_365_370_296} \includegraphics[max width=\textwidth, alt={}, center]{56ca7462-d061-48d3-bc5f-274d925e4e34-4_419_376_370_838} \includegraphics[max width=\textwidth, alt={}, center]{56ca7462-d061-48d3-bc5f-274d925e4e34-4_419_362_370_1400} \includegraphics[max width=\textwidth, alt={}, center]{56ca7462-d061-48d3-bc5f-274d925e4e34-4_421_359_879_580} \includegraphics[max width=\textwidth, alt={}, center]{56ca7462-d061-48d3-bc5f-274d925e4e34-4_419_355_881_1142}

1. One of these variables has the distribution \(\operatorname { Geo } \left( \frac { 1 } { 2 } \right)\). State, with a reason, which variable this is.
2. One of these variables has the distribution \(\mathrm { B } \left( 4 , \frac { 1 } { 2 } \right)\). State, with reasons, which variable this is. \(760 \%\) of the voters at a certain polling station are women. Voters enter the polling station one at a time. The number of voters who enter, up to and including the first woman, is denoted by \(X\).

8 On average, half the plants of a particular variety produce red flowers and the rest produce blue flowers.

1. Ann chooses 8 plants of this variety at random. Find the probability that more than 6 plants produce red flowers.
2. Karim chooses 22 plants of this variety at random.
   1. Find the probability that the number of these plants that produce blue flowers is equal to the number that produce red flowers.
   2. Hence find the probability that the number of these plants that produce blue flowers is greater than the number that produce red flowers.

3

1. A random variable, \(X\), has the distribution \(\mathrm { B } ( 12,0.85 )\). Find
   1. \(\mathrm { P } ( X > 10 )\),
   2. \(\mathrm { P } ( X = 10 )\),
   3. \(\operatorname { Var } ( X )\).
   4. A random variable, \(Y\), has the distribution \(\mathrm { B } \left( 2 , \frac { 1 } { 4 } \right)\). Two independent values of \(Y\) are found. Find the probability that the sum of these two values is 1 .

5 A retail analyst records the numbers of loaves of bread of a particular type bought by a sample of shoppers in a supermarket.

1. Calculate the mean and standard deviation of the numbers of loaves bought per person.
2. Each loaf costs \(\pounds 1.04\). Calculate the mean and standard deviation of the amount spent on loaves per person.

7 Yasmin has 5 coins. One of these coins is biased with P (heads) \(= 0.6\). The other 4 coins are fair. She tosses all 5 coins once and records the number of heads, \(X\).

1. Show that \(\mathrm { P } ( X = 0 ) = 0.025\).
2. Show that \(\mathrm { P } ( X = 1 ) = 0.1375\). The table shows the probability distribution of \(X\).

   \(r\)	0	1	2	3	4	5
   \(\mathrm { P } ( X = r )\)	0.025	0.1375	0.3	0.325	0.175	0.0375

3. Draw a vertical line chart to illustrate the probability distribution.
4. Comment on the skewness of the distribution.
5. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
6. Yasmin tosses the 5 coins three times. Find the probability that the total number of heads is 3 . \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE.}

1 The weights, \(x\) grams, of 100 potatoes are summarised as follows. $$n = 100 \quad \sum x = 24940 \quad \sum x ^ { 2 } = 6240780$$

1. Calculate the mean and standard deviation of \(x\).
2. The weights, \(y\) grams, of the potatoes after they have been peeled are given by the formula \(y = 0.9 x - 15\). Deduce the mean and standard deviation of the weights of the potatoes after they have been peeled.

2 It was stated in 2012 that \(3 \%\) of \(\pounds 1\) coins were fakes. Throughout this question, you should assume that this is still the case.

1. Find the probability that, in a random selection of \(25 \pounds 1\) coins, there is exactly one fake coin. A random sample of \(250 \pounds 1\) coins is selected.
2. Explain why a Poisson distribution is an appropriate approximating distribution for the number of fake coins in the sample.
3. Use a Poisson distribution to find the probability that, in this sample, there are
   (A) exactly 10 fake coins,
   (B) at least 10 fake coins.
4. Use a suitable approximating distribution to find the probability that there are at least 50 fake coins in a sample of 2000 coins. It is known that \(0.2 \%\) of another type of coin are fakes.
5. A random sample of size \(n\) of these coins is taken. Using a Poisson approximating distribution, show that the probability of at most one fake coin in the sample is equal to \(\mathrm { e } ^ { - \lambda } + \lambda \mathrm { e } ^ { - \lambda }\), where \(\lambda = 0.002 n\).
6. Use the approximation \(\mathrm { e } ^ { - \lambda } + \lambda \mathrm { e } ^ { - \lambda } \approx 1 - \frac { \lambda ^ { 2 } } { 2 }\) for small values of \(\lambda\) to estimate the value of \(n\) for which the probability in part ( \(\mathbf { v }\) ) is equal to 0.995 .

2 When a genetic sequence of plant DNA is given a dose of radiation, some of the genes may mutate. The probability that a gene mutates is 0.012 . Mutations occur randomly and independently.

1. Explain the meanings of the terms 'randomly' and 'independently' in this context. A short stretch of DNA containing 20 genes is given a dose of radiation.
2. Find the probability that exactly 1 out of the 20 genes mutates. A longer stretch of DNA containing 500 genes is given a dose of radiation.
3. Explain why a Poisson distribution is an appropriate approximating distribution for the number of genes that mutate.
4. Use this Poisson distribution to find the probability that there are
   (A) exactly two genes that mutate,
   (B) at least two genes that mutate. A third stretch of DNA containing 50000 genes is given a dose of radiation.
5. Use a suitable approximating distribution to find the probability that there are at least 650 genes that mutate.

4 Part of an ecological study involved measuring the heights of trees in a young forest. In order to obtain an estimate of the mean height of all the trees in the forest, a random sample of 70 trees was selected and their heights measured. These heights, \(x\) metres, are summarised by \(\Sigma x = 246.6\) and \(\Sigma x ^ { 2 } = 1183.65\). The mean height of all trees in the forest is denoted by \(\mu\) metres.

1. Calculate a symmetric \(90 \%\) confidence interval for \(\mu\).
2. A student was asked to interpret the interval and said,
   "If 100 independent \(90 \%\) confidence intervals were calculated then 90 of them would contain \(\mu\)." Explain briefly what is wrong with this statement.
3. Four independent \(90 \%\) confidence intervals for \(\mu\) are obtained. Calculate the probability that at least three of the intervals contain \(\mu\).

3 A zoologist is studying the feeding behaviour of a group of 4 gorillas. The random variable \(X\) represents the number of gorillas that are feeding at a randomly chosen moment. The probability distribution of \(X\) is shown in the table below.

1. Find the value of \(p\).
2. Find the expectation and variance of \(X\).
3. The zoologist observes the gorillas on two further occasions. Find the probability that there are at least two gorillas feeding on both occasions.