5.02c Linear coding: effects on mean and variance

1 The random variable \(X\) represents the number of cars arriving at a car wash per 10-minute period. From observations over a number of days, an estimate was made of the probability distribution of \(X\). Table 1 shows this estimated probability distribution. \begin{table}[h] \end{table}

1. In this question you must show detailed reasoning. Use Table 1 to calculate estimates of each of the following.
   You should now assume that \(X\) can be modelled by a Poisson distribution with mean equal to the value which you calculated in part (a).
2. Find each of the following.

4 A fair 8 -sided dice has faces labelled \(1,2 , \ldots , 8\). The random variable \(X\) represents the score when the dice is rolled once.

1. State the distribution of \(X\).
2. Find \(\mathrm { P } ( X < 4 )\).
3. Find each of the following.

1 The random variable \(X\) represents the clutch size (the number of eggs laid) by female birds of a particular species. The probability distribution of \(X\) is given in the table.

1. Find each of the following.
   • \(\mathrm { E } ( X )\)
   • \(\operatorname { Var } ( X )\)
   On average 65\% of eggs laid result in a young bird successfully leaving the nest.
2. 1. Find the mean number of young birds that successfully leave the nest.
   2. Find the standard deviation of the number of young birds that successfully leave the nest.

4 The discrete random variable \(X\) has probability distribution defined by $$\mathrm { P } ( X = r ) = k ( 2 r - 1 ) \quad \text { for } r = 1,2,3,4,5,6 \text {, where } k \text { is a constant. }$$

1. Complete the table in the Printed Answer Booklet giving the probabilities in terms of \(k\).

   \(r\)	1	2	3	4	5	6
   \(\mathrm { P } ( X = r )\)

2. Show that the value of \(k\) is \(\frac { 1 } { 36 }\).
3. Draw a graph to illustrate the distribution.
4. In this question you must show detailed reasoning. Find
   • \(\mathrm { E } ( X )\)
   • \(\operatorname { Var } ( X )\).
   A game consists of a player throwing two fair dice. The score is the maximum of the two values showing on the dice.
5. Show that the probability of a score of 3 is \(\frac { 5 } { 36 }\).
6. Show that the probability distribution for the score in the game is the same as the probability distribution of the random variable \(X\).
7. The game is played three times. Find
   • the mean of the total of the three scores.
   • the variance of the total of the three scores.

1 In a game at a charity fair, a spinner is spun 4 times.
On each spin the chance that the spinner lands on a score of 5 is 0.2 .
The random variable \(X\) represents the number of spins on which the spinner lands on a score of 5 .

1. Find \(\mathrm { P } ( X = 3 )\).
2. Find each of the following.
   • \(\mathrm { E } ( X )\)
   • \(\operatorname { Var } ( X )\)
   One game costs \(\pounds 1\) to play and, for each spin that lands on a score of 5 , the player receives 50 pence.
3. 1. Find the expected total amount of money gained by a player in one game.
   2. Find the standard deviation of the total amount of money gained by a player in one game.

3 A fair four-sided dice has its faces numbered \(0,1,2,3\). The dice is rolled three times. The discrete random variable \(X\) is the sum of the lowest and highest scores obtained.

1. Show that \(\mathrm { P } ( X = 1 ) = \frac { 3 } { 32 }\). The table below shows the probability distribution of \(X\).

   \(r\)	0	1	2	3	4	5	6
   \(\mathrm { P } ( X = r )\)	\(\frac { 1 } { 64 }\)	\(\frac { 3 } { 32 }\)	\(\frac { 13 } { 64 }\)	\(\frac { 3 } { 8 }\)	\(\frac { 13 } { 64 }\)	\(\frac { 3 } { 32 }\)	\(\frac { 1 } { 64 }\)

2. In this question you must show detailed reasoning. Find each of the following.
   • \(\mathrm { E } ( X )\)
   • \(\operatorname { Var } ( X )\)
   • The random variable \(Y\) represents the sum of 10 values of \(X\).
     1. State a property of the 10 values of \(X\) that would make it possible to deduce the standard deviation of \(Y\).
     2. Given that this property holds, determine the standard deviation of \(Y\).

6 The probability distribution of a discrete random variable, \(X\), is shown in the table below.

1. Find \(\mathrm { E } ( X )\) in terms of \(a\) and \(b\).
2. 1. In the case where \(\mathrm { E } ( \mathrm { X } ) = \mathrm { a } + 0.4\), find an expression for \(\operatorname { Var } ( X )\) in terms of \(a\).
   2. In this case, show that the greatest possible value of \(\operatorname { Var } ( X )\) is 0.65 . You must state the associated value of \(a\).
3. You are now given instead that \(\mathrm { E } ( X )\) is not known.
   1. State the least possible value of \(\operatorname { Var } ( X )\).
   2. Give all possible pairs of values of \(a\) and \(b\) which give the least possible value of \(\operatorname { Var } ( X )\) stated in part (c)(i).

1 A quiz team of 4 students is to be selected from a group of 7 girls and 5 boys. The team is selected at random from the students in the group. The number of girls in the team is denoted by the random variable \(X\).

1. Show that \(\mathrm { P } ( X = 4 ) = \frac { 7 } { 99 }\). Table 1 shows the probability distribution of \(X\). \begin{table}[h]

   \(r\)	0	1	2	3	4
   \(\mathrm { P } ( X = r )\)	\(\frac { 1 } { 99 }\)	\(\frac { 14 } { 99 }\)	\(\frac { 42 } { 99 }\)	\(\frac { 35 } { 99 }\)	\(\frac { 7 } { 99 }\)

   \captionsetup{labelformat=empty} \caption{Table 1}
   \end{table}
2. Find each of the following.
   It is decided that the quiz team must have at least 1 girl and at least 1 boy, but the team is still otherwise selected at random.
3. Explain whether \(\mathrm { E } ( X )\) would be smaller than, equal to or larger than the value which you found in part (b).

1 The probability distribution of a discrete random variable \(X\) is given by the formula \(\mathrm { P } ( \mathrm { X } = \mathrm { r } ) = \mathrm { k } \left( ( \mathrm { r } - 1 ) ^ { 2 } + 1 \right)\) for \(r = 1,2,3,4,5\).

1. Show that \(k = \frac { 1 } { 35 }\). The distribution of \(X\) is shown in the table.

   \(r\)	1	2	3	4	5
   \(\mathrm { P } ( \mathrm { X } = \mathrm { r } )\)	\(\frac { 1 } { 35 }\)	\(\frac { 2 } { 35 }\)	\(\frac { 1 } { 7 }\)	\(\frac { 2 } { 7 }\)	\(\frac { 17 } { 35 }\)

2. Comment briefly on the shape of the distribution.
3. Find each of the following.
   The random variable \(Y\) is given by \(Y = 5 X - 10\).
4. Find each of the following.

1 A fair six-sided dice is rolled three times.
The random variable \(X\) represents the lowest of the three scores.
The probability distribution of \(X\) is given by the formula \(\mathrm { P } ( X = r ) = k \left( 127 - 39 r + 3 r ^ { 2 } \right)\) for \(r = 1,2,3,4,5,6\).

1. Complete the copy of the table in the Printed Answer Booklet.

   \(r\)	1	2	3	4	5	6
   \(\mathrm { P } ( X = r )\)	\(91 k\)	\(61 k\)	\(37 k\)

2. Show that \(k = \frac { 1 } { 216 }\).
3. Draw a graph to illustrate the distribution.
4. Comment briefly on the shape of the distribution.
5. In this question you must show detailed reasoning. Find each of the following.

3 The table shows the probability distribution of the random variable \(X\), where \(a\) and \(b\) are constants.

1. Given that \(\mathrm { E } ( X ) = 1.8\), determine the values of \(a\) and \(b\). The random variable \(Y\) is given by \(Y = 10 - 3 X\).
2. Using the values of \(a\) and \(b\) which you found in part (a), find each of the following.

9 The random variable \(X\) has a discrete uniform distribution over the values \(\{ 0,1,2 , \ldots , 20 \}\).

1. Find \(\mathrm { P } ( X \leqslant 7 )\).
2. Find each of the following.
   The spreadsheet shows a simulation of the distribution of \(X\). Each of the 25 rows of the spreadsheet below the heading row shows a simulation of 10 independent values of \(X\) together with the value of the mean of the 10 values, denoted by \(Y\).

   \includegraphics[max width=\textwidth, alt={}]{77eabbd6-a058-457f-9601-d66f3c2db005-07_38_45_880_279}	A	B	C	D	E	F	G	H	I	J	K	L
   1	\(X _ { 1 }\)	\(X _ { 2 }\)	\(X _ { 3 }\)	\(X _ { 4 }\)	\(X _ { 5 }\)	\(X _ { 6 }\)	\(X _ { 7 }\)	\(X _ { 8 }\)	\(X _ { 9 }\)	\(X _ { 10 }\)	\(Y\)
   2	1	6	2	1	18	6	4	9	11	11	6.9
   3	13	14	12	2	4	11	16	0	16	0	8.8
   4	4	17	1	16	4	10	12	2	18	13	9.7
   5	2	8	12	1	4	16	12	2	15	8	8.0
   6	7	15	16	0	4	7	1	13	0	20	8.3
   7	15	13	10	1	12	0	20	15	16	6	10.8
   8	14	13	17	12	2	18	16	18	9	4	12.3
   9	20	2	12	3	17	3	0	18	15	13	10.3
   10	2	12	5	12	2	6	0	9	10	15	7.3
   11	5	11	13	10	9	17	10	4	20	15	11.4
   12	14	9	9	7	6	20	2	2	11	16	9.6
   13	15	19	18	19	7	6	6	20	3	8	12.1
   14	5	10	6	4	1	19	15	8	17	18	10.3
   15	0	3	15	15	11	12	0	3	9	16	8.4
   16	1	12	1	15	0	4	11	11	9	2	6.6
   17	12	5	0	8	3	8	12	19	13	12	9.2
   18	9	5	1	13	5	4	18	1	1	19	7.6
   19	16	2	20	20	12	17	2	7	8	20	12.4
   20	18	17	3	2	8	18	7	0	11	6	9.0
   21	15	10	7	20	4	0	5	6	11	14	9.2
   22	3	9	10	14	2	1	8	6	0	7	6.0
   23	11	10	11	10	19	11	3	7	10	0	9.2
   24	12	14	6	6	5	20	11	18	10	14	11.6
   25	1	11	5	14	11	10	1	1	2	0	5.6
   26	0	14	7	11	18	5	10	20	11	9	10.5
   27

3. Use the spreadsheet to estimate \(\mathrm { P } ( Y \leqslant 7 )\).
4. Explain why the true value of \(\mathrm { P } ( Y \leqslant 7 )\) is less than \(\mathrm { P } ( X \leqslant 7 )\), relating your answer to \(\operatorname { Var } ( X )\) and \(\operatorname { Var } ( Y )\).
5. The random variable \(W\) is the mean of 30 independent values of \(X\). Determine an estimate of \(\mathrm { P } ( W \leqslant 7 )\).

11 The random variable \(X\) takes the value 1 with probability \(p\) and the value 0 with probability \(1 - p\).

1. Find each of the following.
   Use the results of part (a) to prove that

1 The number of insurance policy sales made per month by a salesperson is modelled by the random variable \(X\), with probability distribution shown in the table.

1. Find each of the following.
   The salesperson is paid a basic salary of \(\pounds 1000\) per month plus \(\pounds 500\) for each policy that is sold.
2. Find the mean and standard deviation of the salesperson's monthly salary.

1 In a game at a fair, players choose 4 countries from a list of 10 countries. The names of all 10 countries are then put in a box and the player selects 4 of them at random. The random variable \(X\) represents the number of countries that match those which the player originally chose.

1. Show that the probability that a randomly selected player matches all 4 countries is \(\frac { 1 } { 210 }\). Table 1 shows the probability distribution of \(X\). \begin{table}[h]

   \(r\)	0	1	2	3	4
   \(\mathrm { P } ( X = r )\)	\(\frac { 1 } { 14 }\)	\(\frac { 8 } { 21 }\)	\(\frac { 3 } { 7 }\)	\(\frac { 4 } { 35 }\)	\(\frac { 1 } { 210 }\)

   \captionsetup{labelformat=empty} \caption{Table 1}
   \end{table}
2. Find each of the following.
   Find the mean and standard deviation of the player's loss per game.
3. In order to try to attract more customers, the rules will be changed as follows. The game will still cost \(\pounds 1\) to play. The player will get 25 pence back for every country which is matched, plus an additional bonus of \(\pounds 100\) if all four countries are matched. Find the player's mean gain or loss per game with these new rules.

10 The discrete random variables \(X\) and \(Y\) have distributions as follows: \(X \sim \mathrm {~B} ( 20,0.3 )\) and \(Y \sim \operatorname { Po } ( 3 )\). The spreadsheet in Fig. 10 shows a simulation of the distributions of \(X\) and \(Y\). Each of the 20 rows below the heading row consists of a value of \(X\), a value of \(Y\), and the value of \(X - 2 Y\). \begin{table}[h] \end{table}

1. Use the spreadsheet to estimate each of the following.
   The mean of 50 values of \(X - 2 Y\) is denoted by the random variable \(W\).
2. Calculate an estimate of \(\mathrm { P } ( W > 1 )\).

2 In a game at a charity fair, a player rolls 3 unbiased six-sided dice. The random variable \(X\) represents the difference between the highest and lowest scores.

1. Show that \(\mathrm { P } ( X = 0 ) = \frac { 1 } { 36 }\). The table shows the probability distribution of \(X\).

   \(r\)	0	1	2	3	4	5
   \(\mathrm { P } ( \mathrm { X } = \mathrm { r } )\)	\(\frac { 1 } { 36 }\)	\(\frac { 5 } { 36 }\)	\(\frac { 2 } { 9 }\)	\(\frac { 1 } { 4 }\)	\(\frac { 2 } { 9 }\)	\(\frac { 5 } { 36 }\)

2. Draw a graph to illustrate the distribution.
3. Describe the shape of the distribution.
4. In this question you must show detailed reasoning. Find each of the following.
   As a result of playing the game, the player receives \(30 X\) pence from the organiser of the game.
5. Find the variance of the amount that the player receives.
6. The player pays \(k\) pence to play the game. Given that the average profit made by the organiser is 12.5 pence per game, determine the value of \(k\).

9 The discrete random variable \(X\) has a uniform distribution over the set of all integers between \(- n\) and \(n\) inclusive, where \(n\) is a positive integer.

1. Given that \(n\) is odd, determine \(\mathrm { P } \left( \mathrm { X } > \frac { 1 } { 2 } \mathrm { n } \right)\), giving your answer as a single fraction in terms of \(n\).
2. Determine the variance of the sum of 10 independent values of \(X\), giving your answer in the form \(\mathrm { an } ^ { 2 } + \mathrm { bn }\), where \(a\) and \(b\) are constants.

1. The probability distribution of the discrete random variable \(X\) is

$$P ( X = x ) = \begin{cases} \frac { k } { x } & \text { for } x = 1,2 \text { and } 3 \\ \frac { m } { 2 x } & \text { for } x = 6 \text { and } 9 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) and \(m\) are positive constants.
Given that \(\mathrm { E } ( X ) = 3.8\), find \(\operatorname { Var } ( X )\)

1. Flobee sells tomato seeds in packets, each containing 40 seeds. Flobee advertises that only 4\% of its tomato seeds do not germinate.

Amodita is investigating the germination of Flobee's tomato seeds. She plants 125 packets of Flobee's tomato seeds and records the number of seeds that do not germinate in each packet. Amodita wants to test whether the binomial distribution \(\mathrm { B } ( 40,0.04 )\) is a suitable model for these data. The table below shows the expected frequencies, to 2 decimal places, using this model.

1. Calculate the value of \(r\) and the value of \(s\)
2. Stating your hypotheses clearly, carry out the test at the \(5 \%\) level of significance. You should state the number of degrees of freedom, critical value and conclusion clearly. Amodita believes that Flobee should use a more realistic value for the percentage of their tomato seeds that do not germinate.
   She decides to test the data using a new model \(\mathrm { B } ( 40 , p )\)
3. Showing your working, suggest a more realistic value for \(p\)

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

1. Find \(\mathrm { E } ( X )\) Given that \(\operatorname { Var } ( X ) = 8.79\)
2. find \(\mathrm { E } \left( X ^ { 2 } \right)\) The discrete random variable \(Y\) has probability distribution

   \(y\)	- 2	- 1	0	1	2
   \(\mathrm { P } ( Y = y )\)	\(3 a\)	\(a\)	\(b\)	\(a\)	\(c\)

   where \(a\), \(b\) and \(c\) are constants.
   For the random variable \(Y\) $$\mathrm { P } ( Y \leqslant 0 ) = 0.75 \quad \text { and } \quad \mathrm { E } \left( Y ^ { 2 } + 3 \right) = 5$$
3. Find the value of \(a\), the value of \(b\) and the value of \(c\) The random variable \(W = Y - X\) where \(Y\) and \(X\) are independent.
   The random variable \(T = 3 W - 8\)
4. Calculate \(\mathrm { P } ( W > T )\)

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

1. Find in terms of \(q\)
   1. \(\mathrm { E } ( X )\)
   2. \(\mathrm { E } \left( X ^ { 2 } \right)\) Given that \(\operatorname { Var } ( X ) = 3.66\)
2. show that \(q = 0.3\) In a game, the score is given by the discrete random variable \(X\) Given that games are independent,
3. calculate the probability that after the 4th game has been played, the total score is exactly 20 A round consists of 4 games plus 2 bonus games. The bonus games are only played if after the 4th game has been played the total score is exactly 20 A prize of \(\pounds 10\) is awarded if 6 games are played in a round and the total score for the round is at least 27 Bobby plays 3 rounds.
4. Find the probability that Bobby wins at least \(\pounds 10\)

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

where \(r\) and \(k\) are positive constants.
The standard deviation of \(X\) equals the mean of \(X\) Find the exact value of \(r\)

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

where \(p\) and \(r\) are probabilities.
Given that \(\mathrm { E } ( X ) = 1.95\) find the exact value of \(\mathrm { E } ( \sqrt { X + 1 } )\) giving your answer in the form \(a + b \sqrt { 2 }\) where \(a\) and \(b\) are rational.
(6)

1. Robin shoots 8 arrows at a target each day for 100 days.

The number of times he hits the target each day is summarised in the table below. Misha believes that these data can be modelled by a binomial distribution.

1. State, in context, two assumptions that are implied by the use of this model.
2. Find an estimate for the proportion of arrows Robin shoots that hit the target. Misha calculates expected frequencies, to 2 decimal places, as follows.

   Number of hits	0	1	2	3	4	5	6	7	8
   Expected frequency	2.81	12.67	\(r\)	28.05	19.73	\(s\)	2.50	0.40	0.03

3. Find the value of \(r\) and the value of \(s\) Misha correctly used a suitable test to assess her belief.
4. 1. Explain why she used a test with 3 degrees of freedom.
   2. Complete the test using a \(5 \%\) level of significance. You should clearly state your hypotheses, test statistic, critical value and conclusion.