5.04a Linear combinations: E(aX+bY), Var(aX+bY)

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.

4 A machine manufactures batches of 100 titanium sheets. The thickness of every sheet in a batch is Normally distributed with mean \(\mu \mathrm { mm }\) and standard deviation 0.03 mm . You should assume that each sheet is of uniform thickness and that the thicknesses of different sheets are independent of each other. The values of \(\mu\) for three different batches, A, B and C, are 3.125, 3.117 and 3.109 respectively.

1. Determine the probability that the total thickness of 10 sheets from Batch A is less than 31.0 mm .
2. Determine the probability that, if a single sheet from Batch A is cut into pieces and 10 of the pieces are stacked together, the total thickness of the stack is less than 31.0 mm .
3. Determine the probability that, if one sheet from each of Batches A, B and C are stacked together, the total thickness of the stack is at least 9.4 mm .
4. Determine the probability that the total thickness of 10 sheets from Batch A is less than the total thickness of 10 sheets from Batch B.

8 The random variable \(X\) has a continuous uniform distribution over [0,10].

1. Find the probability that, if two independent values of \(X\) are taken, one is less than 3 and the other is greater than 3 . The random variable \(T\) denotes the sum of 5 independent values of \(X\).
2. State the value of \(\mathrm { P } ( T \leqslant 25 )\). The spreadsheet below shows the heading row and the first 20 data rows from a total of 100 data rows of a simulation of the distribution of \(X\). Each of the 100 rows shows a simulation of 5 independent values of \(X\), together with \(T\), the sum of the 5 values. All of the values have been rounded to 2 decimal places. In column I the spreadsheet shows the number of values of \(T\) that are less than or equal to the corresponding values in column H . For example, there are 75 simulated values of \(T\) that are less than or equal to 30 .

   	A	B	c	D	E	F	G	H	I
   1	\(\mathrm { X } _ { 1 }\)	\(\mathrm { X } _ { 2 }\)	\(\mathrm { X } _ { 3 }\)	\(\mathrm { X } _ { 4 }\)	\(\mathrm { X } _ { 5 }\)	T		t	Number \(\leqslant \mathrm { t }\)
   2	3.73	6.65	4.93	0.41	9.33	25.06		0	0
   3	4.95	6.58	4.48	2.51	7.26	25.79		5	0
   4	8.10	4.87	4.26	3.83	0.79	21.85		10	1
   5	6.70	4.10	5.10	1.82	6.76	24.48		15	4
   6	3.73	8.38	8.49	9.87	1.31	31.79		20	23
   7	3.22	4.36	0.12	1.34	9.49	18.53		25	48
   8	9.17	7.13	5.47	4.35	2.44	28.55		30	75
   9	3.42	1.93	6.04	2.99	8.85	23.24		35	93
   10	0.98	0.68	9.82	9.83	7.28	28.58		40	99
   11	5.86	1.67	7.77	4.08	7.14	26.52		45	100
   12	9.20	0.31	5.82	5.31	6.45	27.10		50	100
   13	7.04	4.30	2.06	0.06	4.16	17.62
   14	0.31	5.02	1.48	5.37	1.77	13.94
   15	3.77	6.04	1.21	7.67	5.01	23.69
   16	1.21	5.54	1.90	1.43	6.91	17.00
   17	9.27	1.98	5.80	9.37	9.34	35.76
   18	4.30	5.66	2.80	1.56	1.19	15.51
   19	7.15	3.19	6.89	5.41	2.18	24.82
   20	6.18	6.32	3.01	6.49	9.12	31.13
   21	5.03	5.99	5.19	6.97	3.55	26.73

3. Use the spreadsheet output to estimate each of the following.
   The random variable \(Y\) is the mean of 100 independent values of \(T\). Determine an estimate of \(\mathrm { P } ( Y > 26 )\).

3 At a launderette the process of cleaning a load of clothes consists of three stages: washing, drying and folding. The times in minutes for each process are modelled by independent Normal distributions with means and standard deviations as shown in the table.

1. Find the probability that drying a randomly chosen load of clothes takes more than 50 minutes.
2. It is given that for \(99 \%\) of loads of clothes the washing time is less than \(k\) minutes. Find the value of \(k\).
3. Determine the probability that the drying time for a randomly chosen load of clothes is less than the total of the washing and folding times.
4. Determine the probability that the mean time for cleaning 5 randomly chosen loads of clothes is less than 90 minutes. You should assume that the time for cleaning any load is independent of the time for cleaning any other load.

10 Ben takes an underground train to work and back home each day. The waiting time is defined as the time from when he reaches the station platform until he boards the train. On his way to work the waiting time is \(X\) minutes, where \(X\) is modelled by a continuous uniform distribution on \([ 0,6 ]\). On his way back from work, the waiting time is \(Y\) minutes, where \(Y\) is modelled by a continuous uniform distribution on [0,4]. Ben's total waiting time for both journeys is \(Z\) minutes, where \(Z = X + Y\). You should assume that \(X\) and \(Y\) are independent.

1. Find \(\mathrm { E } ( \mathrm { Z } )\).
2. Ben thinks that \(Z\) will be well modelled by a continuous uniform distribution on \([ 0,10 ]\). By considering variances, show that he is not correct.
3. Ben's friend Jamila constructs the spreadsheet below, which shows a simulation of 20 values of \(X , Y\) and \(Z\). All of the values have been rounded to 2 decimal places.

   \multirow[b]{3}{*}{ 1 2 }	A	B	C
   	X	Y	Z
   	1.17	3.83	5.01
   3	2.01	0.81	2.82
   4	1.27	1.52	2.78
   5	1.41	3.94	5.35
   6	4.11	2.94	7.05
   7	1.76	0.96	2.72
   8	3.29	0.98	4.27
   9	0.77	0.22	0.99
   10	0.99	1.44	2.43
   11	4.79	2.43	7.22
   12	3.82	3.93	7.75
   13	5.25	2.74	7.99
   14	2.64	0.48	3.12
   15	1.54	2.18	3.72
   16	2.71	1.66	4.36
   17	0.04	3.24	3.28
   18	5.95	3.12	9.07
   19	5.22	1.21	6.42
   20	4.16	0.11	4.27
   21	1.02	0.99	2.01
   22

   Write down an estimate of \(\mathrm { P } ( Z > 6 )\).
4. Use a Normal approximation to determine the probability that Ben's total waiting time when travelling to and from work on 40 days is more than 210 minutes.

11 The discrete random variable \(X\) has a uniform distribution over the set of all integers between 25 and \(n\) inclusive, where \(n\) is a positive integer with \(n > 25\).

1. Determine \(\mathrm { P } \left( \mathrm { X } < \frac { \mathrm { n } + 25 } { 2 } \right)\) in each of the following cases.

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 \(\mathrm { P } ( W > 1 )\).

10 Sarah takes a bus to work each weekday morning and returns each evening. The times in minutes that she has to wait for the bus in the morning and evening are modelled by uniform distributions over the intervals \([ 0,10 ]\) and \([ 0,6 ]\) respectively. The times in minutes for the bus journeys in the morning and evening are modelled by \(\mathrm { N } ( 25,4 )\) and \(\mathrm { N } ( 28,16 )\) respectively. You should assume that all of the times are independent. The total time in minutes that she takes for her two journeys, including the waiting times, is denoted by the random variable \(T\). The spreadsheet below shows the first 20 rows of a simulation of 500 return journeys. It also shows in column H the numbers of values of \(T\) that are less than or equal to the corresponding values in column G. For example, there are 156 out of the 500 simulated values of \(T\) which are less than or equal to 58 minutes. All of the times have been rounded to 2 decimal places.

1. Use the spreadsheet output to estimate each of the following.

2. The probability of winning a certain game at a funfair is \(p\). Aman plays the game 5 times and Boaz plays the game 8 times. The independent random variables \(X\) and \(Y\) denote the number of wins for Aman and Boaz respectively.

1. Given that \(\mathrm { E } ( X Y ) = 6 \cdot 4\), calculate \(p\).
2. Find \(\operatorname { Var } ( X Y )\).

4. The continuous random variable, \(X\), has the following probability density function $$f ( x ) = \begin{cases} k x & \text { for } 0 \leqslant x < 1 \\ k x ^ { 3 } & \text { for } 1 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant. \includegraphics[max width=\textwidth, alt={}, center]{4ecf99c5-c4b3-41b7-a8df-a7c2ca7fcd6a-3_851_935_826_678}

1. Show that \(k = \frac { 4 } { 17 }\).
2. Determine \(\mathrm { E } ( X )\).
3. Calculate \(\mathrm { E } ( 3 X - 1 )\) and \(\operatorname { Var } ( 3 X - 1 )\).

2. The random variables \(X\) and \(Y\) are independent, with \(X\) having mean \(\mu\) and variance \(\sigma ^ { 2 }\), and \(Y\) having mean \(\mu\) and variance \(k \sigma ^ { 2 }\), where \(k\) is a positive constant. Let \(\bar { X }\) denote the mean of a random sample of 20 observations of \(X\), and let \(\bar { Y }\) denote the mean of a random sample of 25 observations of \(Y\).

1. Given that \(T _ { 1 } = \frac { 3 \bar { X } + 7 \bar { Y } } { 10 }\), show that \(T _ { 1 }\) is an unbiased estimator for \(\mu\).
2. Given that \(T _ { 2 } = \frac { \bar { X } + a ^ { 2 } \bar { Y } } { 1 + a } , a > 0\), and \(T _ { 2 }\) is an unbiased estimator for \(\mu\), prove that \(a = 1\).
3. Find and simplify expressions for the variances of \(T _ { 1 }\) and \(T _ { 2 }\).
4. Show that the value of \(k\) for which \(T _ { 1 }\) and \(T _ { 2 }\) are equally good estimators is \(\frac { 5 } { 6 }\).
5. Given that \(T _ { 3 } = ( 1 - \lambda ) \bar { X } + \lambda \bar { Y }\), find an expression for \(\lambda\), in terms of \(k\), for which \(T _ { 3 }\) has the smallest possible variance.

5. The masses, \(X\), in kg, of men who work for a large company are normally distributed with mean 75 and standard deviation 10.

1. Find the probability that the mean mass of a random sample of 5 men is less than 70 kg .
2. The mean mass, in kg , of a random sample of \(n\) men drawn from this distribution is \(\bar { X }\). Given that \(\mathrm { P } ( \bar { X } > 80 )\) is approximately \(0 \cdot 007\), find \(n\). The masses, in kg, of women who work for the company are normally distributed with mean 68 and standard deviation 6 . A lift in the company building will not move if the total mass in the lift is more than 500 kg .
3. A random sample of 3 men and 4 women get in the lift. Find the probability that the lift will not move.
4. State a modelling assumption you have made in calculating your answer for part (c).

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 probability distribution given by

The random variable \(Y = 2 - 5 X\) Given that \(\mathrm { E } ( \mathrm { Y } ) = - 4\) and \(\mathrm { P } ( \mathrm { Y } \geqslant - 3 ) = 0.45\)

1. find the probability distribution of X . Given also that \(\mathrm { E } \left( \mathrm { Y } ^ { 2 } \right) = 75\)
2. find the exact value of \(\operatorname { Var } ( \mathrm { X } )\)
3. Find \(\mathrm { P } ( \mathrm { Y } > \mathrm { X } )\) \section*{Q uestion 2 continued}

1. A biased spinner can land on the numbers \(1,2,3,4\) or 5 with the following probabilities.

The spinner will be spun 80 times and the mean of the numbers it lands on will be calculated. Find an estimate of the probability that this mean will be greater than 3.25
(6)

1. A six-sided die has sides labelled \(1,2,3,4,5\) and 6

The random variable \(S\) represents the score when the die is rolled.
Alicia rolls the die 45 times and the mean score, \(\bar { S }\), is calculated.
Assuming the die is fair and using a suitable approximation,

1. find, to 3 significant figures, the value of \(k\) such that \(\mathrm { P } ( \bar { S } < k ) = 0.05\)
2. Explain the relevance of the Central Limit Theorem in part (a). Alicia considers the following hypotheses: \(\mathrm { H } _ { 0 }\) : The die is fair \(\mathrm { H } _ { 1 }\) : The die is not fair
   If \(\bar { S } < 3.1\) or \(\bar { S } > 3.9\), then \(\mathrm { H } _ { 0 }\) will be rejected.
   Given that the true distribution of \(S\) has mean 4 and variance 3
3. find the power of this test.
4. Describe what would happen to the power of this test if Alicia were to increase the number of rolls of the die.
   Give a reason for your answer.

7 A manufacturer makes two versions of a toy. One version is made out of wood and the other is made out of plastic. The weights, \(W \mathrm {~kg}\), of the wooden toys are normally distributed with mean 2.5 kg and standard deviation 0.7 kg . The weights, \(X \mathrm {~kg}\), of the plastic toys are normally distributed with mean 1.27 kg and standard deviation 0.4 kg . The random variables \(W\) and \(X\) are independent.

1. Find the probability that the weight of a randomly chosen wooden toy is more than double the weight of a randomly chosen plastic toy. The manufacturer packs \(n\) of these wooden toys and \(2 n\) of these plastic toys into the same container. The maximum weight the container can hold is 252 kg . The probability of the contents of this container being overweight is 0.2119 to 4 decimal places.
2. Calculate the value of \(n\).

7 Fence panels come in two sizes, large and small. The lengths of the large panels are normally distributed with mean 198 cm and standard deviation 5 cm . The lengths of the small panels are normally distributed with mean 74 cm and standard deviation 3 cm .

1. Find the probability that the total length of a random sample of 3 large panels is greater than the total length of a random sample of 8 small panels. One large panel and one small panel are selected at random.
2. Find the probability that the length of the large panel is more than \(\frac { 8 } { 3 }\) times the length of the small panel. Rosa needs 1000 cm of fencing. The large panels cost \(\pounds 80\) each and the small panels cost \(\pounds 30\) each. Rosa's plan is to buy 5 large panels and measure the total length. If the total length is less than 1000 cm she will then buy one small panel as well.
3. Calculate whether or not the expected cost of Rosa's plan is cheaper than simply buying 14 small panels.

1. The weights of a particular type of apple, \(A\) grams, and a particular type of orange, \(R\) grams, each follow independent normal distributions.

$$A \sim \mathrm {~N} \left( 160,12 ^ { 2 } \right) \quad R \sim \mathrm {~N} \left( 140,10 ^ { 2 } \right)$$

1. Find the distribution of
   1. \(A + R\)
   2. the total weight of 2 randomly selected apples. A box contains 4 apples and 1 orange only. Jesse selects 2 pieces of fruit at random from the box.
2. Find the probability that the total weight of the 2 pieces of fruit exceeds 310 grams. From a large number of apples and oranges, Celeste selects \(m\) apples and 1 orange at random. The random variable \(W\) is given by $$W = \left( \sum _ { i = 1 } ^ { m } A _ { i } \right) - n \times R$$ where \(n\) is a positive integer.
   Given that the middle \(95 \%\) of the distribution of \(W\) lies between 1100.08 and 1499.92 grams, (c) find the value of \(m\) and the value of \(n\)

1. The random variable \(X \sim \mathrm {~N} \left( 5,0.4 ^ { 2 } \right)\) and the random variable \(Y \sim \mathrm {~N} \left( 8,0.1 ^ { 2 } \right)\) \(X\) and \(Y\) are independent random variables.
   A random sample of \(a\) independent observations is taken from the distribution of \(X\) and one observation is taken from the distribution of \(Y\)

The random variable \(W = X _ { 1 } + X _ { 2 } + X _ { 3 } + \ldots + X _ { a } + b Y\) and has the distribution \(\mathrm { N } \left( 169,2 ^ { 2 } \right)\) Find the value of \(a\) and the value of \(b\)

1. Scaffolding poles come in two sizes, long and short. The length \(L\) of a long pole has the normal distribution \(\mathrm { N } \left( 19.6,0.6 ^ { 2 } \right)\). The length \(S\) of a short pole has the normal distribution N(4.8, 0.32). The random variables \(L\) and \(S\) are independent.

A long pole and a short pole are selected at random.

1. Find the probability that the length of the long pole is more than 4 times the length of the short pole. Show your working clearly. Four short poles are selected at random and placed end to end in a row. The random variable \(T\) represents the length of the row.
2. Find the distribution of \(T\).
3. Find \(\mathrm { P } ( | L - T | < 0.2 )\)

1 The numbers of customers arriving at a ticket desk between 8 a.m. and 9 a.m. on a Monday morning and on a Tuesday morning are denoted by \(X\) and \(Y\) respectively. It is given that \(X \sim \operatorname { Po } ( 17 )\) and \(Y \sim \operatorname { Po } ( 14 )\).

1. Find
   1. \(\mathrm { P } ( X + Y ) > 40\),
   2. \(\operatorname { Var } ( 2 X - Y )\).
   3. State a necessary assumption for your calculations in part (i) to be valid.

3 A discrete random variable \(X\) has the distribution \(\mathrm { U } ( 11 )\).
The mean of 50 observations of \(X\) is denoted by \(\bar { X }\).
Use an approximate method, which should be justified, to find \(\mathrm { P } ( \bar { X } \leqslant 6.10 )\).

9 The continuous random variable \(C\) has the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\). The sum of a random sample of 16 observations of \(C\) is 224.0 .

1. Find an unbiased estimate of \(\mu\).
2. It is given that an unbiased estimate of \(\sigma ^ { 2 }\) is 0.24. Find the value of \(\Sigma c ^ { 2 }\). \(D\) is the sum of 10 independent observations of \(C\).
3. Explain whether \(D\) has a normal distribution. The continuous random variable \(F\) is normally distributed with mean 15.0, and it is known that \(\mathrm { P } ( F < 13.2 ) = 0.115\).
4. Use the unbiased estimates of \(\mu\) and \(\sigma ^ { 2 }\) to find \(\mathrm { P } ( D + F > 157.0 )\). \section*{OCR} \section*{Oxford Cambridge and RSA}