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

7

1. The number of text messages, \(N\), sent by Peter each month on his mobile phone never exceeds 40. When \(0 \leqslant N \leqslant 10\), he is charged for 5 messages.
   When \(10 < N \leqslant 20\), he is charged for 15 messages.
   When \(20 < N \leqslant 30\), he is charged for 25 messages.
   When \(30 < N \leqslant 40\), he is charged for 35 messages.
   The number of text messages, \(Y\), that Peter is charged for each month has the following probability distribution:

   \(\boldsymbol { y }\)	5	15	25	35
   \(\mathbf { P } ( \boldsymbol { Y } = \boldsymbol { y } )\)	0.1	0.2	0.3	0.4

   1. Calculate the mean and the standard deviation of \(Y\).
   2. The Goodtime phone company makes a total charge for text messages, \(C\) pence, each month given by $$C = 10 Y + 5$$ Calculate \(\mathrm { E } ( C )\).
2. The number of text messages, \(X\), sent by Joanne each month on her mobile phone is such that $$\mathrm { E } ( X ) = 8.35 \quad \text { and } \quad \mathrm { E } \left( X ^ { 2 } \right) = 75.25$$ The Newtime phone company makes a total charge for text messages, \(T\) pence, each month given by $$T = 0.4 X + 250$$ Calculate \(\operatorname { Var } ( T )\).

4 A house has a total of five bedrooms, at least one of which is always rented.
The probability distribution for \(R\), the number of bedrooms that are rented at any given time, is given by $$\mathrm { P } ( R = r ) = \begin{cases} 0.5 & r = 1 \\ 0.4 ( 0.6 ) ^ { r - 1 } & r = 2,3,4 \\ 0.0296 & r = 5 \end{cases}$$

1. Complete the table below.
2. Find the probability that fewer than 3 bedrooms are not rented at any given time.
3. 1. Find the value of \(\mathrm { E } ( R )\).
   2. Show that \(\mathrm { E } \left( R ^ { 2 } \right) = 4.8784\) and hence find the value of \(\operatorname { Var } ( R )\).
4. Bedrooms are rented on a monthly basis. The monthly income, \(\pounds M\), from renting bedrooms in the house may be modelled by $$M = 1250 R - 282$$ Find the mean and the standard deviation of \(M\).

   \(\boldsymbol { r }\)	1	2	3	4	5
   \(\mathbf { P } ( \boldsymbol { R } = \boldsymbol { r } )\)	0.5				0.0296

2. Suggest, with reasons, suitable distributions for modelling each of the following:

1. the number of times the letter J occurs on each page of a magazine,
2. the length of string left over after cutting as many 3 metre long pieces as possible from partly used balls of string,
3. the number of heads obtained when spinning a coin 15 times.

6. A teacher is monitoring attendance at lessons in her department. She believes that the number of students absent from each lesson follows a Poisson distribution and wished to test the null hypothesis that the mean is 2.5 against the alternative hypothesis that it is greater than 2.5 She visits one lesson and decides on a critical region of 6 or more students absent.

1. Find the significance level of this test.
2. State any assumptions made in carrying out this test and comment on their validity. The teacher decides to undertake a wider study by looking at a sample of all the lessons that have taken place in the department during the previous four weeks.
3. Suggest a suitable sampling frame. She finds that there have been 96 pupils absent from the 30 lessons in her sample.
4. Using a suitable approximation, test at the \(5 \%\) level of significance the null hypothesis that the mean is 2.5 students absent per lesson against the alternative hypothesis that it is greater than 2.5. You may assume that the number of absences follows a Poisson distribution.
   (6 marks)

2. A driving instructor keeps records of all the learners she has taught. In order to analyse her success rate she wishes to take a random sample of 120 of these learners.

1. Suggest a suitable sampling frame and identify the sampling units. She believes that only 1 in 20 of the people she teaches fail to pass their test in their first two attempts. She decides to use her sample to test whether or not the proportion is different from this.
2. Using a suitable approximation and stating clearly the hypotheses she should use, find the largest critical region for this test such that the probability in each "tail" is less than \(2.5 \%\).
3. State the significance level of this test.

4. A music website is visited by an average of 30 different people per hour on a weekday evening. The site's designer believes that the number of visitors to the site per hour can be modelled by a Poisson distribution.

1. State the conditions necessary for a Poisson distribution to be applicable and comment on their validity in this case. Assuming that the number of visitors does follow a Poisson distribution, find the probability that there will be
2. less than two visitors in a 10 -minute interval,
3. at least ten visitors in a 15-minute interval.
4. Using a suitable approximation, find the probability of the site being visited by more than 100 people between 6 pm and 9 pm on a Thursday evening.
   (5 marks)

6 An aircraft, based at airport A, flies regularly to and from airport B.
The aircraft's flying time, \(X\) minutes, from A to B has a mean of 128 and a variance of 50 .
The aircraft's flying time, \(Y\) minutes, on the return flight from B to A is such that $$\mathrm { E } ( Y ) = 112 , \quad \operatorname { Var } ( Y ) = 50 \quad \text { and } \quad \rho _ { X Y } = - 0.4$$

1. Given that \(F = X + Y\) :
   1. find the mean of \(F\);
   2. show that the variance of \(F\) is 60 .
2. At airport B , the stopover time, \(S\) minutes, is independent of \(F\) and has a mean of 75 and a variance of 36 . Find values for the mean and the variance of:
   1. \(T = F + S\);
   2. \(M = F - 3 S\).
3. Hence, assuming that \(T\) and \(M\) are normally distributed, determine the probability that, on a particular round trip of the aircraft from A to B and back to A :
   1. the time from leaving A to returning to A exceeds 300 minutes;
   2. the stopover time is greater than one third of the total flying time.

7

1. The random variable \(X\) has a Poisson distribution with \(\mathrm { E } ( X ) = \lambda\).
   1. Prove, from first principles, that \(\mathrm { E } ( X ( X - 1 ) ) = \lambda ^ { 2 }\).
   2. Hence deduce that \(\operatorname { Var } ( X ) = \lambda\).
2. The independent Poisson random variables \(X _ { 1 }\) and \(X _ { 2 }\) are such that \(\mathrm { E } \left( X _ { 1 } \right) = 5\) and \(\mathrm { E } \left( X _ { 2 } \right) = 2\). The random variables \(D\) and \(F\) are defined by $$D = X _ { 1 } - X _ { 2 } \quad \text { and } \quad F = 2 X _ { 1 } + 10$$
   1. Determine the mean and the variance of \(D\).
   2. Determine the mean and the variance of \(F\).
   3. For each of the variables \(D\) and \(F\), give a reason why the distribution is not Poisson.
3. The daily number of black printer cartridges sold by a shop may be modelled by a Poisson distribution with a mean of 5 . Independently, the daily number of colour printer cartridges sold by the same shop may be modelled by a Poisson distribution with a mean of 2. Use a distributional approximation to estimate the probability that the total number of black and colour printer cartridges sold by the shop during a 4 -week period ( 24 days) exceeds 175.

6 The table shows the probability distribution for the number of weekday (Monday to Friday) morning newspapers, \(X\), purchased by the Reed household per week.

1. Find values for \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
2. The number of weekday (Monday to Friday) evening newspapers, \(Y\), purchased by the same household per week is such that $$\mathrm { E } ( Y ) = 2.0 , \quad \operatorname { Var } ( Y ) = 1.5 \quad \text { and } \quad \operatorname { Cov } ( X , Y ) = - 0.43$$ Find values for the mean and variance of:
   1. \(S = X + Y\);
   2. \(\quad D = X - Y\).
3. The total cost per week, \(L\), of the Reed household's weekday morning and evening newspapers may be assumed to be normally distributed with a mean of \(\pounds 2.31\) and a standard deviation of \(\pounds 0.89\). The total cost per week, \(M\), of the household's weekend (Saturday and Sunday) newspapers may be assumed to be independent of \(L\) and normally distributed with a mean of \(\pounds 2.04\) and a standard deviation of \(\pounds 0.43\). Determine the probability that the total cost per week of the Reed household's newspapers is more than \(\pounds 5\).

5 In the manufacture of desk drawer fronts, a machine cuts sheets of veneered chipboard into rectangular pieces of width \(W\) millimetres and height \(H\) millimetres. The 4 edges of each of these pieces are then covered with matching veneered tape. The distributions of \(W\) and \(H\) are such that $$\mathrm { E } ( W ) = 350 \quad \operatorname { Var } ( W ) = 5 \quad \mathrm { E } ( H ) = 210 \quad \operatorname { Var } ( H ) = 4 \quad \rho _ { W H } = 0.75$$

1. Calculate the mean and the variance of the length of tape, \(T = 2 W + 2 H\), needed for the edges of a drawer front.
2. A desk has 4 such drawers whose sizes may be assumed to be independent. Given that \(T\) may be assumed to be normally distributed, determine the probability that the total length of tape needed for the edges of the desk's 4 drawer fronts does not exceed 4.5 metres.
   \includegraphics[max width=\textwidth, alt={}]{b855b5b3-097e-4894-aaec-d77f515949b0-13_2484_1709_223_153}

6 The weight, \(X\) grams, of a dressed pheasant may be modelled by a normal random variable with a mean of 1000 and a standard deviation of 120 . Pairs of dressed pheasants are selected for packing into boxes. The total weight of a pair, \(Y = X _ { 1 } + X _ { 2 }\) grams, may be modelled by a normal distribution with a mean of 2000 and a standard deviation of 140 .

1. 1. Show that \(\operatorname { Cov } \left( X _ { 1 } , X _ { 2 } \right) = - 4600\).
   2. Given that \(X _ { 1 } - X _ { 2 }\) may be assumed to be normally distributed, determine the probability that the difference between the weights of a selected pair of dressed pheasants exceeds 250 grams.
2. The weight of a box is independent of the total weight of a pair of dressed pheasants, and is normally distributed with a mean of 500 grams and a standard deviation of 40 grams. Determine the probability that a box containing a pair of dressed pheasants weighs less than 2750 grams.
   \includegraphics[max width=\textwidth, alt={}]{fa3bf9d6-f064-4214-acff-d8b88c33a81e-16_2486_1714_221_153}

6 Alyssa lives in the country but works in a city centre.
Her journey to work each morning involves a car journey, a walk and wait, a train journey, and a walk. Her car journey time, \(U\) minutes, from home to the village car park has a mean of 13 and a standard deviation of 3 . Her time, \(V\) minutes, to walk from the village car park to the village railway station and wait for a train to depart has a mean of 15 and a standard deviation of 6 . Her train journey time, \(W\) minutes, from the village railway station to the city centre railway station has a mean of 24 and a standard deviation of 4 . Her time, \(X\) minutes, to walk from the city centre railway station to her office has a mean of 9 and a standard deviation of 2 . The values of the product moment correlation coefficient for the above 4 variables are $$\rho _ { U V } = - 0.6 \quad \text { and } \quad \rho _ { U W } = \rho _ { U X } = \rho _ { V W } = \rho _ { V X } = \rho _ { W X } = 0$$

1. Determine values for the mean and the variance of:
   1. \(M = U + V\);
   2. \(D = W - 2 U\);
   3. \(T = M + W + X\), given that \(\rho _ { M W } = \rho _ { M X } = 0\).
2. Assuming that the variables \(M , D\) and \(T\) are normally distributed, determine the probability that, on a particular morning:
   1. Alyssa's journey time from leaving home to leaving the village railway station is exactly 30 minutes;
   2. Alyssa's train journey time is more than twice her car journey time;
   3. Alyssa's total journey time is between 50 minutes and 70 minutes.

5 The schedule for an organisation's afternoon meeting is as follows.
Session A (Speaker 1) 2.00 pm to 3.15 pm
Session B (Discussion) 3.15 pm to 3.45 pm
Session C (Speaker 2) \(\quad 3.45 \mathrm { pm }\) to 5.00 pm
Records show that:
the duration, \(X\), of Session A has mean 68 minutes and standard deviation 10 minutes;
the duration, \(Y\), of Session B has mean 25 minutes and standard deviation 5 minutes;
the duration, \(Z\), of Session C has mean 73 minutes and standard deviation 15 minutes;
and that: $$\rho _ { X Z } = 0 \quad \rho _ { X Y } = - 0.8 \quad \rho _ { Y Z } = 0$$

1. Determine the means and the variances of:
   1. \(L = X + Z\);
   2. \(M = X + Y\).
2. Assuming that \(L\) and \(M\) are each normally distributed, determine the probability that:
   1. the total time for the two speaker sessions is less than \(2 \frac { 1 } { 2 }\) hours;
   2. Session C is late in starting.

5 The numbers of daily morning operations, \(X\), and daily afternoon operations, \(Y\), in an operating theatre of a small private hospital can be modelled by the following bivariate probability distribution.

1. 1. State why \(\mathrm { E } ( X ) = 4\) and show that \(\operatorname { Var } ( X ) = 0.9\).
   2. Given that $$\mathrm { E } ( Y ) = 3.7 , \operatorname { Var } ( Y ) = 0.61 \text { and } \mathrm { E } ( X Y ) = 14.4$$ calculate values for \(\operatorname { Cov } ( X , Y )\) and \(\rho _ { X Y }\).
2. Calculate values for the mean and the variance of:
   1. \(T = X + Y\);
   2. \(\quad D = X - Y\).
      [0pt] [4 marks]

6 Population \(A\) has a normal distribution with unknown mean \(\mu _ { A }\) and a variance of 18.8.
Population \(B\) has a normal distribution with unknown mean \(\mu _ { B }\) but with the same variance as Population \(A\). The random variables \(\bar { X } _ { A }\) and \(\bar { X } _ { B }\) denote the means of independent samples, each of size \(n\), from populations \(A\) and \(B\) respectively.

1. Find an expression, in terms of \(n\), for \(\operatorname { Var } \left( \bar { X } _ { A } - \bar { X } _ { B } \right)\).
2. Given that the width of a \(99 \%\) confidence interval for \(\mu _ { A } - \mu _ { B }\) is to be at most 5 , calculate the minimum value for \(n\).
   [0pt] [5 marks]

6

1. The independent random variables \(S\) and \(L\) have means \(\mu _ { S }\) and \(\mu _ { L }\) respectively, and a common variance of \(\sigma ^ { 2 }\). The variable \(\bar { S }\) denotes the mean of a random sample of \(n\) observations on \(S\) and the variable \(\bar { L }\) denotes the mean of a random sample of \(n\) observations on \(L\). Find a simplified expression, in terms of \(\sigma ^ { 2 }\), for the variance of \(\bar { L } - 2 \bar { S }\).
2. A machine fills both small bottles and large bottles with shower gel. It is known that the volume of shower gel delivered by the machine is normally distributed with a standard deviation of 8 ml .
   1. A random sample of 25 small bottles filled by the machine contained a mean volume of \(\bar { s } = 258 \mathrm { ml }\) of shower gel. An independent random sample of 25 large bottles filled by the machine contained a mean volume of \(\bar { l } = 522 \mathrm { ml }\) of shower gel. Investigate, at the \(10 \%\) level of significance, the hypothesis that the mean volume of shower gel in a large bottle is more than twice that in a small bottle.
      [0pt] [7 marks]
   2. Deduce that, for the test of the hypothesis in part (b)(i), the critical value of \(\bar { L } - 2 \bar { S }\) is 4.585 , correct to three decimal places.
      [0pt] [2 marks]
   3. In fact, the mean volume of shower gel in a large bottle exceeds twice that in a small bottle by 10 ml . Determine the probability of a Type II error for a test of the hypothesis in part (b)(i) at the 10\% level of significance, based upon random samples of 25 small bottles and 25 large bottles.
      [0pt] [4 marks]

3. The time that a school pupil spends on French homework each week is normally distributed with a mean of 55 minutes and a standard deviation of 10 minutes. The time that this pupil spends on English homework each week is normally distributed with a mean of 1 hour 30 minutes and a standard deviation of 18 minutes. Find the probability that in a randomly chosen week

1. the pupil spends more than 2 hours in total doing French and English homework,
2. the pupil spends more than twice as long doing English homework as he spends doing French homework.
   (6 marks)

3 At a pottery which manufactures mugs, it is known that \(5 \%\) of mugs are faulty. The mugs are produced in batches of 20 . Faults are modelled as occurring randomly and independently. The number of faulty mugs in a batch is denoted by the random variable \(X\).

1. Determine \(\mathrm { P } ( X \geqslant 2 )\).
2. Find \(\operatorname { Var } ( X )\). Independently of the mugs, the pottery also manufactures cups, and it is known that \(7 \%\) of cups are faulty. The cups are produced in batches of 30 . Faults are modelled as occurring randomly and independently. The number of faulty cups in a batch is denoted by the random variable \(Y\).
3. Determine the standard deviation of \(X + Y\). When 10 batches of cups have been produced, a sample of 15 cups is tested to ensure that the handles of the cups are properly attached.
4. Explain why it might not be sensible to select a sample of 15 cups from the same batch.

2 A supermarket sells oranges. Their weights are modelled by the random variable \(X\) which has a Normal distribution with mean 345 grams and standard deviation 15 grams. When the oranges have been peeled, their weights in grams, \(Y\), are modelled by \(Y = 0.7 X\).

1. Find the probability that a randomly chosen peeled orange weighs less than 250 grams. I randomly choose 5 oranges to buy.
2. Find the probability that the total weight of the 5 unpeeled oranges is at least 1800 grams.
3. I peel three of the oranges and leave the remaining two unpeeled. Find the probability that the total weight of the two unpeeled oranges is greater than the total weight of the three peeled ones.

1 In a quiz a contestant is asked up to four questions. The contestant's turn ends once the contestant gets a question wrong or has answered all four questions. The probability that a particular contestant gets any question correct is 0.6 , independently of other questions. The discrete random variable \(X\) models the number of questions which the contestant gets correct in a turn.

1. Show that \(\mathrm { P } ( X = 4 ) = 0.1296\). The probability distribution of \(X\) is shown in Fig. 1.1. \begin{table}[h]

   \(r\)	0	1	2	3	4
   \(\mathrm { P } ( X = r )\)	0.4	0.24	0.144	0.0864	0.1296

   \captionsetup{labelformat=empty} \caption{Fig. 1.1}
   \end{table}
2. Find each of the following.
   The number of points that a contestant scores is as shown in Fig. 1.2. \begin{table}[h]

   Number of questions correct	Number of points scored
   0 or 1	0
   2	2
   3	3
   4	5

   \captionsetup{labelformat=empty} \caption{Fig. 1.2}
   \end{table} The discrete random variable \(Y\) models the number of points which the contestant scores.
3. Without doing any working, explain whether each of the following will be less than, equal to or greater than the corresponding value for \(X\).

6 The random variable \(X\) has a uniform distribution over the values \(\{ 1,4,7 , \ldots , 3 n - 2 \}\), where \(n\) is a positive integer.

1. Determine \(\operatorname { Var } ( X )\) in terms of \(n\).
2. Given that \(n = 100\), find the probability that \(X\) is within one standard deviation of the mean.

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 \(Y\).

7 The discrete random variable \(X\) has a uniform distribution over the set of all integers between 100 and \(n\) inclusive, where \(n\) is a positive integer with \(n > 100\).

1. Given that \(n\) is even, determine \(\mathrm { P } \left( \mathrm { X } < \frac { 100 + \mathrm { n } } { 2 } \right)\).
2. Determine the variance of the sum of 50 independent values of \(X\), giving your answer in the form \(\mathrm { a } \left( \mathrm { n } ^ { 2 } + \mathrm { bn } + \mathrm { c } \right)\), where \(a , b\) and \(c\) are constants.

3 The weights of bananas sold by a supermarket are modelled by a Normal distribution with mean 205 g and standard deviation 11 g .

1. Find the probability that the total weight of 5 randomly selected bananas is at least 1 kg . When a banana is peeled the change in its weight is modelled as being a reduction of \(35 \%\).
2. Find the probability that the weight of a randomly selected peeled banana is at most 150 g Andy makes smoothies. Each smoothie is made using 2 peeled bananas and 20 strawberries from the supermarket, all the items being randomly chosen. The weight of a strawberry is modelled by a Normal distribution with mean 22.5 g and standard deviation 2.7 g .
3. Find the probability that the total weight of a smoothie is less than 700 g .

2 A manufacturer is testing how long coloured LED lights will last before the battery runs out, using two different battery types. The times in hours before the battery runs out are modelled by independent Normal distributions with means and standard deviations as shown in the table.

1. In a particular test, a battery of type A is used and the time taken for it to run out is recorded. This process is repeated until a total of 5 randomly selected batteries have been used. Determine the probability that the total time the 5 batteries last is at least 120 hours.
2. In a similar test, 3 randomly selected batteries of type A are used, one after the other. Then 2 randomly selected batteries of type B are used, one after the other. Determine the probability that the 3 type A batteries last longer in total than the 2 type B batteries.
3. Explain why it is necessary that the Normal distributions are independent in order to be able to find the probability in part (b).