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

4 The volume, in millilitres, of a small cup of coffee has the distribution \(\mathrm { N } ( 103.4,10.2 )\). The volume of a large cup of coffee is 1.5 times the volume of a small cup of coffee.

1. Find the mean and standard deviation of the volume of a large cup of coffee.
2. Find the probability that the total volume of a randomly chosen small cup of coffee and a randomly chosen large cup of coffee is greater than 250 ml .

1 At an internet café, the charge for using a computer is 5 cents per minute. The number of minutes for which people use a computer has mean 23 and standard deviation 8.

1. Find, in cents, the mean and standard deviation of the amount people pay when using a computer.
2. Each day, 15 people use computers independently. Find, in cents, the mean and standard deviation of the total amount paid by 15 people.

1 A fair six-sided die is thrown 20 times and the number of sixes, \(X\), is recorded. Another fair six-sided die is thrown 20 times and the number of odd-numbered scores, \(Y\), is recorded. Find the mean and standard deviation of \(X + Y\).

4 The masses, in grams, of large bags of sugar and small bags of sugar are denoted by \(X\) and \(Y\) respectively, where \(X \sim \mathrm {~N} \left( 5.1,0.2 ^ { 2 } \right)\) and \(Y \sim \mathrm {~N} \left( 2.5,0.1 ^ { 2 } \right)\). Find the probability that the mass of a randomly chosen large bag is less than twice the mass of a randomly chosen small bag.

6 The battery in Sue's phone runs out at random moments. Over a long period, she has found that the battery runs out, on average, 3.3 times in a 30-day period.

1. Find the probability that the battery runs out fewer than 3 times in a 25-day period.
2. (a) Use an approximating distribution to find the probability that the battery runs out more than 50 times in a year ( 365 days).
   (b) Justify the approximating distribution used in part (ii)(a).
3. Independently of her phone battery, Sue's computer battery also runs out at random moments. On average, it runs out twice in a 15-day period. Find the probability that the total number of times that her phone battery and her computer battery run out in a 10-day period is at least 4 .

6 The masses, in kilograms, of cartons of sugar and cartons of flour have the distributions \(\mathrm { N } \left( 78.8,12.6 ^ { 2 } \right)\) and \(\mathrm { N } \left( 62.0,10.0 ^ { 2 } \right)\) respectively.

1. The standard load for a certain crane is 8 cartons of sugar and 3 cartons of flour. The maximum load that can be carried safely by the crane is 900 kg . Stating a necessary assumption, find the percentage of standard loads that will exceed the maximum safe load.
2. Find the probability that a randomly chosen carton of sugar has a smaller mass than a randomly chosen carton of flour.

4 Each year a transport firm uses \(X\) litres of gasoline and \(Y\) litres of diesel fuel, where \(X\) and \(Y\) have the independent distributions \(X \sim \mathrm {~N} ( 10700,950 ) ^ { 2 }\) and \(Y \sim \mathrm {~N} \left( 13400,1210 ^ { 2 } \right)\).

1. Find the probability that in a randomly chosen year the firm uses more gasoline than diesel fuel.
   The costs per litre of gasoline and diesel fuel are \\(0.80 and \\)0.85 respectively.
2. Find the probability that the total cost of gasoline and diesel fuel in a randomly chosen year is between \(\\) 20000\( and \)\\( 22000\).

5 A teacher models the numbers of girls and boys who arrive late for her class on any day by the independent random variables \(G \sim \operatorname { Po } ( 0.10 )\) and \(B \sim \operatorname { Po } ( 0.15 )\) respectively.

1. Find the probability that during a randomly chosen 2-day period no girls arrive late.
2. Find the probability that during a randomly chosen 5-day period the total number of students who arrive late is less than 3 .
3. It is given that the values of \(\mathrm { P } ( G = r )\) and \(\mathrm { P } ( B = r )\) for \(r \geqslant 3\) are very small and can be ignored. Find the probability that on a randomly chosen day more girls arrive late than boys.
   Following a timetable change the teacher claims that on average more students arrive late than before the change. During a randomly chosen 5-day period a total of 4 students are late.
4. Test the teacher's claim at the \(5 \%\) significance level.

6 The numbers of barrels of oil, in millions, extracted per day in two oil fields \(A\) and \(B\) are modelled by the independent random variables \(X\) and \(Y\) respectively, where \(X \sim \mathrm {~N} \left( 3.2,0.4 ^ { 2 } \right)\) and \(Y \sim \mathrm {~N} \left( 4.3,0.6 ^ { 2 } \right)\). The income generated by the oil from the two fields is \(\\) 90\( per barrel for \)A\( and \)\\( 95\) per barrel for \(B\).

1. Find the mean and variance of the daily income, in millions of dollars, generated by field \(A\). [3]
2. Find the probability that the total income produced by the two fields in a day is at least \(\\) 670$ million.

1

1. 1. A random variable \(X\) has the distribution \(\mathrm { B } ( 2540,0.001 )\). Use the Poisson approximation to the binomial distribution to find \(\mathrm { P } ( X > 1 )\).
   2. Explain why the Poisson approximation is appropriate in this case.
2. Two independent random variables, \(S\) and \(T\), have distributions \(\operatorname { Po } ( 2.1 )\) and \(\operatorname { Po } ( 3.5 )\) respectively. Find the mean and standard deviation of \(S + T\).

5 The marks in paper 1 and paper 2 of an examination are denoted by \(X\) and \(Y\) respectively, where \(X\) and \(Y\) have the independent continuous distributions \(\mathrm { N } \left( 56,6 ^ { 2 } \right)\) and \(\mathrm { N } \left( 43,5 ^ { 2 } \right)\) respectively.

1. Find the probability that a randomly chosen paper 1 mark is more than a randomly chosen paper 2 mark.
2. Each candidate's overall mark is \(M\) where \(M = X + 1.5 Y\). The minimum overall mark for grade A is 135 . Find the proportion of students who gain a grade A .

4 Small drops of two liquids, \(A\) and \(B\), are randomly and independently distributed in the air. The average numbers of drops of \(A\) and \(B\) per cubic centimetre of air are 0.25 and 0.36 respectively.

1. A sample of \(10 \mathrm {~cm} ^ { 3 }\) of air is taken at random. Find the probability that the total number of drops of \(A\) and \(B\) in this sample is at least 4 .
2. A sample of \(100 \mathrm {~cm} ^ { 3 }\) of air is taken at random. Use an approximating distribution to find the probability that the total number of drops of \(A\) and \(B\) in this sample is less than 60 .

5 The times, in months, taken by a builder to build two types of house, \(P\) and \(Q\), are represented by the independent variables \(T _ { 1 } \sim \mathrm {~N} \left( 2.2,0.4 ^ { 2 } \right)\) and \(T _ { 2 } \sim \mathrm {~N} \left( 2.8,0.5 ^ { 2 } \right)\) respectively.

1. Find the probability that the total time taken to build one house of each type is less than 6 months.
2. Find the probability that the time taken to build a type \(Q\) house is more than 1.2 times the time taken to build a type \(P\) house.

3 Sugar and flour for making cakes are measured in cups. The mass, in grams, of one cup of sugar has the distribution \(\mathrm { N } ( 250,10 )\). The mass, in grams, of one cup of flour has the independent distribution \(\mathrm { N } ( 160,9 )\). Each cake contains 2 cups of sugar and 5 cups of flour. Find the probability that the total mass of sugar and flour in one cake exceeds 1310 grams.

7 The independent random variables \(X\) and \(Y\) have the distributions \(\operatorname { Po } ( 2.1 )\) and \(\operatorname { Po } ( 3.5 )\) respectively.

1. Find \(\mathrm { P } ( X + Y = 3 )\).
2. Given that \(X + Y = 3\), find \(\mathrm { P } ( X = 2 )\).
3. A random sample of 100 values of \(X\) is taken. Find the probability that the sample mean is more than 2.2.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

5 The masses, in grams, of large boxes of chocolates and small boxes of chocolates have the distributions \(\mathrm { N } ( 325,6.1 )\) and \(\mathrm { N } ( 167,5.6 )\) respectively.

1. Find the probability that the total mass of 10 randomly chosen large boxes of chocolates is less than 3240 g .
2. Find the probability that the mass of a randomly chosen large box of chocolates is more than twice the mass of a randomly chosen small box of chocolates.

1 The random variable \(X\) has mean 2.4 and variance 3.1.

1. The random variable \(Y\) is the sum of four independent values of \(X\). Find the mean and variance of \(Y\).
2. The random variable \(Z\) is defined by \(Z = 4 X - 3\). Find the mean and variance of \(Z\).

3 Tien throws a ball. The distance it travels can be modelled by a normal distribution with mean 20 m and variance \(9 \mathrm {~m} ^ { 2 }\). His younger sister Su Chen also throws a ball and the distance her ball travels can be modelled by a normal distribution with mean 14 m and variance \(12 \mathrm {~m} ^ { 2 }\). Su Chen is allowed to add 5 metres on to her distance and call it her 'upgraded distance'. Find the probability that Tien's distance is larger than Su Chen's upgraded distance.

4 The weights of men follow a normal distribution with mean 71 kg and standard deviation 7 kg . The weights of women follow a normal distribution with mean 57 kg and standard deviation 5 kg . The total weight of 5 men and 2 women chosen randomly is denoted by \(X \mathrm {~kg}\).

1. Show that \(\mathrm { E } ( X ) = 469\) and \(\operatorname { Var } ( X ) = 295\).
2. The total weight of 4 men and 3 women chosen randomly is denoted by \(Y \mathrm {~kg}\). Find the mean and standard deviation of \(X - Y\) and hence find \(\mathrm { P } ( X - Y > 22 )\).

4 The cost of electricity for a month in a certain town under scheme \(A\) consists of a fixed charge of 600 cents together with a charge of 5.52 cents per unit of electricity used. Stella uses scheme \(A\). The number of units she uses in a month is normally distributed with mean 500 and variance 50.41.

1. Find the mean and variance of the total cost of Stella's electricity in a randomly chosen month. Under scheme \(B\) there is no fixed charge and the cost in cents for a month is normally distributed with mean 6600 and variance 421. Derek uses scheme \(B\).
2. Find the probability that, in a randomly chosen month, Derek spends more than twice as much as Stella spends.

7

1. Random variables \(Y\) and \(X\) are related by \(Y = a + b X\), where \(a\) and \(b\) are constants and \(b > 0\). The standard deviation of \(Y\) is twice the standard deviation of \(X\). The mean of \(Y\) is 7.92 and is 0.8 more than the mean of \(X\). Find the values of \(a\) and \(b\).
2. Random variables \(R\) and \(S\) are such that \(R \sim \mathrm {~N} \left( \mu , 2 ^ { 2 } \right)\) and \(S \sim \mathrm {~N} \left( 2 \mu , 3 ^ { 2 } \right)\). It is given that \(\mathrm { P } ( R + S > 1 ) = 0.9\).
   1. Find \(\mu\).
   2. Hence find \(\mathrm { P } ( S > R )\).

5 The marks of candidates in Mathematics and English in 2009 were represented by the independent random variables \(X\) and \(Y\) with distributions \(\mathrm { N } \left( 28,5.6 ^ { 2 } \right)\) and \(\mathrm { N } \left( 52,12.4 ^ { 2 } \right)\) respectively. Each candidate's marks were combined to give a final mark \(F\), where \(F = X + \frac { 1 } { 2 } Y\).

1. Find \(\mathrm { E } ( F )\) and \(\operatorname { Var } ( F )\).
2. The final marks of a random sample of 10 candidates from Grinford in 2009 had a mean of 49. Test at the 5\% significance level whether this result suggests that the mean final mark of all candidates from Grinford in 2009 was lower than elsewhere.

4 The masses, in milligrams, of three minerals found in 1 tonne of a certain kind of rock are modelled by three independent random variables \(P , Q\) and \(R\), where \(P \sim \mathrm {~N} \left( 46,19 ^ { 2 } \right) , Q \sim \mathrm {~N} \left( 53,23 ^ { 2 } \right)\) and \(R \sim \mathrm {~N} \left( 25,10 ^ { 2 } \right)\). The total value of the minerals found in 1 tonne of rock is modelled by the random variable \(V\), where \(V = P + Q + 2 R\). Use the model to find the probability of finding minerals with a value of at least 93 in a randomly chosen tonne of rock.

3 Three coats of paint are sprayed onto a surface. The thicknesses, in millimetres, of the three coats have independent distributions \(\mathrm { N } \left( 0.13,0.02 ^ { 2 } \right) , \mathrm { N } \left( 0.14,0.03 ^ { 2 } \right)\) and \(\mathrm { N } \left( 0.10,0.01 ^ { 2 } \right)\). Find the probability that, at a randomly chosen place on the surface, the total thickness of the three coats of paint is less than 0.30 millimetres.

3 The cost of hiring a bicycle consists of a fixed charge of 500 cents together with a charge of 3 cents per minute. The number of minutes for which people hire a bicycle has mean 142 and standard deviation 35.

1. Find the mean and standard deviation of the amount people pay when hiring a bicycle.
2. 6 people hire bicycles independently. Find the mean and standard deviation of the total amount paid by all 6 people.