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

2 An examination consists of a written paper and a practical test. The written paper marks ( \(M\) ) have mean 54.8 and standard deviation 16.0. The practical test marks ( \(P\) ) are independent of the written paper marks and have mean 82.4 and standard deviation 4.8. The final mark is found by adding \(75 \%\) of \(M\) to \(25 \%\) of \(P\). Find the mean and standard deviation of the final marks for the examination. [3]

6 The random variable \(T\) denotes the time, in seconds, for 100 m races run by Tania. \(T\) is normally distributed with mean \(\mu\) and variance \(\sigma ^ { 2 }\). A random sample of 40 races run by Tania gave the following results. $$n = 40 \quad \Sigma t = 560 \quad \Sigma t ^ { 2 } = 7850$$

1. Calculate unbiased estimates of \(\mu\) and \(\sigma ^ { 2 }\).
   The random variable \(S\) denotes the time, in seconds, for 100 m races run by Suki. \(S\) has the independent distribution \(\mathrm { N } ( 14.2,0.3 )\).
2. Using your answers to part (a), find the probability that, in a randomly chosen 100 m race, Suki's time will be at least 0.1 s more than Tania's time.

5 In a certain large document, typing errors occur at random and at a constant mean rate of 0.2 per page.

1. Find the probability that there are fewer than 3 typing errors in 10 randomly chosen pages.
2. Use an approximating distribution to find the probability that there are more than 50 typing errors in 200 randomly chosen pages.
   In the same document, formatting errors occur at random and at a constant mean rate of 0.3 per page.
3. Find the probability that the total number of typing and formatting errors in 20 randomly chosen pages is between 8 and 11 inclusive.

3 Drops of water fall randomly from a leaking tap at a constant average rate of 5.2 per minute.

1. Find the probability that at least 3 drops fall during a randomly chosen 30 -second period.
2. Use a suitable approximating distribution to find the probability that at least 650 drops fall during a randomly chosen 2-hour period.

4 Each month a company sells \(X \mathrm {~kg}\) of brown sugar and \(Y \mathrm {~kg}\) of white sugar, where \(X\) and \(Y\) have the independent distributions \(\mathrm { N } \left( 2500,120 ^ { 2 } \right)\) and \(\mathrm { N } \left( 3700,130 ^ { 2 } \right)\) respectively.

1. Find the mean and standard deviation of the total amount of sugar that the company sells in 3 randomly chosen months.
   The company makes a profit of \(\\) 1.50\( per kilogram of brown sugar sold and makes a loss of \)\\( 0.20\) per kilogram of white sugar sold.
2. Find the probability that, in a randomly chosen month, the total profit is less than \(\\) 3000$.

4 The masses, in kilograms, of chemicals \(A\) and \(B\) produced per day by a factory are modelled by the independent random variables \(X\) and \(Y\) respectively, where \(X \sim \mathrm {~N} ( 10.3,5.76 )\) and \(Y \sim \mathrm {~N} ( 11.4,9.61 )\). The income generated by the chemicals is \(\\) 2.50\( per kilogram for \)A\( and \)\\( 3.25\) per kilogram for \(B\).

1. Find the mean and variance of the daily income generated by chemical \(A\). \includegraphics[max width=\textwidth, alt={}, center]{d42b3c4d-c426-4231-a35a-cac80dbdf82c-06_56_1566_495_333}
2. Find the probability that, on a randomly chosen day, the income generated by chemical \(A\) is greater than the income generated by chemical \(B\).

6 A factory makes loaves of bread in batches. One batch of loaves contains \(X\) kilograms of dried yeast and \(Y\) kilograms of flour, where \(X\) and \(Y\) have the independent distributions \(\mathrm { N } \left( 0.7,0.02 ^ { 2 } \right)\) and \(\mathrm { N } \left( 100.0,3.0 ^ { 2 } \right)\) respectively. Dried yeast costs \(\\) 13.50\( per kilogram and flour costs \)\\( 0.90\) per kilogram. For making one batch of bread the total of all other costs is \(\\) 55\(. The factory sells each batch of bread for \)\\( 200\). Find the probability that the profit made on one randomly chosen batch of bread is greater than \(\\) 40$. [7]

2 The masses, in kilograms, of small and large bags of wheat have the independent distributions \(\mathrm { N } ( 16.0,0.4 )\) and \(\mathrm { N } ( 51.0,0.9 )\) respectively. Find the probability that the total mass of 3 randomly chosen small bags is greater than the mass of one randomly chosen large bag. \includegraphics[max width=\textwidth, alt={}, center]{acd6f1c9-bbaf-40ca-b5cb-8466ddb9f596-04_2720_38_109_2010}

4 The lifetimes, in hours, of Longlive light bulbs and Enerlow light bulbs have the independent distributions \(\mathrm { N } \left( 1020,45 ^ { 2 } \right)\) and \(\mathrm { N } \left( 2800,52 ^ { 2 } \right)\) respectively.

1. Find the probability that the total of the lifetimes of five randomly chosen Longlive bulbs is less than 5200 hours.
2. Find the probability that the lifetime of a randomly chosen Enerlow bulb is at least three times that of a randomly chosen Longlive bulb.

2 In athletics matches the triple jump event consists of a hop, followed by a step, followed by a jump. The lengths covered by Albert in each part are independent normal variables with means \(3.5 \mathrm {~m} , 2.9 \mathrm {~m}\), 3.1 m and standard deviations \(0.3 \mathrm {~m} , 0.25 \mathrm {~m} , 0.35 \mathrm {~m}\) respectively. The length of the triple jump is the sum of the three parts.

1. Find the mean and standard deviation of the length of Albert's triple jumps.
2. Find the probability that the mean of Albert's next four triple jumps is greater than 9 m .

3 The independent random variables \(X\) and \(Y\) are such that \(X\) has mean 8 and variance 4.8 and \(Y\) has a Poisson distribution with mean 6. Find

1. \(\mathrm { E } ( 2 X - 3 Y )\),
2. \(\operatorname { Var } ( 2 X - 3 Y )\).

5 A clock contains 4 new batteries each of which gives a voltage which is normally distributed with mean 1.54 volts and standard deviation 0.05 volts. The voltages of the batteries are independent. The clock will only work if the total voltage is greater than 5.95 volts.

1. Find the probability that the clock will work.
2. Find the probability that the average total voltage of the batteries of 20 clocks chosen at random exceeds 6.2 volts.

2 A mathematics module is assessed by an examination and by coursework. The examination makes up \(75 \%\) of the total assessment and the coursework makes up \(25 \%\). Examination marks, \(X\), are distributed with mean 53.2 and standard deviation 9.3. Coursework marks, \(Y\), are distributed with mean 78.0 and standard deviation 5.1. Examination marks and coursework marks are independent. Find the mean and standard deviation of the combined mark \(0.75 X + 0.25 Y\).

4 A certain make of washing machine has a wash-time with mean 56.9 minutes and standard deviation 4.8 minutes. A certain make of tumble dryer has a drying-time with mean 61.1 minutes and standard deviation 6.3 minutes. Both times are normally distributed and are independent of each other. Find the probability that a randomly chosen wash-time differs by more than 3 minutes from a randomly chosen drying-time.

2 The random variable \(X\) has the distribution \(\mathrm { N } \left( 3.2,1.2 ^ { 2 } \right)\). The sum of 60 independent observations of \(X\) is denoted by \(S\). Find \(\mathrm { P } ( S > 200 )\).

3 The lengths of red pencils are normally distributed with mean 6.5 cm and standard deviation 0.23 cm .

1. Two red pencils are chosen at random. Find the probability that their total length is greater than 12.5 cm . The lengths of black pencils are normally distributed with mean 11.3 cm and standard deviation 0.46 cm .
2. Find the probability that the total length of 3 red pencils is more than 6.7 cm greater than the length of 1 black pencil.

5 Fiona and Jhoti each take one shower per day. The times, in minutes, taken by Fiona and Jhoti to take a shower are represented by the independent variables \(F \sim \mathrm {~N} \left( 12.2,2.8 ^ { 2 } \right)\) and \(J \sim \mathrm {~N} \left( 11.8,2.6 ^ { 2 } \right)\) respectively. Find the probability that, on a randomly chosen day,

1. the total time taken to shower by Fiona and Jhoti is less than 30 minutes,
2. Fiona takes at least twice as long as Jhoti to take a shower.

2 The independent random variables \(X\) and \(Y\) have the distributions \(\mathrm { N } ( 6.5,14 )\) and \(\mathrm { N } ( 7.4,15 )\) respectively. Find \(\mathrm { P } ( 3 X - Y < 20 )\).

3 Weights of cups have a normal distribution with mean 91 g and standard deviation 3.2 g . Weights of saucers have an independent normal distribution with mean 72 g and standard deviation 2.6 g . Cups and saucers are chosen at random to be packed in boxes, with 6 cups and 6 saucers in each box. Given that each empty box weighs 550 g , find the probability that the total weight of a box containing 6 cups and 6 saucers exceeds 1550 g .

5 Packets of cereal are packed in boxes, each containing 6 packets. The masses of the packets are normally distributed with mean 510 g and standard deviation 12 g . The masses of the empty boxes are normally distributed with mean 70 g and standard deviation 4 g .

1. Find the probability that the total mass of a full box containing 6 packets is between 3050 g and 3150 g .
2. A packet and an empty box are chosen at random. Find the probability that the mass of the packet is at least 8 times the mass of the empty box.

1 The mean and variance of the random variable \(X\) are 5.8 and 3.1 respectively. The random variable \(S\) is the sum of three independent values of \(X\). The independent random variable \(T\) is defined by \(T = 3 X + 2\).

1. Find the variance of \(S\).
2. Find the variance of \(T\).
3. Find the mean and variance of \(S - T\).

3 The daily times, in minutes, that Yu Ming takes showering, getting dressed and having breakfast are independent and have the distributions \(\mathrm { N } \left( 9,2.2 ^ { 2 } \right) , \mathrm { N } \left( 8,1.3 ^ { 2 } \right)\) and \(\mathrm { N } \left( 17,2.6 ^ { 2 } \right)\) respectively. The total daily time that Yu Ming takes for all three activities is denoted by \(T\) minutes.

1. Find the mean and variance of \(T\).
2. Yu Ming notes the value of \(T\) on each day in a random sample of 70 days and calculates the sample mean. Find the probability that the sample mean is between 33 and 35 .

7 The independent variables \(X\) and \(Y\) are such that \(X \sim \mathrm {~B} ( 10,0.8 )\) and \(Y \sim \mathrm { Po } ( 3 )\). Find

1. \(\mathrm { E } ( 7 X + 5 Y - 2 )\),
2. \(\operatorname { Var } ( 4 X - 3 Y + 3 )\),
3. \(\mathrm { P } ( 2 X - Y = 18 )\).

1 The independent random variables \(X\) and \(Y\) have standard deviations 3 and 6 respectively. Calculate the standard deviation of \(4 X - 5 Y\).

3 In a golf tournament, the number of times in a day that a 'hole-in-one' is scored is denoted by the variable \(X\), which has a Poisson distribution with mean 0.15 . Mr Crump offers to pay \(\\) 200$ each time that a hole-in-one is scored during 5 days of play. Find the expectation and variance of the amount that Mr Crump pays.