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

9 The continuous random variable \(X\) has distribution function F given by $$\mathrm { F } ( x ) = \begin{cases} 0 & x < 2 , \\ \frac { 1 } { 8 } x - \frac { 1 } { 4 } & 2 \leqslant x \leqslant 10 , \\ 1 & x > 10 . \end{cases}$$ Find the value of \(k\) for which \(\mathrm { P } ( X \geqslant k ) = 0.6\). The random variable \(Y\) is defined by \(Y = 2 \ln X\). Find the distribution function of \(Y\). Find the probability density function of \(Y\) and sketch its graph.

3 On a certain road \(20 \%\) of the vehicles are trucks, \(16 \%\) are buses and the remainder are cars.

1. A random sample of 11 vehicles is taken. Find the probability that fewer than 3 are buses.
2. A random sample of 125 vehicles is now taken. Using a suitable approximation, find the probability that more than 73 are cars.

5 In the holidays Martin spends \(25 \%\) of the day playing computer games. Martin's friend phones him once a day at a randomly chosen time.

1. Find the probability that, in one holiday period of 8 days, there are exactly 2 days on which Martin is playing computer games when his friend phones.
2. Another holiday period lasts for 12 days. State with a reason whether it is appropriate to use a normal approximation to find the probability that there are fewer than 7 days on which Martin is playing computer games when his friend phones.
3. Find the probability that there are at least 13 days of a 40-day holiday period on which Martin is playing computer games when his friend phones.

2 The number of orders arriving at a shop during an 8-hour working day is modelled by the random variable \(X\) with distribution \(\operatorname { Po } ( 25.2 )\).

1. State two assumptions that are required for the Poisson model to be valid in this context.
2. 1. Find the probability that the number of orders that arrive in a randomly chosen 3-hour period is between 3 and 5 inclusive.
   2. Find the probability that, in two randomly chosen 1 -hour periods, exactly 1 order will arrive in one of the 1 -hour periods, and at least 2 orders will arrive in the other 1 -hour period. [4]
3. The shop can only deal with a maximum of 120 orders during any 36-hour period. Use a suitable approximating distribution to find the probability that, in a randomly chosen 36-hour period, there will be too many orders for the shop to deal with.

5 The masses, in grams, of large and small packets of Maxwheat cereal have the independent distributions \(\mathrm { N } \left( 410.0,3.6 ^ { 2 } \right)\) and \(\mathrm { N } \left( 206.0,3.7 ^ { 2 } \right)\) respectively.

1. Find the probability that a randomly chosen large packet has a mass that is more than double the mass of a randomly chosen small packet.
   The packets are placed in boxes. The boxes are identical in appearance. \(60 \%\) of the boxes contain exactly 10 randomly chosen large packets. 40\% of the boxes contain exactly 20 randomly chosen small packets.
2. Find the probability that a randomly chosen box contains packets with a total mass of more than 4080 grams.

5 Each week a sports team plays one home match and one away match. In their home matches they score goals at a constant average rate of 2.1 goals per match. In their away matches they score goals at a constant average rate of 0.8 goals per match. You may assume that goals are scored at random times and independently of one another.

1. A week is chosen at random.
   1. Find the probability that the team scores a total of 4 goals in their two matches.
   2. Find the probability that the team scores a total of 4 goals, with more goals scored in the home match than in the away match.
2. Use a suitable approximating distribution to find the probability that the team scores fewer than 25 goals in 10 randomly chosen weeks.
3. Justify the use of the approximating distribution used in part (b).

5

1. The random variable \(X\) has the distribution \(\operatorname { Po } ( \lambda )\).
   1. State the values that \(X\) can take.
      It is given that \(\mathrm { P } ( X = 1 ) = 3 \times \mathrm { P } ( X = 0 )\).
   2. Find \(\lambda\).
   3. Find \(\mathrm { P } ( 4 \leqslant X \leqslant 6 )\).
2. The random variable \(Y\) has the distribution \(\operatorname { Po } ( \mu )\) where \(\mu\) is large. Using a suitable approximating distribution, it is found that \(\mathrm { P } ( Y < 46 ) = 0.0668\), correct to 4 decimal places. Find \(\mu\).

6 Between 7 p.m. and 11 p.m., arrivals of patients at the casualty department of a hospital occur at random at an average rate of 6 per hour.

1. Find the probability that, during any period of one hour between 7 p.m. and 11 p.m., exactly 5 people will arrive.
2. A patient arrives at exactly 10.15 p.m. Find the probability that at least one more patient arrives before 10.35 p.m.
3. Use a suitable approximation to estimate the probability that fewer than 20 patients arrive at the casualty department between 7 p.m. and 11 p.m. on any particular night.

4 The independent random variables \(X\) and \(Y\) have distributions \(\operatorname { Po } ( 2 )\) and \(\mathrm { B } \left( 20 , \frac { 1 } { 4 } \right)\) respectively.

1. Find the mean and standard deviation of \(X - 3 Y\).
2. Find \(\mathrm { P } ( Y = 15 X )\).

4 A certain train journey takes place every day throughout the year. The time taken, in minutes, for the journey is normally distributed with variance 11.2.

1. The mean time for a random sample of \(n\) of these journeys was found. A \(94 \%\) confidence interval for the population mean time was calculated and was found to have a width of 1.4076 minutes, correct to 4 decimal places. Find the value of \(n\).
2. A passenger noted the times for 50 randomly chosen journeys in January, February and March. Give a reason why this sample is unsuitable for use in finding a confidence interval for the population mean time.
3. A researcher took 4 random samples and a \(94 \%\) confidence interval for the population mean was found from each sample. Find the probability that exactly 3 of these confidence intervals contain the true value of the population mean.

5 Large packets of rice are packed in cartons, each containing 20 randomly chosen packets. The masses of these packets are normally distributed with mean 1010 g and standard deviation 3.4 g . The masses of the cartons, when empty, are independently normally distributed with mean 50 g and standard deviation 2.0 g .

1. Find the variance of the masses of full cartons.
   Small packets of rice are packed in boxes. The total masses of full boxes are normally distributed with mean 6730 g and standard deviation 15.0 g . The masses of the boxes and cartons are distributed independently of each other.
2. Find the probability that the mass of a randomly chosen full carton is more than three times the mass of a randomly chosen full box.

5

1. Two random variables \(X\) and \(Y\) have the independent distributions \(\mathrm { N } ( 7,3 )\) and \(\mathrm { N } ( 6,2 )\) respectively. A random value of each variable is taken. Find the probability that the two values differ by more than 2 .
2. Each candidate's overall score in a science test is calculated as follows. The mark for theory is denoted by \(T\), the mark for practical is denoted by \(P\), and the overall score is given by \(T + 1.5 P\). The variables \(T\) and \(P\) are assumed to be independent with distributions \(\mathrm { N } ( 62,158 )\) and \(\mathrm { N } ( 42,108 )\) respectively. You should assume that no continuity corrections are needed when using these distributions.
   1. A pass is awarded to candidates whose overall score is at least 90 . Find the proportion of candidates who pass.
   2. Comment on the assumption that the variables \(T\) and \(P\) are independent.

4 The mass, in tonnes, of steel produced per day at a factory is normally distributed with mean 65.2 and standard deviation 3.6. It can be assumed that the mass of steel produced each day is independent of other days. The factory makes \(\\) 50$ profit on each tonne of steel produced. Find the probability that the total profit made in a randomly chosen 7-day week is less than \(\\) 22000$.

5 Sales of cell phones at a certain shop occur singly, randomly and independently.

1. State one further condition that must be satisfied for the number of sales in a certain time period to be well modelled by a Poisson distribution.
   The average number of sales per hour is 1.2 .
   Assume now that a Poisson distribution is a suitable model.
2. Find the probability that the number of sales during a randomly chosen 12 -hour period will be more than 12 and less than 16 .
3. Use a suitable approximating distribution to find the probability that the number of sales during a randomly chosen 1-month period (140 hours) will be less than 150 .

1 A random variable \(X\) has the distribution \(\mathrm { Po } ( 145 )\).

1. Use a suitable approximating distribution to calculate \(\mathrm { P } ( X \leqslant 150 )\).
2. Justify the use of your approximating distribution in this case.

4 A random variable \(X\) has the distribution \(\mathrm { N } ( 10,12 )\). Two independent values of \(X\), denoted by \(X _ { 1 }\) and \(X _ { 2 }\), are chosen at random.

1. Write down the value of \(\mathrm { P } \left( X _ { 1 } > X _ { 2 } \right)\).
2. Find \(\mathrm { P } \left( X _ { 1 } > 2 X _ { 2 } - 3 \right)\).

3 The masses in kilograms of large and small bags of cement have the independent distributions \(\mathrm { N } ( 50,2.4 )\) and \(\mathrm { N } ( 26,1.8 )\) respectively. Find the probability that the total mass of 5 randomly chosen large bags of cement is greater than the total mass of 10 randomly chosen small bags of cement. \includegraphics[max width=\textwidth, alt={}, center]{7c078a14-98f9-4292-ae76-a2642238176f-04_2714_34_143_2012} \includegraphics[max width=\textwidth, alt={}, center]{7c078a14-98f9-4292-ae76-a2642238176f-05_2724_35_136_20}

6 The volumes, in millilitres, of large and small cups of tea are modelled by the distributions \(\mathrm { N } ( 200,30 )\) and \(\mathrm { N } ( 110,20 )\) respectively.

1. Find the probability that the total volume of a randomly chosen large cup of tea and a randomly chosen small cup of tea is less than 300 ml .
2. Find the probability that the volume of a randomly chosen large cup of tea is more than twice the volume of a randomly chosen small cup of tea.

5 The volumes, in litres, of juice in large and small bottles have the distributions \(\mathrm { N } ( 5.10,0.0102 )\) and \(\mathrm { N } ( 2.51,0.0036 )\) respectively.

1. Find the probability that the total volume of juice in 3 randomly chosen large bottles and 4 randomly chosen small bottles is less than 25.5 litres.
2. Find the probability that the volume of juice in a randomly chosen large bottle is at least twice the volume of juice in a randomly chosen small bottle.

5 The heights of buildings in a large city are normally distributed with mean 18.3 m and standard deviation 2.5 m .

1. Find the probability that the total height of 5 randomly chosen buildings in the city is more than 95 m .
2. Find the probability that the difference between the heights of two randomly chosen buildings in the city is less than 1 m .

2 Each day Samuel travels from \(A\) to \(B\) and from \(B\) to \(C\). He then returns directly from \(C\) to \(A\). The times, in minutes, for these three journeys have the independent distributions \(\mathrm { N } \left( 20,2 ^ { 2 } \right) , \mathrm { N } \left( 18,1.5 ^ { 2 } \right)\) and \(\mathrm { N } \left( 30,1.8 ^ { 2 } \right)\), respectively. Find the probability that, on a randomly chosen day, the total time for his two journeys from \(A\) to \(B\) and \(B\) to \(C\) is less than the time for his return journey from \(C\) to \(A\). [5]

4 The marks, \(x\), of a random sample of 50 students in a test were summarised as follows. $$n = 50 \quad \Sigma x = 1508 \quad \Sigma x ^ { 2 } = 51825$$

1. Calculate unbiased estimates of the population mean and variance.
2. Each student's mark is scaled using the formula \(y = 1.5 x + 10\). Find estimates of the population mean and variance of the scaled marks, \(y\).

5 The thickness of books in a large library is normally distributed with mean 2.4 cm and standard deviation 0.3 cm .

1. Find the probability that the total thickness of 6 randomly chosen books is more than 16 cm .
2. Find the probability that the thickness of a book chosen at random is less than 1.1 times the thickness of a second book chosen at random.

5 Each box of Fruity Flakes contains \(X\) grams of flakes and \(Y\) grams of fruit, where \(X\) and \(Y\) are independent random variables, having distributions \(\mathrm { N } ( 400,50 )\) and \(\mathrm { N } ( 100,20 )\) respectively. The weight of each box, when empty, is exactly 20 grams. A full box of Fruity Flakes is chosen at random.

1. Find the probability that the total weight of the box and its contents is less than 530 grams.
2. Find the probability that the weight of flakes in the box is more than 4.1 times the weight of fruit in the box.

5 Cans of drink are packed in boxes, each containing 4 cans. The weights of these cans are normally distributed with mean 510 g and standard deviation 14 g . The weights of the boxes, when empty, are independently normally distributed with mean 200 g and standard deviation 8 g .

1. Find the probability that the total weight of a full box of cans is between 2200 g and 2300 g .
2. Two cans of drink are chosen at random. Find the probability that they differ in weight by more than 20 g .