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

4 The independent random variables \(X\) and \(Y\) have normal distributions where \(X \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\) and \(Y \sim \mathrm {~N} \left( 3 \mu , 4 \sigma ^ { 2 } \right)\). Two random samples each of size \(n\) are taken, one from each of these normal populations.

1. Show that \(a \bar { X } + b \bar { Y }\) is an unbiased estimator of \(\mu\) provided that \(a + 3 b = 1\), where \(a\) and \(b\) are constants and \(\bar { X }\) and \(\bar { Y }\) are the respective sample means. In the remainder of the question assume that \(a \bar { X } + b \bar { Y }\) is an unbiased estimator of \(\mu\).
2. Show that \(\operatorname { Var } ( a \bar { X } + b \bar { Y } )\) can be written as \(\frac { \sigma ^ { 2 } } { n } \left( 1 - 6 b + 13 b ^ { 2 } \right)\).
3. The value of the constant \(b\) can be varied. Find the value of \(b\) that gives the minimum of \(\operatorname { Var } ( a \bar { X } + b \bar { Y } )\), and hence find the minimum of \(\operatorname { Var } ( a \bar { X } + b \bar { Y } )\) in terms of \(\sigma\) and \(n\).

4 The independent random variables \(X\) and \(Y\) have normal distributions where \(X \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\) and \(Y \sim \mathrm {~N} \left( 3 \mu , 4 \sigma ^ { 2 } \right)\). Two random samples each of size \(n\) are taken, one from each of these normal populations.

1. Show that \(a \bar { X } + b \bar { Y }\) is an unbiased estimator of \(\mu\) provided that \(a + 3 b = 1\), where \(a\) and \(b\) are constants and \(\bar { X }\) and \(\bar { Y }\) are the respective sample means. In the remainder of the question assume that \(a \bar { X } + b \bar { Y }\) is an unbiased estimator of \(\mu\).
2. Show that \(\operatorname { Var } ( \overline { a X } + b \bar { Y } )\) can be written as \(\frac { \sigma ^ { 2 } } { n } \left( 1 - 6 b + 13 b ^ { 2 } \right)\).
3. The value of the constant \(b\) can be varied. Find the value of \(b\) that gives the minimum of \(\operatorname { Var } ( a \bar { X } + b \bar { Y } )\), and hence find the minimum of \(\operatorname { Var } ( a \bar { X } + b \bar { Y } )\) in terms of \(\sigma\) and \(n\).

4 The independent random variables \(X\) and \(Y\) have normal distributions where \(X \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\) and \(Y \sim \mathrm {~N} \left( 3 \mu , 4 \sigma ^ { 2 } \right)\). Two random samples each of size \(n\) are taken, one from each of these normal populations.

1. Show that \(a \bar { X } + b \bar { Y }\) is an unbiased estimator of \(\mu\) provided that \(a + 3 b = 1\), where \(a\) and \(b\) are constants and \(\bar { X }\) and \(\bar { Y }\) are the respective sample means. In the remainder of the question assume that \(a \bar { X } + b \bar { Y }\) is an unbiased estimator of \(\mu\).
2. Show that \(\operatorname { Var } ( a \bar { X } + b \bar { Y } )\) can be written as \(\frac { \sigma ^ { 2 } } { n } \left( 1 - 6 b + 13 b ^ { 2 } \right)\).
3. The value of the constant \(b\) can be varied. Find the value of \(b\) that gives the minimum of \(\operatorname { Var } ( a \bar { X } + b \bar { Y } )\), and hence find the minimum of \(\operatorname { Var } ( a \bar { X } + b \bar { Y } )\) in terms of \(\sigma\) and \(n\).

The masses, in kilograms, of large and small sacks of flour have the distributions \(\text{N}(55, 3^2)\) and \(\text{N}(27, 2.5^2)\) respectively.

1. Some sacks are loaded onto a boat. The maximum load of flour that the boat can carry safely is 340 kg. Find the probability that the boat can carry safely 3 randomly chosen large sacks of flour and 6 randomly chosen small sacks of flour. [5]
2. Find the probability that the mass of a randomly chosen large sack of flour is greater than the total mass of two randomly chosen small sacks of flour. [5]

\(X\) is a random variable with distribution B(10, 0.2). A random sample of 160 values of \(X\) is taken.

1. Find the approximate distribution of the sample mean, including the values of the parameters. [3]
2. Hence find the probability that the sample mean is less than 1.8. [3]

The masses, in grams, of small and large bags of flour have the distributions N(510, 100) and N(1015, 324) respectively. André selects 4 small bags of flour and 2 large bags of flour at random.

1. Find the probability that the total mass of these 6 bags of flour is less than 4130 g. [5]
2. Find the probability that the total mass of the 4 small bags is more than the total mass of the 2 large bags. [5]

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 \(X \sim\) N(10.3, 5.76) and \(Y \sim\) 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 \(A\). [2]
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]

The weights of bags of fuel have mean 3.2 kg and standard deviation 0.04 kg. The total weight of a random sample of three bags is denoted by \(T\) kg. Find the mean and standard deviation of \(T\). [4]

Bags of sugar are packed in boxes, each box containing 20 bags. The masses of the boxes, when empty, are normally distributed with mean 0.4 kg and standard deviation 0.01 kg. The masses of the bags are normally distributed with mean 1.02 kg and standard deviation 0.03 kg.

1. Find the probability that the total mass of a full box of 20 bags is less than 20.6 kg. [5]
2. Two full boxes are chosen at random. Find the probability that they differ in mass by less than 0.02 kg. [5]

Cars arrive at a motorway toll booth at an average rate of 150 per hour.

1. Suggest a suitable distribution to model the number of cars arriving at the toll booth, \(X\), per minute. [2]
2. State clearly any assumptions you have made by suggesting this model. [2]

Using your model,

1. find the probability that in any given minute
   1. no cars arrive,
   2. more than 3 cars arrive.
   [3]
2. In any given 4 minute period, find \(m\) such that P(\(X > m\)) = 0.0487 [3]
3. Using a suitable approximation find the probability that fewer than 15 cars arrive in any given 10 minute period. [6]

A garden centre sells canes of nominal length 150 cm. The canes are bought from a supplier who uses a machine to cut canes of length \(L\) where \(L \sim \mathrm{N}(\mu, 0.3^2)\).

1. Find the value of \(\mu\), to the nearest 0.1 cm, such that there is only a 5\% chance that a cane supplied to the garden centre will have length less than 150 cm. [4]

A customer buys 10 of these canes from the garden centre.

1. Find the probability that at most 2 of the canes have length less than 150 cm. [3]

Another customer buys 500 canes.

1. Using a suitable approximation, find the probability that fewer than 35 of the canes will have length less than 150 cm. [6]

1. Write down the condition needed to approximate a Poisson distribution by a Normal distribution. [1]

The random variable \(Y \sim \text{Po}(30)\).

1. Estimate P(\(Y > 28\)). [6]

1. State two conditions under which a random variable can be modelled by a binomial distribution. [2]

In the production of a certain electronic component it is found that 10% are defective. The component is produced in batches of 20.

1. Write down a suitable model for the distribution of defective components in a batch. [1]

Find the probability that a batch contains

1. no defective components, [2]
2. more than 6 defective components. [2]
3. Find the mean and the variance of the defective components in a batch. [2]

A supplier buys 100 components. The supplier will receive a refund if there are more than 15 defective components.

1. Using a suitable approximation, find the probability that the supplier will receive a refund. [4]

Minor defects occur in a particular make of carpet at a mean rate of 0.05 per m\(^2\).

1. Suggest a suitable model for the distribution of the number of defects in this make of carpet. Give a reason for your answer.

A carpet fitter has a contract to fit this carpet in a small hotel. The hotel foyer requires 30 m\(^2\) of this carpet. Find the probability that the foyer carpet contains

1. exactly 2 defects, [3]
2. more than 5 defects. [3]

The carpet fitter orders a total of 355 m\(^2\) of the carpet for the whole hotel.

1. Using a suitable approximation, find the probability that this total area of carpet contains 22 or more defects. [6]

A continuous random variable \(X\) has probability density function f(\(x\)) where $$f(x) = \begin{cases} kx^n & 0 \leq x \leq 1 \\ 0 & \text{otherwise} \end{cases}$$ where \(k\) and \(n\) are positive integers.

1. Find \(k\) in terms of \(n\). [3]
2. Find E(\(X\)) in terms of \(n\). [3]
3. Find E(\(X^2\)) in terms of \(n\). [2]

Given that \(n = 2\)

1. find Var(3\(X\)). [3]

A farm produces potatoes. The potatoes are packed into sacks. The weight of a sack of potatoes is modelled by a normal distribution with mean 25.6 kg and standard deviation 0.24 kg

1. Find the probability that two randomly chosen sacks of potatoes differ in weight by more than 0.5 kg [6]

Sacks of potatoes are randomly selected and packed onto pallets. The weight of an empty pallet is modelled by a normal distribution with mean 20.0 kg and standard deviation 0.32 kg Each full pallet of potatoes holds 30 sacks of potatoes.

1. Find the probability that the total weight of a randomly chosen full pallet of potatoes is greater than 785 kg [5]

A fair six-sided die is labelled with the numbers 1, 2, 3, 4, 5 and 6. The die is rolled 40 times and the score, \(S\), for each roll is recorded.

1. Find the mean and the variance of \(S\). [2]
2. Find an approximation for the probability that the mean of the 40 scores is less than 3 [3]

The weights of a group of males are normally distributed with mean 80 kg and standard deviation 2.6 kg. A random sample of 10 of these males is selected.

1. Write down the distribution of \(M\), the mean weight, in kg, of this sample. [2]
2. Find P(\(M < 78.5\)). [3]

The weights of a group of females are normally distributed with mean 59 kg and standard deviation 1.9 kg. A random sample of 6 of the males and 4 of the females enters a lift that can carry a maximum load of 730 kg.

1. Find the probability that the maximum load will be exceeded when these 10 people enter the lift. [5]

The weights of a group of males are normally distributed with mean 80 kg and standard deviation 2.6 kg. A random sample of 10 of these males is selected.

1. Write down the distribution of \(\bar{M}\), the mean weight, in kg, of this sample. [2]
2. Find P(\(\bar{M} < 78.5\)). [3]

The weights of a group of females are normally distributed with mean 59 kg and standard deviation 1.9 kg. A random sample of 6 of the males and 4 of the females enters a lift that can carry a maximum load of 730 kg.

1. Find the probability that the maximum load will be exceeded when these 10 people enter the lift. [5]

The random variable \(A\) is defined as $$A = 4X - 3Y$$ where \(X \sim \text{N}(30, 3^2)\), \(Y \sim \text{N}(20, 2^2)\) and \(X\) and \(Y\) are independent. Find

1. E(\(A\)), [2]
2. Var(\(A\)). [3]

The random variables \(Y_1\), \(Y_2\), \(Y_3\) and \(Y_4\) are independent and each has the same distribution as \(Y\). The random variable \(B\) is defined as $$B = \sum_{i=1}^{4} Y_i$$

1. Find P(\(B > A\)). [6]

The lifetimes of batteries from manufacturer \(A\) are normally distributed with mean 20 hours and standard deviation 5 hours when used in a camera.

1. Find the mean and standard deviation of the total lifetime of a pack of 6 batteries from manufacturer \(A\). [2]

Judy uses a camera that takes one battery at a time. She takes a pack of 6 batteries from manufacturer \(A\) to use in her camera on holiday.

1. Find the probability that the batteries will last for more than 110 hours on her holiday. [2]

The lifetimes of batteries from manufacturer \(B\) are normally distributed with mean 35 hours and standard deviation 8 hours when used in a camera.

1. Find the probability that the total lifetime of a pack of 6 batteries from manufacturer \(A\) is more than 4 times the lifetime of a single battery from manufacturer \(B\) when used in a camera. [6]

The weights of eggs are normally distributed with mean 60g and standard deviation 5g Sairah chooses 2 eggs at random.

1. Find the probability that the difference in weight of these 2 eggs is more than 2g (5) Sairah is packing eggs into cartons. The weight of an empty egg carton is normally distributed with mean 40g and standard deviation 1.5g
2. Find the distribution of the total weight of a carton filled with 12 randomly chosen eggs. (3)
3. Find the probability that a randomly chosen carton, filled with 12 randomly chosen eggs, weighs more than 800g (2)

The random variable \(R\) is defined as \(R = X + 4Y\) where \(X \sim \text{N}(8, 2^2)\), \(Y \sim \text{N}(14, 3^2)\) and \(X\) and \(Y\) are independent. Find

1. E\((R)\), [2]
2. Var\((R)\), [3]
3. P\((R < 41)\) [3]

The random variables \(Y_1\), \(Y_2\) and \(Y_3\) are independent and each has the same distribution as \(Y\). The random variable \(S\) is defined as $$S = \sum_{i=1}^{3} Y_i - \frac{1}{2}X.$$

1. Find Var\((S)\). [4]

The three tasks most frequently carried out in a garage are \(A\), \(B\) and \(C\). For each of the tasks the times, in minutes, taken by the garage mechanics are assumed to be normally distributed with means and standard deviations given in the following table. Assuming that the times for the three tasks are independent, calculate the probability that

1. the total time taken by a single randomly chosen mechanic to carry out all three tasks lies between 533 and 655 minutes, [5]
2. a randomly chosen mechanic takes longer to carry out task \(B\) than task \(C\). [5]

The number of cars passing a point on a single-track one-way road during a one-minute period is denoted by \(X\). Cars pass the point at random intervals and the expected value of \(X\) is denoted by \(\lambda\).

1. State, in the context of the question, two conditions needed for \(X\) to be well modelled by a Poisson distribution. [2]
2. At a quiet time of the day, \(\lambda = 6.50\). Assuming that a Poisson distribution is valid, calculate P\((4 \leq X < 8)\). [3]
3. At a busy time of the day, \(\lambda = 30\).
   1. Assuming that a Poisson distribution is valid, use a suitable approximation to find P\((X > 35)\). Justify your approximation. [6]
   2. Give a reason why a Poisson distribution might not be valid in this context when \(\lambda = 30\). [1]