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

1. The discrete random variable \(X\) has the probability distribution

1. Show that \(k = 0.1\) Find
2. \(\mathrm { E } ( X )\)
3. \(\mathrm { E } \left( X ^ { 2 } \right)\)
4. \(\operatorname { Var } ( 2 - 5 X )\) Two independent observations \(X _ { 1 }\) and \(X _ { 2 }\) are made of \(X\).
5. Show that \(\mathrm { P } \left( X _ { 1 } + X _ { 2 } = 4 \right) = 0.1\)
6. Complete the probability distribution table for \(X _ { 1 } + X _ { 2 }\)

   \(y\)	2	3	4	5	6	7	8
   \(\mathrm { P } \left( X _ { 1 } + X _ { 2 } = y \right)\)	0.01	0.04	0.10		0.25	0.24

7. Find \(\mathrm { P } \left( 1.5 < X _ { 1 } + X _ { 2 } \leqslant 3.5 \right)\)

7. The random variable \(X\) has probability distribution

1. Given that \(\mathrm { E } ( X ) = 4.5\), write down two equations involving \(p\) and \(q\). Find
2. the value of \(p\) and the value of \(q\),
3. \(\mathrm { P } ( 4 < X \leqslant 7 )\). Given that \(\mathrm { E } \left( X ^ { 2 } \right) = 27.4\), find
4. \(\operatorname { Var } ( X )\),
5. \(\mathrm { E } ( 19 - 4 X )\),
6. \(\operatorname { Var } ( 19 - 4 X )\).

6. The discrete random variable \(X\) has probability function $$\mathrm { P } ( X = x ) = \left\{ \begin{array} { c l } a ( 3 - x ) & x = 0,1,2 \\ b & x = 3 \end{array} \right.$$

1. Find \(\mathrm { P } ( X = 2 )\) and complete the table below.

   \(x\)	0	1	2	3
   \(\mathrm { P } ( X = x )\)	\(3 a\)	\(2 a\)		\(b\)

   Given that \(\mathrm { E } ( X ) = 1.6\)
2. Find the value of \(a\) and the value of \(b\). Find
3. \(\mathrm { P } ( 0.5 < X < 3 )\),
4. \(\mathrm { E } ( 3 X - 2 )\).
5. Show that the \(\operatorname { Var } ( X ) = 1.64\)
6. Calculate \(\operatorname { Var } ( 3 X - 2 )\).

3. The discrete random variable \(X\) has probability distribution given by where \(a\) is a constant.

1. Find the value of \(a\).
2. Write down \(\mathrm { E } ( X )\).
3. Find \(\operatorname { Var } ( X )\). The random variable \(Y = 6 - 2 X\)
4. Find \(\operatorname { Var } ( Y )\).
5. Calculate \(\mathrm { P } ( X \geqslant Y )\).

1. The weight of coffee in glass jars labelled 100 g is normally distributed with mean 101.80 g and standard deviation 0.72 g . The weight of an empty glass jar is normally distributed with mean 260.00 g and standard deviation 5.45 g . The weight of a glass jar is independent of the weight of the coffee it contains.

Find the probability that a randomly selected jar weighs less than 266 g and contains less than 100 g of coffee. Give your answer to 2 significant figures.
(8 marks)

6

1. Explain what you understand by the sampling distribution of a statistic. At Sam's cafe a standard breakfast consists of 6 breakfast items. Customers can then choose to upgrade to a medium breakfast by adding 1 extra breakfast item or they can upgrade to a large breakfast by adding 2 extra breakfast items. Standard, medium and large breakfasts are sold in the ratio \(6 : 3 : 2\) respectively. A random sample of 2 customers is taken from customers who have bought a breakfast from Sam's cafe on a particular day.
2. Find the sampling distribution for the total number, \(T\), of breakfast items bought by these 2 customers. Show your working clearly.
3. Find \(\mathrm { E } ( T )\)

1. The continuous random variable \(G\) has probability density function \(\mathrm { f } ( \mathrm { g } )\) given by

$$f ( g ) = \begin{cases} \frac { 1 } { 15 } ( g + 3 ) & - 1 < g \leqslant 2 \\ \frac { 3 } { 20 } & 2 < g \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$

1. Sketch the graph of \(\mathrm { f } ( \mathrm { g } )\)
2. Find \(\mathrm { P } ( ( 1 \leqslant 2 G \leqslant 6 ) \mid G \leqslant 2 )\) The continuous random variable \(H\) is such that \(\mathrm { E } ( H ) = 12\) and \(\operatorname { Var } ( H ) = 2.4\)
3. Find \(\mathrm { E } \left( 2 H ^ { 2 } + 3 G + 3 \right)\) Show your working clearly.
   (Solutions relying on calculator technology are not acceptable.)

7. The continuous random variable \(X\) is uniformly distributed over the interval \([ a , b ]\)

1. Find an expression, in terms of \(a\) and \(b\), for \(\mathrm { E } ( 3 - 2 X )\)
2. Find \(\mathrm { P } \left( X > \frac { 1 } { 3 } b + \frac { 2 } { 3 } a \right)\) Given that \(\mathrm { E } ( X ) = 0\)
3. find an expression, in terms of \(b\) only, for \(\mathrm { E } \left( 3 X ^ { 2 } \right)\) Given also that the range of \(X\) is 18
4. find \(\operatorname { Var } ( X )\)
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4 A bag contains 50 counters, each with one of the numbers 4,7 or 10 written on it in the ratio \(2 : 3 : 5\) respectively. A random sample of 2 counters is taken from the bag. The numbers on the 2 counters are recorded as \(D _ { 1 }\) and \(D _ { 2 }\) The random variable \(M\) represents the mean of \(D _ { 1 }\) and \(D _ { 2 }\)

1. Show that \(\mathrm { P } ( M = 4 ) = \frac { 9 } { 245 }\)
2. Find the sampling distribution of \(M\) A random sample of \(n\) sets of 2 counters is taken. The random variable \(T\) represents the number of these \(n\) sets of 2 counters that have a mean of 4 Given that each set of 2 counters is replaced after it is drawn,
3. calculate the minimum value of \(n\) such that \(\mathrm { P } ( T = 0 ) < 0.15\)

2. A continuous random variable \(X\) has cumulative distribution function $$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x < 1 \\ \frac { 1 } { 5 } ( x - 1 ) & 1 \leqslant x \leqslant 6 \\ 1 & x > 6 \end{array} \right.$$

1. Find \(\mathrm { P } ( X > 4 )\)
2. Write down the value of \(\mathrm { P } ( X \neq 4 )\)
3. Find the probability density function of \(X\), specifying it for all values of \(x\)
4. Write down the value of \(\mathrm { E } ( X )\)
5. Find \(\operatorname { Var } ( X )\)
6. Hence or otherwise find \(\mathrm { E } \left( 3 X ^ { 2 } + 1 \right)\)

3. Explain what you understand by

1. a statistic,
2. a sampling distribution. A factory stores screws in packets. A small packet contains 100 screws and a large packet contains 200 screws. The factory keeps small and large packets in the ratio 4:3 respectively.
3. Find the mean and the variance of the number of screws in the packets stored at the factory. A random sample of 3 packets is taken from the factory and \(Y _ { 1 } , Y _ { 2 }\) and \(Y _ { 3 }\) denote the number of screws in each of these packets.
4. List all the possible samples.
5. Find the sampling distribution of \(\bar { Y }\)

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4. The number of houses sold by an estate agent follows a Poisson distribution, with a mean of 2 per week.

1. Find the probability that in the next 4 weeks the estate agent sells,
   1. exactly 3 houses,
   2. more than 5 houses. The estate agent monitors sales in periods of 4 weeks.
2. Find the probability that in the next twelve of these 4 week periods there are exactly nine periods in which more than 5 houses are sold. The estate agent will receive a bonus if he sells more than 25 houses in the next 10 weeks.
3. Use a suitable approximation to estimate the probability that the estate agent receives a bonus.

7. As part of a selection procedure for a company, applicants have to answer all 20 questions of a multiple choice test. If an applicant chooses answers at random the probability of choosing a correct answer is 0.2 and the number of correct answers is represented by the random variable \(X\).

1. Suggest a suitable distribution for \(X\).
   (2) Each applicant gains 4 points for each correct answer but loses 1 point for each incorrect answer. The random variable \(S\) represents the final score, in points, for an applicant who chooses answers to this test at random.
2. Show that \(S = 5 X - 20\)
3. Find \(\mathrm { E } ( S )\) and \(\operatorname { Var } ( S )\). An applicant who achieves a score of at least 20 points is invited to take part in the final stage of the selection process.
4. Find \(\mathrm { P } ( S \geqslant 20 )\) (4) Cameron is taking the final stage of the selection process which is a multiple choice test consisting of 100 questions. He has been preparing for this test and believes that his chance of answering each question correctly is 0.4
5. Using a suitable approximation, estimate the probability that Cameron answers more than half of the questions correctly.

1. The number of defects per metre in a roll of cloth has a Poisson distribution with mean 0.25

Find the probability that

1. a randomly chosen metre of cloth has 1 defect,
2. the total number of defects in a randomly chosen 6 metre length of cloth is more than 2 A tailor buys 300 metres of cloth.
3. Using a suitable approximation find the probability that the tailor's cloth will contain less than 90 defects.

A continuous random variable \(X\) is uniformly distributed over the interval [ \(b , 4 b\) ] where \(b\) is a constant.

1. Write down \(\mathrm { E } ( X )\).
2. Use integration to show that \(\operatorname { Var } ( X ) = \frac { 3 b ^ { 2 } } { 4 }\).
3. Find \(\operatorname { Var } ( 3 - 2 X )\). Given that \(b = 1\) find
4. the cumulative distribution function of \(X , \mathrm {~F} ( x )\), for all values of \(x\),
5. the median of \(X\).
   

A telesales operator is selling a magazine. Each day he chooses a number of people to telephone. The probability that each person he telephones buys the magazine is 0.1

1. Suggest a suitable distribution to model the number of people who buy the magazine from the telesales operator each day.
2. On Monday, the telesales operator telephones 10 people. Find the probability that he sells at least 4 magazines.
3. Calculate the least number of people he needs to telephone on Tuesday, so that the probability of selling at least 1 magazine, on that day, is greater than 0.95 A call centre also sells the magazine. The probability that a telephone call made by the call centre sells a magazine is 0.05 The call centre telephones 100 people every hour.
4. Using a suitable approximation, find the probability that more than 10 people telephoned by the call centre buy a magazine in a randomly chosen hour.

In a call centre, the number of telephone calls, \(X\), received during any 10 -minute period follows a Poisson distribution with mean 9

1. Find
   1. \(\mathrm { P } ( X > 5 )\)
   2. \(\mathrm { P } ( 4 \leqslant X < 10 )\) The length of a working day is 7 hours.
2. Using a suitable approximation, find the probability that there are fewer than 370 telephone calls in a randomly selected working day. A week, consisting of 5 working days, is selected at random.
3. Find the probability that in this week at least 4 working days have fewer than 370 telephone calls.

6. On a typical weekday morning customers arrive at a village post office independently and at a rate of 3 per 10 minute period. Find the probability that

1. at least 4 customers arrive in the next 10 minutes,
2. no more than 7 customers arrive between 11.00 a.m. and 11.30 a.m. The period from 11.00 a.m. to 11.30 a.m. next Tuesday morning will be divided into 6 periods of 5 minutes each.
3. Find the probability that no customers arrive in at most one of these periods. The post office is open for \(3 \frac { 1 } { 2 }\) hours on Wednesday mornings.
4. Using a suitable approximation, estimate the probability that more than 49 customers arrive at the post office next Wednesday morning. END

1. A dog breeder claims that the mean weight of male Great Dane dogs is 20 kg more than the mean weight of female Great Dane dogs.

Tammy believes that the mean weight of male Great Dane dogs is more than 20 kg more than the mean weight of female Great Dane dogs. She takes random samples of 50 male and 50 female Great Dane dogs and records their weights. The results are summarised below, where \(x\) denotes the weight, in kg , of a male Great Dane dog and \(y\) denotes the weight, in kg, of a female Great Dane dog. $$\sum x = 3610 \quad \sum x ^ { 2 } = 260955.6 \quad \sum y = 2585 \quad \sum y ^ { 2 } = 133757.2$$

1. Find unbiased estimates for the mean and variance of the weights of
   1. the male Great Dane dogs,
   2. the female Great Dane dogs.
2. Stating your hypotheses clearly, carry out a suitable test to assess Tammy's belief. Use a \(5 \%\) level of significance and state your critical value.
3. For the test in part (b), state whether or not it is necessary to assume that the weights of the Great Dane dogs are normally distributed. Give a reason for your answer.
4. State an assumption you have made in carrying out the test in part (b).

1. Secondary schools in a region conduct ability testing at the start of Year 7 and the start of Year 8. Each year a regional education officer randomly selects 240 Year 7 students and 240 Year 8 students from across the region. The results for last year are summarised in the table below.

The regional education officer claims that there is no difference between the mean scores of these two year groups.

1. Test the regional education officer's claim at the \(1 \%\) significance level. You should state your hypotheses, test statistic and critical value clearly.
2. Explain the significance of the Central Limit Theorem in part (a).

5 Claire grows strawberries on her farm. She wants to compare two brands of fertiliser, brand \(A\) and brand \(B\). She grows two sets of plants of the same variety of strawberries under the same conditions, fertilising one set with brand \(A\) and the other with brand \(B\). The yields per plant, in grams, from each set of plants are summarised below.

1. Stating your hypotheses clearly, carry out a suitable test to assess whether the mean yield from plants using fertiliser \(A\) is greater than the mean yield from plants using fertiliser \(B\).
   Use a 1\% level of significance and state your test statistic and critical value. The total cost of fertiliser \(A\) for Claire's 50 plants was \(\pounds 75\) The total cost of fertiliser \(B\) for Claire's 40 plants was \(\pounds 50\) Claire sells all her strawberries at \(\pounds 3\) per kilogram.
2. Use this information, together with your answer in part (a), to advise Claire on which of the two brands of fertiliser she should use next year in order to maximise her expected profit per plant, giving a reason for your answer.

1. A professor claims that undergraduates studying History have a typing speed of more than 15 words per minute faster than undergraduates studying Maths.

A sample is taken of 38 undergraduates studying History and 45 undergraduates studying Maths. The typing speed, \(x\) words per minute, of each undergraduate is recorded. The results are summarised in the table below.

1. Use a suitable test, at the \(5 \%\) level of significance, to investigate the professor's claim.
   State clearly your hypotheses, test statistic and critical value.
2. State two assumptions you have made in carrying out the test in part (a).

1. Small containers and large containers are independently filled with fruit juice.

The amounts of fruit juice in small containers are normally distributed with mean 180 ml and standard deviation 4.5 ml The amounts of fruit juice in large containers are normally distributed with mean 330 ml and standard deviation 6.7 ml The random variable \(W\) represents the total amount of fruit juice in a random sample of 2 small containers minus the amount of fruit juice in 1 randomly selected large container. \(W \sim \mathrm {~N} ( a , b )\) where \(a\) and \(b\) are positive constants.

1. Find the value of \(a\) and the value of \(b\)
2. Find the probability that a randomly chosen large container of fruit juice contains more than 1.8 times the amount of fruit juice in a randomly chosen small container. A random sample of 3 small containers of fruit juice is taken.
3. Find the probability that the first container of fruit juice in this sample contains at least 5 ml more than the mean amount of fruit juice in all 3 small containers.

7. The random variable \(X\) is defined as $$X = 4 Y - 3 W$$ where \(Y \sim \mathrm {~N} \left( 40,3 ^ { 2 } \right) , W \sim \mathrm {~N} \left( 50,2 ^ { 2 } \right)\) and \(Y\) and \(W\) are independent.

1. Find \(\mathrm { P } ( X > 25 )\) The random variables \(Y _ { 1 } , Y _ { 2 }\) and \(Y _ { 3 }\) are independent and each has the same distribution as \(Y\). The random variable \(A\) is defined as $$A = \sum _ { i = 1 } ^ { 3 } Y _ { i }$$ The random variable \(C\) is such that \(C \sim \mathrm {~N} \left( 115 , \sigma ^ { 2 } \right)\) Given that \(\mathrm { P } ( A - C < 0 ) = 0.2\) and that \(A\) and \(C\) are independent,
2. find the variance of \(C\).

6. The random variable \(W\) is defined as $$W = 3 X - 4 Y$$ where \(X \sim \mathrm {~N} \left( 21,2 ^ { 2 } \right)\) and \(Y \sim \mathrm {~N} \left( 8.5 , \sigma ^ { 2 } \right)\) and \(X\) and \(Y\) are independent.
Given that \(\mathrm { P } ( W < 44 ) = 0.9\)

1. find the value of \(\sigma\), giving your answer to 2 decimal places. The random variables \(A _ { 1 } , A _ { 2 }\) and \(A _ { 3 }\) each have the same distribution as \(A\), where \(A \sim \mathrm {~N} \left( 28,5 ^ { 2 } \right)\) The random variable \(B\) is defined as $$B = 2 X + \sum _ { i = 1 } ^ { 3 } A _ { i }$$ where \(X , A _ { 1 } , A _ { 2 }\) and \(A _ { 3 }\) are independent.
2. Find \(\mathrm { P } ( B \leqslant 145 \mid B > 120 )\)