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OCR S4 2007 June Q5
12 marks Standard +0.3
5 The continuous random variable \(X\) has probability density function given by $$\mathrm { f } ( x ) = \begin{cases} \frac { 1 } { ( \alpha - 1 ) ! } x ^ { \alpha - 1 } \mathrm { e } ^ { - x } & x \geqslant 0 \\ 0 & x < 0 \end{cases}$$ where \(\alpha\) is a positive integer.
  1. Explain how you can deduce that \(\int _ { 0 } ^ { \infty } x ^ { \alpha - 1 } \mathrm { e } ^ { - x } \mathrm {~d} x = ( \alpha - 1 )\) !.
  2. Write down an integral for the moment generating function \(\mathrm { M } _ { X } ( t )\) of \(X\) and show, by using the substitution \(x = \frac { u } { 1 - t }\), that \(\mathrm { M } _ { X } ( t ) = ( 1 - t ) ^ { - \alpha }\).
  3. Use the moment generating function to find, in terms of \(\alpha\),
    1. \(\mathrm { E } ( X )\),
    2. \(\operatorname { Var } ( X )\).
OCR S4 2007 June Q6
15 marks Standard +0.3
6 The discrete random variable \(X\) takes the values 0 and 1 with \(\mathrm { P } ( X = 0 ) = q\) and \(\mathrm { P } ( X = 1 ) = p\), where \(p + q = 1\).
  1. Write down the probability generating function of \(X\). The sum of \(n\) independent observations of \(X\) is denoted by \(S\).
  2. Write down the probability generating function of \(S\), and name the distribution of \(S\).
  3. Use the probability generating function of \(S\) to find \(\mathrm { E } ( S )\) and \(\operatorname { Var } ( S )\).
  4. The independent random variables \(Y\) and \(Z\) are such that \(Y\) has the distribution \(\mathrm { B } \left( 10 , \frac { 1 } { 2 } \right)\), and \(Z\) has probability generating function \(\mathrm { e } ^ { - ( 1 - t ) }\). Find the probability that the sum of one random observation of \(Y\) and one random observation of \(Z\) is equal to 2 .
OCR S4 2007 June Q7
15 marks Standard +0.3
7 The continuous random variable \(X\) has a uniform distribution over the interval \([ 0 , \theta ]\) so that the probability density function is given by $$f ( x ) = \begin{cases} \frac { 1 } { \theta } & 0 \leqslant x \leqslant \theta \\ 0 & \text { otherwise } \end{cases}$$ where \(\theta\) is a positive constant. A sample of \(n\) independent observations of \(X\) is taken and the sample mean is denoted by \(\bar { X }\).
  1. The estimator \(T _ { 1 }\) is defined by \(T _ { 1 } = 2 \bar { X }\). Show that \(T _ { 1 }\) is an unbiased estimator of \(\theta\). It is given that the probability density function of the largest value, \(U\), in the sample is $$g ( u ) = \begin{cases} \frac { n u ^ { n - 1 } } { \theta ^ { n } } & 0 \leqslant u \leqslant \theta \\ 0 & \text { otherwise } \end{cases}$$
  2. Find \(\mathrm { E } ( U )\) and show that \(\operatorname { Var } ( U ) = \frac { n \theta ^ { 2 } } { ( n + 1 ) ^ { 2 } ( n + 2 ) }\).
  3. The estimator \(T _ { 2 }\) is defined by \(T _ { 2 } = \frac { n + 1 } { n } U\). Given that \(T _ { 2 }\) is also an unbiased estimator of \(\theta\), show that \(T _ { 2 }\) is a more efficient estimator than \(T _ { 1 }\) for \(n > 1\).
OCR S4 2008 June Q1
7 marks Standard +0.3
1 For the mutually exclusive events \(A\) and \(B , \mathrm { P } ( A ) = \mathrm { P } ( B ) = x\), where \(x \neq 0\).
  1. Show that \(x \leqslant \frac { 1 } { 2 }\).
  2. Show that \(A\) and \(B\) are not independent. The event \(C\) is independent of \(A\) and also independent of \(B\), and \(\mathrm { P } ( C ) = 2 x\).
  3. Show that \(\mathrm { P } ( A \cup B \cup C ) = 4 x ( 1 - x )\).
OCR S4 2008 June Q2
8 marks Standard +0.3
2 Part of Helen's psychology dissertation involved the reaction times to a certain stimulus. She measured the reaction times of 30 randomly selected students, in seconds correct to 2 decimal places. The results are shown in the following stem-and-leaf diagram.
1412
1524
16036
17157
1834579
19246789
2001345789
217
Key: 18 | 3 means 1.83 seconds Helen wishes to test whether the population median time exceeds 1.80 seconds.
  1. Give a reason why the Wilcoxon signed-rank test should not be used.
  2. Carry out a suitable non-parametric test at the \(5 \%\) significance level.
OCR S4 2008 June Q3
11 marks Standard +0.3
3 From the records of Mulcaster United Football Club the following distribution was suggested as a probability model for future matches. \(X\) and \(Y\) denoted the numbers of goals scored by the home team and the away team respectively.
\(X\)
\cline { 2 - 5 } \multicolumn{1}{c}{}0123
00.110.040.060.08
10.080.050.120.05
20.050.080.070.03
30.030.060.070.02
Use the model to find
  1. \(\mathrm { E } ( X )\),
  2. the probability that the away team wins a randomly chosen match,
  3. the probability that the away team wins a randomly chosen match, given that the home team scores. One of the directors, an amateur statistician, finds that \(\operatorname { Cov } ( X , Y ) = 0.007\). He states that, as this value is very close to zero, \(X\) and \(Y\) may be considered to be independent.
  4. Comment on the director's statement.
OCR S4 2008 June Q4
7 marks Standard +0.3
4 William takes a bus regularly on the same journey, sometimes in the morning and sometimes in the afternoon. He wishes to compare morning and afternoon journey times. He records the journey times on 7 randomly chosen mornings and 8 randomly chosen afternoons. The results, each correct to the nearest minute, are as follows, where M denotes a morning time and A denotes an afternoon time.
MAAMMMMMMAAAAAA
192022242526283031333537383942
William wishes to test for a difference between the average times of morning and afternoon journeys.
  1. Given that \(s _ { M } ^ { 2 } = 16.5\) and \(s _ { A } ^ { 2 } = 64.5\), with the usual notation, explain why a \(t\)-test is not appropriate in this case.
  2. William chooses a non-parametric test at the \(5 \%\) significance level. Carry out the test, stating the rejection region.
OCR S4 2008 June Q5
11 marks Standard +0.8
5 The discrete random variable \(X\) has moment generating function \(\frac { 1 } { 4 } \mathrm { e } ^ { 2 t } + a \mathrm { e } ^ { 3 t } + b \mathrm { e } ^ { 4 t }\), where \(a\) and \(b\) are constants. It is given that \(\mathrm { E } ( X ) = 3 \frac { 3 } { 8 }\).
  1. Show that \(a = \frac { 1 } { 8 }\), and find the value of \(b\).
  2. Find \(\operatorname { Var } ( X )\).
  3. State the possible values of \(X\).
OCR S4 2008 June Q6
15 marks Challenging +1.8
6 The continuous random variable \(Y\) has cumulative distribution function given by $$\mathrm { F } ( y ) = \begin{cases} 0 & y < a , \\ 1 - \frac { a ^ { 3 } } { y ^ { 3 } } & y \geqslant a , \end{cases}$$ where \(a\) is a positive constant. A random sample of 3 observations, \(Y _ { 1 } , Y _ { 2 } , Y _ { 3 }\), is taken, and the smallest is denoted by \(S\).
  1. Show that \(\mathrm { P } ( S > s ) = \left( \frac { a } { s } \right) ^ { 9 }\) and hence obtain the probability density function of \(S\).
  2. Show that \(S\) is not an unbiased estimator of \(a\), and construct an unbiased estimator, \(T _ { 1 }\), based on \(S\). It is given that \(T _ { 2 }\), where \(T _ { 2 } = \frac { 2 } { 9 } \left( Y _ { 1 } + Y _ { 2 } + Y _ { 3 } \right)\), is another unbiased estimator of \(a\).
  3. Given that \(\operatorname { Var } ( Y ) = \frac { 3 } { 4 } a ^ { 2 }\) and \(\operatorname { Var } ( S ) = \frac { 9 } { 448 } a ^ { 2 }\), determine which of \(T _ { 1 }\) and \(T _ { 2 }\) is the more efficient estimator.
  4. The values of \(Y\) for a particular sample are 12.8, 4.5 and 7.0. Find the values of \(T _ { 1 }\) and \(T _ { 2 }\) for this sample, and give a reason, unrelated to efficiency, why \(T _ { 1 }\) gives a better estimate of \(a\) than \(T _ { 2 }\) in this case.
OCR S4 2008 June Q7
13 marks Challenging +1.2
7 The probability generating function of the random variable \(X\) is given by $$\mathrm { G } ( t ) = \frac { 1 + a t } { 4 - t }$$ where \(a\) is a constant.
  1. Find the value of \(a\).
  2. Find \(\mathrm { P } ( X = 3 )\). The sum of 3 independent observations of \(X\) is denoted by \(Y\). The probability generating function of \(Y\) is denoted by \(\mathrm { H } ( t )\).
  3. Use \(\mathrm { H } ( t )\) to find \(\mathrm { E } ( Y )\).
  4. By considering \(\mathrm { H } ( - 1 ) + \mathrm { H } ( 1 )\), show that \(\mathrm { P } ( Y\) is an even number \() = \frac { 62 } { 125 }\).
OCR S4 2011 June Q1
6 marks Standard +0.8
1 The random variable \(X\) has the distribution \(\mathrm { B } ( n , p )\).
  1. Show, from the definition, that the probability generating function of \(X\) is \(( q + p t ) ^ { n }\), where \(q = 1 - p\).
  2. The independent random variable \(Y\) has the distribution \(\mathrm { B } ( 2 n , p )\) and \(T = X + Y\). Use probability generating functions to determine the distribution of \(T\), giving its parameters.
OCR S4 2011 June Q2
8 marks Standard +0.3
2 A botanist believes that some species of plants produce more flowers at high altitudes than at low altitudes. In order to investigate this belief the botanist randomly samples 11 species of plants each of which occurs at both altitudes. The numbers of flowers on the plants are shown in the table.
Species1234567891011
Number of flowers at low altitude534729654112
Number of flowers at high altitude161081416202115212
  1. Use the Wilcoxon signed rank test at the 5\% significance level to test the botanist's belief.
  2. Explain why the Wilcoxon rank sum test should not be used for this test.
OCR S4 2011 June Q3
10 marks Standard +0.8
3 For the events \(A\) and \(B , \mathrm { P } ( A ) = \mathrm { P } ( B ) = \frac { 3 } { 4 }\) and \(\mathrm { P } \left( A \mid B ^ { \prime } \right) = \frac { 1 } { 2 }\).
  1. Find \(\mathrm { P } ( A \cap B )\). For a third event \(C , \mathrm { P } ( C ) = \frac { 1 } { 4 }\) and \(C\) is independent of the event \(A \cap B\).
  2. Find \(\mathrm { P } ( A \cap B \cap C )\).
  3. Given that \(\mathrm { P } ( C \mid A ) = \lambda\) and \(\mathrm { P } ( B \mid C ) = 3 \lambda\), and that no event occurs outside \(A \cup B \cup C\), find the value of \(\lambda\).
OCR S4 2011 June Q4
10 marks Standard +0.8
4 The discrete random variable \(X\) has moment generating function \(\left( \frac { 1 } { 4 } + \frac { 3 } { 4 } \mathrm { e } ^ { t } \right) ^ { 3 }\).
  1. Find \(\mathrm { E } ( X )\).
  2. Find \(\mathrm { P } ( X = 2 )\).
  3. Show that \(X\) can be expressed as a sum of 3 independent observations of a random variable \(Y\). Obtain the probability distribution of \(Y\), and the variance of \(Y\).
OCR S4 2011 June Q5
11 marks Standard +0.3
5 A test was carried out to compare the breaking strengths of two brands of elastic band, \(A\) and \(B\), of the same size. Random samples of 6 were selected from each brand and the breaking strengths were measured. The results, in suitable units and arranged in ascending order for each brand, are as follows.
Brand \(A :\)5.68.79.210.711.212.6
Brand \(B :\)10.111.612.012.212.913.5
  1. Give one advantage that a non-parametric test might have over a parametric test in this context.
  2. Carry out a suitable Wilcoxon test at the \(5 \%\) significance level of whether there is a difference between the average breaking strengths of the two brands.
  3. An extra elastic band of brand \(B\) was tested and found to have a breaking strength exceeding all of the other 12 bands. Determine whether this information alters the conclusion of your test.
OCR S4 2011 June Q6
13 marks Standard +0.3
6 A City Council comprises 16 Labour members, 14 Conservative members and 6 members of Other parties. A sample of two members was chosen at random to represent the Council at an event. The number of Labour members and the number of Conservative members in this sample are denoted by \(L\) and \(C\) respectively. The joint probability distribution of \(L\) and \(C\) is given in the following table. \(C\)
\(L\)
012
0\(\frac { 1 } { 42 }\)\(\frac { 16 } { 105 }\)\(\frac { 4 } { 21 }\)
1\(\frac { 2 } { 15 }\)\(\frac { 16 } { 45 }\)0
2\(\frac { 13 } { 90 }\)00
  1. Verify the two non-zero probabilities in the table for which \(C = 1\).
  2. Find the expected number of Conservatives in the sample.
  3. Find the expected number of Other members in the sample.
  4. Explain why \(L\) and \(C\) are not independent, and state what can be deduced about \(\operatorname { Cov } ( L , C )\).
OCR S4 2011 June Q7
14 marks Challenging +1.2
7 The continuous random variable \(U\) has unknown mean \(\mu\) and known variance \(\sigma ^ { 2 }\). In order to estimate \(\mu\), two random samples, one of 4 observations of \(U\) and the other of 6 observations of \(U\), are taken. The sample means are denoted by \(\bar { U } _ { 4 }\) and \(\bar { U } _ { 6 }\) respectively. One estimator \(S\), given by \(S = \frac { 1 } { 2 } \left( \bar { U } _ { 4 } + \bar { U } _ { 6 } \right)\), is proposed.
  1. Show that \(S\) is unbiased and find \(\operatorname { Var } ( S )\) in terms of \(\sigma ^ { 2 }\). A second estimator \(T\) of the form \(a \bar { U } _ { 4 } + b \bar { U } _ { 6 }\) is proposed, where \(a\) and \(b\) are chosen such that \(T\) is an unbiased estimator for \(\mu\) with the smallest possible variance.
  2. Find the values of \(a\) and \(b\) and the corresponding variance of \(T\).
  3. State, giving a reason, which of \(S\) and \(T\) is the better estimator.
  4. Compare the efficiencies of this preferred estimator and the mean of all 10 observations.
OCR S4 2012 June Q1
5 marks Challenging +1.2
1 Independent random variables \(X\) and \(Y\) have distributions \(\mathrm { B } ( 7 , p )\) and \(\mathrm { B } ( 8 , p )\) respectively.
  1. Explain why \(X + Y \sim \mathrm {~B} ( 15 , p )\).
  2. Find \(\mathrm { P } ( X = 2 \mid X + Y = 5 )\).
OCR S4 2012 June Q2
6 marks Standard +0.3
2 The continuous random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} 4 x e ^ { - 2 x } & x \geqslant 0 \\ 0 & \text { otherwise } \end{cases}$$
  1. Show that the moment generating function ( mgf ) of \(X\) is $$\frac { 4 } { ( 2 - t ) ^ { 2 } } , \text { where } | t | < 2$$
  2. Explain why the \(\operatorname { mgf }\) of \(- X\) is \(\frac { 4 } { ( 2 + t ) ^ { 2 } }\).
  3. Two random observations of \(X\) are denoted by \(X _ { 1 }\) and \(X _ { 2 }\). What is the \(\operatorname { mgf }\) of \(X _ { 1 } - X _ { 2 }\) ?
OCR S4 2012 June Q3
9 marks Standard +0.3
3 Because of the large number of students enrolled for a university geography course and the limited accommodation in the lecture theatre, the department provides a filmed lecture. Students are randomly assigned to two groups, one to attend the lecture theatre and the other the film. At the end of term the two groups are given the same examination. The geography professor wishes to test whether there is a difference in the performance of the two groups and selects the marks of two random samples of students, 6 from the group attending the lecture theatre and 7 from the group attending the films. The marks for the two samples, ordered for convenience, are shown below.
Lecture theatre:303648515962
Filmed lecture:40495256636468
  1. Stating a necessary assumption, carry out a suitable non-parametric test, at the \(10 \%\) significance level, for a difference between the median marks of the two groups.
  2. State conditions under which a two-sample \(t\)-test could have been used.
  3. Assuming that the tests in parts (i) and (ii) are both valid, state with a reason which test would be preferable.
OCR S4 2012 June Q4
12 marks Standard +0.8
4 The random variable \(U\) has the distribution \(\operatorname { Geo } ( p )\).
  1. Show, from the definition, that the probability generating function ( pgf ) of \(U\) is given by $$G _ { U } ( t ) = \frac { p t } { 1 - q t } , \text { for } | t | < \frac { 1 } { q } ,$$ where \(q = 1 - p\).
  2. Explain why the condition \(| t | < \frac { 1 } { q }\) is necessary.
  3. Use the pgf to obtain \(\mathrm { E } ( U )\). Each packet of Corn Crisp cereal contains a voucher and \(20 \%\) of the vouchers have a gold star. When 4 gold stars have been collected a gift can be claimed. Let \(X\) denote the number of packets bought by a family up to and including the one from which the \(4 ^ { \text {th } }\) gold star is obtained.
  4. Obtain the pgf of \(X\).
  5. Find \(\mathrm { P } ( X = 6 )\).
OCR S4 2012 June Q5
11 marks Standard +0.3
5 A one-tail sign test of a population median is to be carried out at the \(5 \%\) significance level using a sample of size \(n\).
  1. Show by calculation that the test can never result in rejection of the null hypothesis when \(n = 4\). The coach of a college swimming team expects Elena, the best 50 m freestyle swimmer, to have a median time less than 30 seconds. Elena found from records of her previous 72 swims that 44 were less than 30 seconds and 28 were greater than 30 seconds.
  2. Stating a necessary assumption, test at the \(5 \%\) significance level whether Elena's median time for the 50 m freestyle is less than 30 seconds.
OCR S4 2012 June Q6
12 marks Standard +0.8
6 The random variables \(S\) and \(T\) are independent and have joint probability distribution given in the table.
\(S\)
\cline { 2 - 5 }012
\cline { 2 - 5 }1\(a\)0.18\(b\)
20.080.120.20
\cline { 2 - 5 }
\cline { 2 - 5 }
  1. Show that \(a = 0.12\) and find the value of \(b\).
  2. Find \(\mathrm { P } ( T - S = 1 )\).
  3. Find \(\operatorname { Var } ( T - S )\).
OCR S4 2012 June Q7
12 marks Challenging +1.2
7 The continuous random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { 1 } { 4 } ( 1 + a x ) & - 2 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) is a constant.
  1. Show that \(| a | \leqslant \frac { 1 } { 2 }\).
  2. Find \(\mathrm { E } ( X )\) in terms of \(a\).
  3. Construct an unbiased estimator \(T _ { 1 }\) of \(a\) based on one observation \(X _ { 1 }\) of \(X\).
  4. A second observation \(X _ { 2 }\) is taken. Show that \(T _ { 2 }\), where \(T _ { 2 } = \frac { 3 } { 8 } \left( X _ { 1 } + X _ { 2 } \right)\), is also an unbiased estimator of a.
  5. Given that \(\operatorname { Var } ( X ) = \sigma ^ { 2 }\), determine which of \(T _ { 1 }\) and \(T _ { 2 }\) is the better estimator.
OCR S4 2012 June Q8
5 marks Standard +0.8
8 Events \(A\) and \(B\) are such that \(\mathrm { P } ( A ) = 0.3\) and \(\mathrm { P } ( A \mid B ) = 0.6\).
  1. Show that \(\mathrm { P } ( B ) \leqslant 0.5\).
  2. Given also that \(\mathrm { P } ( A \cup B ) = x\), find \(\mathrm { P } ( B )\) in terms of \(x\).