Question 1
A number, π, of contestants are registered to take part in an archery contest. The distance
between the centre of the target and the point that the π π‘β archerβs arrow hits is given by the
random variable ππ , for π = 1, β¦ , π. The random variables π1 , β¦ , ππ are independent and
identically distributed, each following an exponential distribution with a mean of 10cm. Each
archer has one shot and the archer whose arrow hits closest to the centre of the target wins the
contest.
a) Determine the probability density function of the winnerβs distance from the centre of
the target, that is, the density of the random variable
π = πππ{π1 , β¦ , ππ }.
(3 marks)
b) Because of an outbreak of food poisoning in the hotel where the contestants are
staying, it is possible that not all registered archers can participate in the contest. The
number of archers taking part is given by the discrete random variable π, such that
π(π = 10) = 0.8, π(π = 9) = 0.1, π(π = 8) = 0.1
and π is independent of π1 , π2 , β¦ Hence, the winnerβs distance from the centre of the
target is given by the random variable π = πππ{π1 , β¦ , ππ }. Using the properties of
conditional expectation, calculate πΈ(π).
(4 marks)
Total: 7 marks
Question 2
Consider the random variables π1 and π2 , with means πΈ(π1 ) = πΈ(π2 ) = 0 and variances
πππ(π1 ) = πππ(π2 ) = 1. The random variables follow the bivariate normal distribution, which
means that their joint probability density function is given by
1
π₯12 β 2π₯1 π₯2 π + π₯22
π(π₯1 , π₯2 ) =
ππ₯π {β
}, π₯1 , π₯2 β β
2(1 β π2 )
2πβ1 β π2
where π β (β1,1). You can take as given that each of π1 , π2 follows a standard normal
distribution and that their correlation coefficient is π.
a) Show that if π = 0, the random variables π1 , π2 are statistically independent.
(1 mark)
b) Show that the conditional density ππ1 |π2 takes the following form:
2
1
1 π₯1 β π₯2 π
ππ1 |π2 (π₯1 |π₯2 ) =
ππ₯π {β (
) }
2
2 β1 β π2
β2πβ1 β π
(1 mark)
1
c) For π = 0.99, state the value of πππ(π2 |π1 = π₯) for some π₯ and interpret it.
d) Define the random variable π3 = (π1
all steps, and interpret your finding.
)2
(2 mark)
. Show that πΆππ£(π1 , π3 ) = 0, carefully justifying
(4 marks)
Total: 8 marks
Question 3
Four university lecturers (A, B, D, and C) teach four modules each within a given academic
year. The sample mean and variance of each lecturerβs module evaluation score, calculated
across each lecturerβs modules, are given in the table below.
Number of modules
Lecturer A
4
Lecturer B
4
Lecturer C
4
Lecturer D
4
Average score
Variance of scores
2.60
0.2196
3.13
0.3751
3.56
0.1851
3.92
0.2416
a) Perform a one-way Analysis of Variance for the above data, stating clearly the
hypotheses tested and reporting your test result at the 5% significance level. (You may
assume that all assumptions of the one-way ANOVA model are satisfied. You are given
the following critical values of the F distribution, one of which will be needed to answer
this question: πΉ3,12,0.025 = 4.474, πΉ12,3,0.05 = 8.745, πΉ3,12,0.05 = 3.490, πΉ4,12,0.05 =
3.259.)
(4 marks)
b) After being called in by his Head of Department to discuss his low feedback scores,
Lecturer A claims that the reason his scores are comparatively low is that his class sizes
were large. The following scatter-plot shows all lecturersβ scores plotted against the sizes
of the four classes they each taught, together with the line of best fit, obtained via simple
regression model of the form
ππ = π½0 + π½1 π₯π + ππ , ππ βΌ π(0, π 2 ),
where ππ are the evaluation scores for individual modules and π₯π are the corresponding
class sizes.
i.
The estimate of the variance π 2 in the simple linear regression model is π 2 =
0.2744. Calculate the values of π
2 and of the correlation coefficient of the
evaluation scores with the class size.
(4 marks)
2
Figure 1
From the plot, estimate the value of the intercept, π1 . You are given that the
standard error of π1 is π π΅1 = 0.0039. Calculate a 95% confidence interval and
state your conclusion. (You may assume that the relevant critical value of the
t distribution is approximately 2.)
(3 marks)
The Head of Department is not convinced that class size explains poor evaluation
scores. She states that it may just be a coincidence that the worst performing lecturers
teach larger classes. Explain what further analysis could be carried out to explore the
issue further.
(2 marks)
Total: 13 marks
ii.
c)
3
Question 4
Let the random variable π represent the effort that a randomly chosen actuarial science student
puts towards studying for a statistics module (on a scale from 0 to 5) and the random variable π
represent that studentβs final exam mark. Assume that the conditional expectation of π given
π = π₯ be given by the following formula:
πΈ(π|π = π₯) = 20 + 10π₯ + 20 β
tanh(π₯ β 2),
where
π 2π₯ β 1
tanh(π₯) = 2π₯
π +1
is the hyperbolic tangent function. The graph of the function π(π₯) = πΈ(π|π = π₯) is represented
by the solid line in Figure 2 below.
a) Let π be normally distributed, with mean equal to 2 and standard deviation equal to
0.5. Provide a simulation algorithm for calculating numerically the unconditional
expectation πΈ(π), starting from π standard normal observations π§1 , β¦ , π§π .
(5 marks)
b) Calculate πΈ(π|π = 0) and interpret the result.
(1 mark)
c) An education researcher, who is not aware of the formula for πΈ(π|π = π₯) given above,
tries to understand the relationship between studentsβ effort and their final exam mark.
The researcher manages to collect data from (π, π) for 20 students. The data are shown
in Figure 2 as points. The lecturer is fitting two regression models to the data, with
prediction equations:
Model 1: π¦Μ = β23.88 + 32.56 β
π₯
Model 2: π¦Μ = 19.83 β
π₯
Model 1 is a standard linear regression model. Model 2 is fitted by fixing the intercept to
2
zero, that is, 19.83 is the minimiser of the expression min β20
π=1(π¦π β π β
π₯π ) .
b
Draw the regression lines corresponding to the two models on (a print-out of) Figure 2
and include this in your coursework submission.
(3 marks)
d) Comment on the rationale behind Model 2. Explain whether Model 1 or Model 2 is
preferable.
(3 marks)
Total: 12 marks
4
Figure 2
5
6
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