Summer 2015 examination
ST102
Elementary Statistical Theory
Suitable for all candidates
Instructions to candidates
This paper contains seven questions. Answer BOTH questions from Section A, and THREE
questions from Section B. All questions carry equal numbers of marks.
If you answer more than three questions from Section B, only your BEST 3 answers will count
towards the final mark.
This examination counts for 100% of the final grade.
Full working must be shown to gain all marks for each question.
Time allowed -
Reading Time:
Writing Time:
None
3 hours
You are supplied with:
Murdoch & Barnes Statistical Tables, 4th edition
Formulae Sheets (at the end of this paper)
You may also use:
No additional materials
Calculators:
Calculators are allowed in this examination
©LSE ST 2015/ST102
Page 1 of 10
SECTION A
Answer BOTH questions from Section A. Both questions carry equal numbers of marks.
1. (a) The amount of coffee dispensed into a coffee cup by a coffee machine follows a
normal distribution with mean 150 ml and standard deviation 10 ml. The coffee is
sold at the price of £1 per cup. However, the coffee cups are marked at the 137 ml
level, and any cup with coffee below this level will be given away free of charge. The
amounts of coffee dispensed in different cups are independent of each other.
i. Find the probability that the total amount of coffee in 5 cups exceeds 700 ml.
(� marks)
ii. Find the probability that the difference in the amounts of coffee in 2 cups is
smaller than 20 ml.
(� marks)
iii. Find the probability that one cup is filled below the level of 137 ml.
iv. Find the expected income from selling one cup of coffee.
(� mark)
(� marks)
v. Find the probability that a customer pays at most £4 for purchasing 5 cups of
coffee.
(� marks)
(b) An employer is about to hire one new graduate from a group of n candidates. The n
candidates can be ranked in a unilateral order according to their abilities (i.e. no two
candidates have the same ability). The employer proceeds according to both of the
following rules:
• Each candidate is interviewed in succession (in a random order ) and a decision
is made whether to hire the candidate at the end of each interview.
• Having rejected m 1 candidates (m > 1), the employer can hire the mth
candidate only if the ability of the mth candidate is better than those of the
previous m 1 candidates.
We are told that a candidate was hired at the end of the ith interview. Find the
probability that this hired person was the candidate with the highest ability. State this
probability for i = 1, 2, k and n (where k < n).
(� marks)
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2. (a) A random sample {X1 , X2 , . . . , Xn } is drawn from the distribution with the
following probability density function:
f (x; ✓) =
for x
p
✓
p e ✓ x
2 x
0, and 0 otherwise.
i. Find the maximum likelihood estimator for ✓.
(8 marks)
ii. Suppose n = 4 and we observe x1 = 4.1, x2 = 7.3, x3 = 6.5 and x4 = 8.8.
Based on this random sample, calculate the maximum likelihood estimate of
✓/2.
(� marks)
(b) Let {X1 , X2 , . . . , Xn } be a random sample from a population with mean µ and
variance 2 < 1. The method of moments estimator for µ is X̄.
i. Derive the method of moments estimator for
2.
(� marks)
ii. Determine the bias of your derived estimator.
iii. Propose an appropriate unbiased estimator for
that it is unbiased.
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(� marks)
2.
You do not need to verify
(� mark)
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SECTION B
Answer THREE questions from Section B. All questions carry equal numbers of marks.
If you answer more than three questions from Section B, only your BEST 3 answers will count
towards the final mark.
3. (a) Consider a probability density function of the following form:
(
↵k↵ /x↵+1 for x
k
f (x) =
0
otherwise
where ↵ > 0 is a parameter, and k > 0 (the smallest possible value of X) is a
known number.
i. Derive the cumulative distribution function, i.e. F (x).
ii. Derive the expected value of X, i.e. E(X).
(� marks)
(� marks)
(b) Consider a random variable X with the Laplace distribution. The probability density
function is:
1
f (x) = e |x| for
1 < x < 1.
2
Show that the moment generating function of X is:
MX (t) =
1
1
t2
and give the interval around 0 which t must be in for MX (t) to be well-defined.
(8 marks)
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4. (a) Z1 , Z2 , . . . , Z5 are independent standard normal random variables.
distribution of:
i.
5
P
i=1
ii.
5
P
i=1
State the
ai Zi , where {ai } are arbitrary constants
Zi2 .
Find the values of k such that:
iii. P (kZ1 + Z2  4) = 0.8413
iv. P (Z12 + Z22  7.378) = k
v. P Z12 + Z22 + Z32  k(Z42 + Z52 ) = 0.95.
(�� marks)
(b) Y is a random variable with expected value zero, P (Y = 1) = 0.2 and
P (Y = 2) = 0.1. It is known that Y takes just one other value besides 1 and 2.
i. What is the other value that Y takes?
ii. What is the variance of Y ?
(� marks)
(c) Let X be the amount of money won or lost in betting £5 on red in roulette, such that:
P (X = 5) =
18
38
and P (X =
5) =
20
38
.
If a gambler bets on red 100 times, use the central limit theorem to estimate the
probability that those wagers result in less than £50 in losses.
(6 marks)
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5. Consider two random variables X and Y . They both take the values 1, 0 and 1. The
joint probabilities for each pair are given by the following table, with ✓ 2 R a parameter.
Y = 1
Y =0
Y =1
X= 1
0.1 + ✓
0.1 + ✓
0.1 + ✓
X=0
0.1
0.2 6✓
0.1
X=1
0.3 + 3✓
0
0
(a) What is the range of values which the parameter ✓ can take for the above table to
correspond to a probability table?
(� marks)
(b) Calculate the marginal probabilities and the expected values of X and Y .
(� marks)
(c) Provide an unbiased estimator of ✓ based on Y but not on X. Also, give an unbiased
estimator of ✓ based on X but not on Y .
(� marks)
(d) Define U = min(X, Y ), i.e. U takes the minimum of X and Y . Calculate the
covariance of U and X.
(� marks)
(e) Are there any values of ✓ such that U and X are independent random variables?
Explain your answer.
(� marks)
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6. (a) Suppose {X1 , X2 , . . . , Xn } and {Y1 , Y2 , . . . , Ym } are two independent random
2 ) and N (µ , 2 ) with sample sizes n and
samples from, respectively, N (µX , X
Y
Y
m such that n 6= m. We are interested in testing the following hypotheses:
H0 : µX
µY =
0
vs.
H1 : µ X
µY 6=
0.
2 = 2 is unknown, derive the appropriate test statistic and explain
Assuming that X
Y
why it has its particular distribution under H0 .
You may state without proof the sampling distributions of the sample means X̄ and Ȳ
2 / 2 and (m
and also the sampling distributions of (n 1)SX
1)SY2 / Y2 , noting
X
that all these random variables are independent of each other.
(�� marks)
(b) Two independent samples from normally distributed populations yield the following
results:
P
Sample 1 n = 8
(x
x̄)2 = 21.2
P i
Sample 2 m = 9
(yi ȳ)2 = 29.8
Test at the 5% signficance level whether the population variances are the same
based on the above data. Make sure you state the hypotheses, the test statistic and
its distribution under the null hypothesis.
(� marks)
(c) Define the term ‘p-value’ and briefly explain how it is used in statistical hypothesis
testing when testing the value of the parameter ✓.
(� marks)
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7. (a) Consider the linear model yi = xi + "i , where E("i ) = 0, Var("i ) =
Cov("i , "j ) = 0 for i 6= j, and x1 , x2 , . . . , xn are constants.
Find the least squares estimator of
for .
2 > 0,
, and show that it is an unbiased estimator
(� marks)
(b) Let the observations (yi , xi ), for i = 1, 2, . . . , n be taken from the linear regression
model:
yi = 0 + 1 xi + "i
where the "i s are independent and N (0,
2 ) with
2 > 0 unknown.
i. Let ybi = b0 + b1 xi , where b0 and b1 are the least squares estimators of
and 1 , respectively. Show that:
n
X
(b
yi
i=1
ȳ) = ( b1 )2
2
n
X
(xi
0
x̄)2
i=1
where ȳ and x̄ denote, respectively, the sample means of the yi s and xi s.
(� marks)
For n = 12, the following summary statistics are available:
X
X
X
xi = 1.06,
yi = 13.14,
xi yi = 8.39
i
i
X
x2i = 14.09,
i
ii. Calculate b0 , b1 and b 2 .
i
X
yi2 = 27.80.
i
iii. For x = 0.5, calculate a 95% confidence interval for µ(x) =
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(6 marks)
0 +
1 x.
(� marks)
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