Preparatory work for Mathematics and Statistics
Many modules assume a good knowledge of mathematics and some statistics. Mathematical
tools will be used in Macroeconomics 2 and Microeconomics 2, and Econometrics 1
assumes additionally some background in statistics. As part of your preparation for the
course you should spend some time before starting the programme revising your
mathematics.
Prerequisites
You should be familiar with the following: basic algebra: variables, formulae, rules for
manipulation of algebraic expressions, solving linear equations in one variable and
simultaneous linear equations, quadratic equations; functions of one variable; logarithms;
exponential functions; differential calculus: the derivative of a function of one variable, rules
of differentiation (xn, log(x), ln(x), ex, ax, the chain rule, the product rule, the quotient rule);
integration: indefinite integral, definite integral, areas under graphs of functions; curve
sketching, turning points and inflexion points of functions of one variable, asymptotes.
Suggested Reading
“Essential Mathematics for Economic Analysis” by Knut Sydsæter and Peter Hammond,
Third Edition, Prentice Hall.
Also, you might want to look at some statistical appendices you find in some econometrics
text books. For example, “Introductory Econometrics: A Modern Approach” by Wooldridge,
4th edition, has a nice chapter. However, note, the 5th edition does not have this.
Further reading
You may wish to have a look at free courseware uploaded by MIT:
http://ocw.mit.edu/courses/mathematics/
Useful modules include single variable calculus, differentiation and linear algebra.
1
Maths Exercises
Please review the following exercises before the start of the term:
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Examples 1 to 4 in pages 11 and 12 (algebraic expressions).
Examples 3 and 4 in pages 19 and 20 (fractional powers).
Examples 1 and 2 in pages 30 and 31 (absolute values).
Examples 1 and 2 in pages 36 and 37 (simple equations).
Example 1 in page 38 (simple macroeconomic model).
Example 2 in page 42 (quadratic equation).
Example 1 in page 46 (simple system of two equations).
Examples 2 to 4 in pages 82 and 83 (univariate functions).
Examples 1 and 2 in pages 87 and 88 (graph of univariate functions).
Examples 1 and 2 in pages 90 and 91 (linear functions).
Example 3 in pages 122 (logs).
Example 2 in page 148 (cirlcle).
Example 1 and 2 in pages 158 and 159 (derivatives).
Example 3 and 4 in pages 172 and 173 (limits).
Examples 2 and 3 in page 176 (derivates).
Example 3 in page 179 (derivative).
Example 5 in page 182 (derivative)
All exercises are fully solved in the text.
2
Statistics Exercises
PLEASE BRING YOUR SOLUTIONS TO YOUR PRE-SESSIONAL CLASSES AT THE
START OF THE COURSE.
Exercise 1: Descriptive Statistics
A random sample of incomes in a community yields the following values (in Thousands of £ per
year): 42, 37, 78, 38, 35, 52.
1. Judging from this sample, the income random variable is:
(a) symmetric.
(b) negatively skewed.
(c) positively skewed.
(d) normally distributed.
(e) chi-squared distributed.
2. The sample variance of the income random variable is:
(a) 47
(b) 222.667
(c) 16.346
(d) 14.922
(e) 267.2
3.
The sample standard deviation of the income random variable is:
(a) 47
(b) 222.667
(c) 16.346
(d) 14.922
(e) 267.2
4.
What is an appropriate measure of central tendency for the income random variable?
(a) 40
(b) 47
(c) 38
(d) 42
(e) 56.4
5.
A colleague has added a 7th observation to the above sample. The sample mean when
calculated on all 7 observations is 49. What is the income value of the observation that
was added?
(a) 47
(b) 49
(c) 2
(d) 61
(e) 343
3
6.
7.
You obtained a set of 40 observations and calculated a sample standard deviation of
3.564. What is the population variance for the same 40 observations?
(a) 12.702
(b) 495.382
(c) 3.512
(d) 13.028
(e) 12.385
The table below reports the A-level score in the best 3 A-levels for all leavers from a
UK ‘old’ University in 1993, broken down by gender (where 10 points are awarded for
a grade A, 8 points for a grade B, 6=C, 4=D and 2=E):
A-level score
14
16
18
20
22
24
26
28
30
(a)
(b)
(c)
(d)
(e)
(f)
8.
Calculate the mean.
Calculate the variance and standard deviation.
Calculate the median.
Calculate the mode.
Calculate the interquartile range.
What do you think are the interesting points about the distribution of A-level
scores for males and females.
For a sample of 35 companies, the table below shows the percentage change in
output over the last 3 months, grouped into classes.
% change
Number of companies
(a)
(b)
(c)
(d)
(e)
(f)
9.
No. females
1591
2413
3262
3846
4289
4241
3819
3063
3057
-2-0
13
0-2
4
2-4
8
4-6
7
6-8
3
Calculate the mean percentage change in output.
Calculate the median.
Find the modal class.
Calculate the variance.
Calculate the standard deviation.
Calculate the interquartile range.
In question 2 suppose the 1st interval width were -4-0% instead of -2-0% and the last
interval were 6-10% instead of 6-8% what would be the effect on the mean and
variance of the distribution. In question 2, suppose an innovation in technology means
all output grows by 2 percentage points more than previously, what would be the effect
on the mean and variance of the distribution. If alternatively as a result of the new
technology output increases by 20%, what would be the effect on the mean and
variance of the distribution.
4
Exercise 2: Probability
This information relates to question 1-3. The frequencies of the number of GCSE passes at
grades A to C, derived from Youth Cohort Survey data for 2004, for boys and girls:
No. of GCSE passes
0
1-4
5-6
7-10
11-17
Boys
883
1056
528
2120
707
Girls
717
1281
625
2952
1136
Note: the numbers of GCSE passes shown are inclusive, so, for example, 1-4 means 1, 2, 3
and 4 passes.
1.
What proportion of pupils obtained 5 or 6 GCSE passes at grades A to C?
(a) 0.1.
(b) 0.093.
(c) 0.422.
(d) 0.096.
(e) 0.167.
2.
What proportion of boys got at least 7 GCSE passes at grades A to C?
(a) 0.866.
(b) 0.534.
(c) 0.609.
(d) 0.830.
(e) 0.847.
3.
What is the probability that the pupil is female?
(a) 0.169.
(b) 0.134.
(c) 0.154.
(d) 0.384.
(e) 0.616.
This information relates to question 4-6. Consider the experiment of rolling a single dice.
Events A and B are defined as
A = {an odd numbered face is showing}
B = {faces 1 or 2 or 3 or 4 are showing}
4.
Which one of the following statements is true?
(a) A and B are mutually exclusive events.
(b) A and B are disjoint events.
(c) B is a subset of A.
(d) A is the complementary event to B.
(e) A and B are independent events.
5
5.
What is the probability of the event A  B ?
(a) 3/4.
(b) 2/3.
(c) 1/2.
(d) 1/3.
(e) 1/6.
6.
What is the probability of the event A  B ?
(a) 3/4.
(b) 2/3.
(c) 1/2.
(d) 1/3.
(e) 1/6.
This information relates to questions 7-11. We have a sample of 150 students where we
have information on the kind of school they go to: Type of school: S1=State non-selective,
S2=State Selective or S3=Private Selective and how they travel to school. Type of transport:
T1=Bike, T2=Foot, T3=Public Transport or T4=Car:
Transport – T
T1
T2
T3
T4
Total
7.
8.
9.
School – S
S2
9
9
24
6
48
S1
12
12
33
6
63
What is the value for P(S2)?
(a) 48
(b) 0.68
(c) 0.32
(d) 0.06
(e) None of the above
What is the value for PS1  T3  ?
(a) 0.22
(b) 0.42
(c) 0.1764
(d) 26.46
(e) None of the above
What is the value for PT1 | S 3  ?
(a) 0.02
(b) 0.125
(c) 0.0416
(d) 0.16
(e) None of the above
6
S3
3
15
6
15
39
Total
24
36
63
27
150
10.
How many students would you expect to go to a private School and use public transport if
the two variables T and S were independent?
(a) 6
(d) 4.26
(c) 10.92
(d) 16.38
(e) None of the above
11.
The Government Economic Service is concerned about the Economics and
Econometric skills of its employees. 40% of these employees signed up to Economic
classes and 50% to Econometric classes. Of those signing up for Economic classes,
30% signed up for Econometric classes.
(a) What is the probability that a randomly selected employee signed up for both
classes?
(b) What is the probability that a randomly selected worked who signed up for the
econometrics classes also signed up for the Economics classes?
(c) What is the probability that a randomly chosen worker signed up for at least one
of these two classes?
(d) Are the two events statistically independent?
12.
The alumni association at a leading UK university solicits donations by telephone. It is
estimated that for any individual the probability of an instant donation was 0.05, 0.20
give no immediate donation, but a request for further information through mail and 0.75
have no interest. Mailed information is sent to all persons requesting it and 0.15 of
these eventually give a donation. An operator makes a sequence of calls, which can
be assumed to be independent.
(a)
(b)
What is the probability of a donation?
What is the probability that an instant credit card donation is immediately
preceded by at least 4 no interest calls?
[Hint for this question you need to remember that:
a  ar  ar 2   ar   a / (1  r ) for r  1 ]
7
Exercise 3: Discrete and joint distributions
This information relates to questions 1-2. The joint probability distribution of the random
variables X and Y, as given in the table
X
Y
0
0.37
0.39
1
0.17
0.07
1.
The expected value and variance of X is (to 3 d.p.)
(a) 0.46, 0.46
(b) 0.24, 0.24
(c) 0.24, 0.248
(d) 0.46, 0.248
(e) 0.24, 0.182
2.
The expected value and variance of Y is (to 3 d.p.)
(a) 0.46, 0.46.
(b) 0.24, 0.24.
(c) 0.24, 0.248.
(d) 0.46, 0.248.
(e) 0.24, 0.182.
This information relates to questions 3-5. The discrete random variable X with probability
distribution given by
Values of X
Probabilities
-3
1/3
-1
1/6
1
1/6
3.
The variance var(X) of X is:
(a) 8.
(b) 0.
(c) 11.
(d) 6.333.
(e) 1.
4.
If Z is defined as X  Z 2  1 , the expected value of Z is
(a) -1.
(b) 14.222.
(c) 42.667.
(d) 6.333.
(e) 5.333.
5.
If Z is defined as X  Z 2  1 , the variance of Z is
(a) -1.
(b) 14.222.
(c) 42.667.
(d) 6.333.
(e) 5.333.
8
3
1/3
This information relates to questions 3-5. A random sample of size 2 is drawn from the
probability distribution of the random variable X with probability distribution
Values of X
Probabilities
-2
0.4
0
0.2
2
0.4
Drawing a random sample of size 2 from the population represented by this random variable.
6.
What are the probabilities of the samples (-2, 0) and (-2, 2)?
(a) 2/25 and 4/25.
(b) 4/25 and 2/25.
(c) 1/25 and 2/25.
(d) 4/25 and 4/25.
(e) 2/25 and 1/25.
7.
What is the value of the sample mean X in each of the samples (-2, 0) and (-2, 2)?
(a) -2 and 0.
(b) -2 and -1.
(c) -1 and 2.
(d) 0 and -1.
(e) -1 and 0.
8.
What is the relationship between the population mean and the expected value of the
sample mean X ? Choose just one of the alternatives that best reflects these
properties.
(a) The sample mean is derived from the population mean.
(b) There is no relationship.
(c) The population mean is larger than the expected value of the sample mean X .
(d) The population mean is smaller than the expected value of the sample mean X .
(e) They have equal values.
9.
A contractor estimates the probabilities for the number of days required to complete a
certain type of construction project as follows:
Table 1: Distribution of days of completion
Time (Days)
Probability
(a)
(b)
(c)
(d)
(e)
(f)
1
0.05
2
0.20
3
0.35
4
0.30
5
0.10
What is the probability a randomly chosen project will take less than 3 days to
complete?
Find the expected time to complete.
Find the variance of time required to complete a project.
The contractor’s cost is made up of two parts – a fixed cost of £10,000 plus
£1,000 for each day taken to complete the project. Find the mean and standard
deviation of total project cost.
If 3 projects are undertaken, what is the probability that at least 2 of them will
take at least 4 days to complete, assuming independence of individual project
completion times.
Consider the new distribution, which is the old distribution plus 10 days:
Time (Days)
Probability
11
0.05
12
0.20
13
0.35
9
14
0.30
15
0.10
What is the expected time and variance for time to complete this project and how
do these relate to the answers to (a) and (b), respectively.
(g)
Consider the new distribution, which is the old distribution times 5 days:
Time (Days)
Probability
5
0.05
10
0.20
15
0.35
20
0.30
25
0.10
What is the expected time and variance for time to complete this project and how do
these relate to the answers to (a) and (b), respectively.
(h)
Define E ( X ) 
k
 p x as the expected value of the random variable X and
i 1
i i
k
V ( X )  E[ X  E ( X )]2  E ( X 2 )  E ( X ) 2   pi ( xi  E ( X )) 2 as the variance of
i 1
that distribution. Show the in general:
(i) E (a  X )  a  E ( X ) , (ii) V (a  X )  V ( X ) , (iii) E (aX )  aE ( X ) , (iv)
V (aX )  a 2V ( X )
10.
Consider the following bivariate distribution between Y and X
x
-1
0
1
1
0.2
0.1
0.1
y
2
0.1
0.0
0.2
3
0.0
0.2
0.1
(a)
(b)
(c)
(d)
(e)
Calculate E ( X ), E (Y )
Calculate V ( X ), V (Y ) and cov( X , Y )
Write out the univariate distribution of X+Y. Using this univariate distribution
calculate E ( X  Y ), V ( X  Y ) . How do these relate to your answers to parts (a)
and (b)?
Write out the univariate distribution of X-Y. Using this univariate distribution
calculate E ( X  Y ), V ( X  Y ) . How do these relate to your answers to parts (a)
and (b)?
Define the joint probability density function for the random variables X, Y as
p ( X , Y ) . Define the marginal density of X [Y] as p(x) [p(y)], as p(x) [p(y)], as
p ( X )   p ( x, y ) [ p(Y )   p( x, y) ]. Now E ( X )   xp( y ) , E (Y )   yp ( y ) ,
y
y
x
V ( X )   ( x  E ( X )) p( x) , V (Y )   ( y  E (Y )) p( y ) and
2
2
y
x
cov( X , Y )   ( x  E ( X ))( y  E (Y )) p ( x, y ) . Show that:
x
y
(i) E ( X  Y )  E ( X )  E (Y ) , (ii) V ( X  Y )  V ( X )  V (Y )  2 cov( X , Y )
10
y
Exercise 4: Special and Probability distributions
The following information relates to questions 1-2. The probability mass function of the
random variable X, the number of successes in 6 Bernoulli trials, where the success
probability is = 0.4. The probability mass function of X is
P( X )  P( X  x)  6C x (0.4) x (0.6)6 x x  0,1, 2,3, 4,5, 6
1.
What is Pr( X  1)
(a) 0.047.
(b) 0.311.
(c) 0.187.
(d) 0.544.
(e) 0.746.
2.
What is Pr( X  2) (to 3 d.p.)
(a) 0.047.
(b) 0.311.
(c) 0.187.
(d) 0.544.
(e) 0.746.
The following information relates to questions 3-4. The random variable V with probability
density function given by
3.
1
-1    1

f ( )   2
0 otherwise
What is Pr(V  1) ?
(a)
(b)
(c)
(d)
(e)
4.
0.5.
2.
1.
-1.
0.
What is the variance, var(V), of V?
(a) 1/8.
(b) 1/2.
(c) 1/3.
(d) 1/10.
(e) 1/12.
The following information relates to questions 5-6. The random variable X with probability
distribution N(-2, 4).
5.
What is Pr( X  1) (to 3 d.p.)?
(a) 0.841.
(b) 0.309.
(c) 0.159.
(d) 0.955.
(e) 0.691.
6.
What is the value of x which satisfies Pr( X  x)  0.05 (to 3 d.p.)?
11
(a)
(b)
(c)
(d)
(e)
2.326.
1.96.
-5.29.
1.29.
1.645.
7.
What is the value of x which satisfies Pr( X  x)  0.05 (to 3 d.p.)?
(a) 2.326.
(b) 1.96.
(c) -5.29.
(d) 1.29.
(e) 1.645.
8.
Suppose the probability is 0.5 that the value of the $US will rise against the Japanese
yen over any given week, and that the outcomes week on week are independent.
(a) What is the probability that the value of the $US will rise against the Japanese
yen in a majority of the weeks over a period of 7 weeks?
(b) What is the probability that the value of the $US will rise against the Japanese
yen in each of the 7 weeks?
(c) What is the probability that the value of the $US will rise against the Japanese
yen more than once in the 7 weeks?
9.
A university health centre receives walk-in patients at an average rate of 5 per minute,
during mid-day hours.
(a) Find the probability that there will be fewer than 2 walk-in patients in a particular
mid-day hour.
(b) Find the probability that there will be more than 8 walk-in patients in a particular
mid-day hour.
(c) Find the probability that there will be between 2 and 5 walk-in patients in a
particular mid-day hour.
10.
Anticipated consumer demand for a product next month can be represented by a
normal distribution with mean 1,200 units and standard deviation, 100 units.
(a) What is the probability that sales will exceed 1000 units?
(b) What is the probability that sales will be between 1100 and 1300 units?
(c) The probability is 0.10 that sales will be more than how many units?
11.
A lecturer found that time spent by students on an exercise sheet follow a normal
distribution with mean 60 minutes and standard deviation 18 minutes.
(a) The probability is 0.9 that a randomly chosen student spends more than how
many minutes on this exercise sheet.
(b) The probability is 0.8 that a randomly chosen student spends less than how
many minutes on this exercise sheet.
(c) Two students are chosen at random. What is the probability that at least one of
them spends at least 75 minutes on the exercise sheet?
12