University of London The London School of Economics and Political Science
Homework exercises
ST 102

Winter 2014
MA 103 Slides 4 a
Introduction to Abstract Mathematics
Fibonacci Numbers
1
The Fibonacci numbers
The Fibonacci numbers go back to Leonardo of Pisa
(c. 11701250), who was better known as Fibonacci.
Fibonacci helped bring the HinduArabic numeral system to
University of London The London School of Economics and Political Science
ST 102

Spring 2016
ST102
Elementary Statistical Theory
Chapter 3:
Random variables
Dr James Abdey
Department of Statistics
London School of Economics and Political Science
ST102 Elementary Statistical Theory
Dr James Abdey
MT 2015
Random variables
1
Introduction
In Chapter
University of London The London School of Economics and Political Science
ST 102

Spring 2016
ST102
Elementary Statistical Theory
Chapter 5:
Multivariate random variables
Dr James Abdey
Department of Statistics
London School of Economics and Political Science
ST102 Elementary Statistical Theory
Dr James Abdey
MT 2015
Multivariate random variables
University of London The London School of Economics and Political Science
ST 102

Spring 2016
ST102 Class 5 Additional exercises
1. A discrete random variable X has possible values 0, 1, 2, . . . , n, where n is a known integer.
The probability function of X is:
(
n
x
nx
for x = 0, 1, 2, . . . , n
x (1 )
p(x) =
0
otherwise
where nx = n!/[x! (n x)
University of London The London School of Economics and Political Science
ST 102

Spring 2016
ST102 Class 4 Solutions to Additional exercises
1. Let:
i = P (the best candidate is hired  the hiring occurs in the ith interview).
With n candidates, after i 1 rejections there are n (i 1) = n i + 1 remaining
candidates. Since candidates are selected f
University of London The London School of Economics and Political Science
ST 102

Spring 2016
ST102 Class 7 Additional exercises
1. A discrete random variable X has possible values 0, 1, 2, . . . , and the probability function:
(
e x /x! for x = 0, 1, 2, . . .
p(x) =
0
otherwise
where > 0 is a parameter. Show that E(X) = by determining
P
x p(x).
2
University of London The London School of Economics and Political Science
ST 102

Spring 2016
ST102 Class 4 Additional exercises
1. 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 employe
University of London The London School of Economics and Political Science
ST 102

Spring 2016
ST102 Class 6 Solutions to Additional exercises
1. Clearly, f (x) 0 since 1/x2 0 for x 1. Also:
Z
1
1
dx =
= 1.
x2
x 1
1
Hence f (x) is a valid pdf. However, the expected value would be:
Z
Z
1
1
E(X) =
x 2 dx =
dx = [ln x]
1
x
x
1
1
but this quantity
University of London The London School of Economics and Political Science
ST 102

Spring 2016
ST102 Class 7 Solutions to Additional exercises
1. We have:
E(X) =
X
x p(x) =
x=0
X
x
e x
x!
x
e x
x!
x=0
=
X
x=1
=
x1
X
e
x=1
=
(x 1)!
y
X
e
y=0
y!
= 1
=
where we replace x 1 with y. The result follows from the fact that
P
(e y )/y! is the
y=0
sum
University of London The London School of Economics and Political Science
Homework exercises
ST 102

Fall 2013
(33) Introduction to linear regression analysis
ST102 PROOFS AND DEFINITIONS
(3) Introduction to probability theory
Axiom (1): 0 for all events A
! =
(! )
=
!
!
2
( )
!
!
2 +
+
!
!
!
!
(22) Point estimation I
Mean absolute deviation, =
University of London The London School of Economics and Political Science
Homework exercises
ST 102

Spring 2015
ST107 Exercise 10
1. Explain the main types of variables which statistical methods are used to study.
What are the dierences between association and correlation as statistical terms?
2. A researcher into the use of computational aids conducts a survey in
University of London The London School of Economics and Political Science
Homework exercises
ST 102

Spring 2015
ST107 Outline solutions to Exercise 9
1. (a) This question requires the test of a single proportion. We test the hypotheses:
H0 : = 0.01
vs.
H1 : > 0.01.
We use the following test statistic:
Z=
P
(1 )/n
N (0, 1)
approximately, since n is large. Under H0
University of London The London School of Economics and Political Science
Homework exercises
ST 102

Spring 2015
ST107 Exercise 9
1. An accounting rm wishes to test the claim that no more than 1% of a large number
of transactions contains errors. In order to test this claim, the rm examines a
random sample of 144 transactions and nds that exactly 3 of these are in e
University of London The London School of Economics and Political Science
Homework exercises
ST 102

Spring 2015
ST107 Outline solutions to Exercise 10
1. There are two main types of variable: measurable variables (such as distances, weights,
temperatures, frequencies and sums of money), and categorical variables, which cannot
be measured.
Categorical variables can
University of London The London School of Economics and Political Science
ST 102

Spring 2016
ST102
Elementary Statistical Theory
Chapter 4:
Common distributions of random variables
Dr James Abdey
Department of Statistics
London School of Economics and Political Science
ST102 Elementary Statistical Theory
Dr James Abdey
MT 2015
Common distributio