University of London The London School of Economics and Political Science
PROBABILITY AND STATISTIC INF
STATS ST202

Spring 2016
PSTAT 207:
Minimal sucient statistics
11/6/2006
Functions of sucient statistics
Any 1to1 function of a sucient statistic involves no loss of information and therefore is also
sucient.
Proof: If S is a 1to1 function of T then there is a function S 1 .
University of London The London School of Economics and Political Science
PROBABILITY AND STATISTIC INF
STATS ST202

Spring 2016
MA212: Further Mathematical Methods (Calculus) 201516
Exercises 1: Assumed background
The lectures do not cover practical techniques for integration of functions of a single variable as
students on this course are supposed to be skilled in this already. H
University of London The London School of Economics and Political Science
PROBABILITY AND STATISTIC INF
STATS ST202

Spring 2016
Mathematical Methods, Lecture Thirteen
13. Developing Geometric Insight, 1 of 2
13.1 Visualising the set R2 using position vectors
In Lecture Notes 3, we dened the set R2 as the set of all pairs (a, b) where a and b
are elements of R:
R2 = cfw_(a, b)  a
University of London The London School of Economics and Political Science
PROBABILITY AND STATISTIC INF
STATS ST202

Spring 2016
Mathematical Methods, Lecture TwentyFour
24. Linear Transformations, 2 of 6
In this section we focus on linear transformations T : V ! W between nitedimensional vector spaces V and W and show that any such transformations can
be represented by a matrix.
University of London The London School of Economics and Political Science
PROBABILITY AND STATISTIC INF
STATS ST202

Spring 2016
Welcome to MA 212
Further Mathematical Methods
Jozef Skokan
Calculus
Columbia House COL.3.04
Adam Ostaszewski
Linear Algebra
Columbia House COL.4.06
Department of Mathematics
London School of Economics and Political Science
Lectures
Calculus: weeks 110 in
University of London The London School of Economics and Political Science
PROBABILITY AND STATISTIC INF
STATS ST202

Spring 2016
Chapter 17. Orthogonal Matrices and Symmetries of Space
Take a random matrix, say
1 2 3
A = 4 5 6 ,
7 8 9
and compare the lengths of e1 and Ae1 . The vector e1 has length 1, while Ae1 =
(1, 4, 7) has length 66. Most matrices are like this: They change the
University of London The London School of Economics and Political Science
PROBABILITY AND STATISTIC INF
STATS ST202

Spring 2016
Mathematical Methods, Lecture TwentyTwo
22. Vector Spaces associated with Matrices, 2 of 2
In this section, we complete the presentation of the vector spaces N (A), CS(A) and
RS(A) associated with a matrix A. We start by establishing certain relationship
University of London The London School of Economics and Political Science
PROBABILITY AND STATISTIC INF
STATS ST202

Spring 2016
Mathematical Methods, Lecture Twenty
20. Vector Spaces, 4 of 4
In this section, we develop the concepts of angle and length, so that these concepts
can be applied to a general vector space V where angles and lengths do not arise
in a natural way. For exam
University of London The London School of Economics and Political Science
PROBABILITY AND STATISTIC INF
STATS ST202

Spring 2016
Mathematical Methods, Lecture Nineteen
19. Vector Spaces, 3 of 4
19.1 Theorems on linear span and linear independence
In this subsection, we collect four theorems relating to the linear span and the linear
independence or dependence of a set of vectors cf
University of London The London School of Economics and Political Science
PROBABILITY AND STATISTIC INF
STATS ST202

Spring 2016
Mathematical Methods, Lecture Eighteen
18. Vector Spaces, 2 of 4
18.1 Linear span
Recall that by a linear combination of vectors v1 , v2 , . . . , vk we mean a vector v of
the form
v = 1 v1 + 2 v2 + + k vk ,
for some constants i 2 R.
Now suppose that V is
University of London The London School of Economics and Political Science
PROBABILITY AND STATISTIC INF
STATS ST202

Spring 2016
ST202 P ROBABILITY, D ISTRIBUTION T HEORY, AND I NFERENCE
L ECTURE N OTES
Matteo Barigozzi
Michaelmas Term 20152016
This version October 7, 2015
Introductory Material
Course Aims and Objectives
The rst part of the course aims to convey a thorough underst
University of London The London School of Economics and Political Science
PROBABILITY AND STATISTIC INF
STATS ST202

Spring 2016
Problem Set 1  Due on Wednesday Oct 7 at 12:00
Ex. 1 An urn contains ve balls numbered 1 to 5. We extract two balls with replacement.
a) Draw a diagram to represent the sample space for this experiment. Mark the following events on your diagram:
(a) E1 r
University of London The London School of Economics and Political Science
PROBABILITY AND STATISTIC INF
STATS ST202

Spring 2016
ST202
P ROBABILITY
D ISTRIBUTION T HEORY
AND I NFERENCE
Matteo Barigozzi
Michaelmas Term 20152016
This version September 30, 2015
Introductory Material
Course Aims and Objectives
The rst part of the course aims to convey a thorough understanding of proba
University of London The London School of Economics and Political Science
PROBABILITY AND STATISTIC INF
STATS ST202

Spring 2016
SOLUTION FOR HOMEWORK 4, STAT 6331
Welcome to your fourth homework. Reminder: if you nd a mistake/misprint, do not
email or call me. Write it down on the rst page of your solutions and you may give yourself
a partial credit but keep in mind that the tota
University of London The London School of Economics and Political Science
PROBABILITY AND STATISTIC INF
STATS ST202

Spring 2016
Aspirin and gastric ulcer
Aspirin (acetylsalicyclic acid)
Can be absorbed by
epithelial cells in stomach
Induces gastric ulcer
through cyclooxygenasedependent and local effects
Inhibits the production of
the paracrine prostaglandin
through suppressing