Revision of Basic Probability
The fundamental idea of probability is that chance can be
measured on a scale which runs from zero, which represents
impossibility, to one, which represents certainty.
Sample space, : the set of all outcomes of an experiment
YP: STAT2001/3101, 2016 2017
1
STAT2001: Probability and Inference (0.5 unit)
STAT3101: Probability and Statistics II (0.5 unit)
Lecturer: Yvo Pokern
Aims of course
To continue the study of probability and statistics beyond the basic concepts introduced
i
STAT2002: Linear models and the analysis of variance
STAT7101: Further statistical methods and computing
Chapter 3 of course notes
Connections between the models
Normal population (NP):
Yi = + i
(i = 1, . . . , n) ,
where 1 , . . . , n are i.i.d. with 1 N
STAT2002: Linear models and the analysis of variance
STAT7101: Further statistical methods and computing
Chapter 2 of notes
Statistical models
What do we mean by a statistical model?
Why do we need statistical models?
How do we choose an appropriate stati
YP: STAT2001/3101, 2016-2017
4
Exercise Sheet 1
The exercise sheet is subdivided into three parts:
Part A contains warm-up questions you should do in your own time.
You should solve all questions of Part B and put your solutions in the locker in the und
YP: STAT2001/3101, 2016-2017
7
Exercise Sheet 2
The exercise sheet is subdivided into three parts:
Part A contains warm-up questions you should do in your own time.
You should solve all questions of Part B and put your solutions in the locker in the und
YP: STAT2001/3101, 2015-2016
1
Exercise Sheet 0: Get Ready!
Mathematics background
For this course you will need a knowledge of basic mathematical tools. The following
is a list of mathematical topics that will be encountered during the course, either in
STAT2002/STAT7101
Autumn term 20162017
Exercises - Sheet 4
To be discussed in the problem class on Friday 4th November. Please note that you
are expected to work on the exercises below before the problem class.
1. In 2006 and 2007 the students of the STAT
STAT2002/STAT7101
Autumn term 20162017
Exercises - Sheet 3
To be discussed in the problem class on Friday 28th October. Please note that you are
expected to work on the exercises below before the problem class.
The following data set gives the fat (in gra
STAT2002/STAT7101
Autumn term 20162017
Exercises - Sheet 2
To be discussed in the computer workshop on Friday 21st October. Please note that
you are expected to work on the exercises below before the workshop.
In a study of the effect of hormones on the g
STAT2002/STAT7101
Autumn term 20162017
Exercises - Sheet 1
To be discussed in the problem class on Friday 14 October. Please note that you are
expected to work on the exercises below before the problem class.
A researcher has tested two treatments, A and
Probability
3105
Based on lectures by
Dr N Sidorova
typed by John Sylvester
Jan 2012
Contents
1 Rigorous set up
1.1 Probability space, events and random variables . . . . . . . . . . . . . . . .
1.2 Distribution function . . . . . . . . . . . . . . . . .
Concepts from Last Lecture
Condence Interval in a Frequentist Framework
Pivotal quantity
1
7.2 Likelihood-based condence intervals
Suppose that is the maximum likelihood estimator of the unknown parameter .
We know:
asymptotically,
N(, 1/I()
I()( ) N(
Concepts from last lecture
Hypothesis Testing
Null and Alternative
Type I and II errors
size and power of a test
1
Classical approach to hypothesis testing:
State of Nature
= 0
d0
= 1
Type II error with prob.
Decision
d1 Type I error with prob.
d0
3 Likelihood and Suciency
Chapter 3 of Lecture Notes focuses on the important basic statistical concepts of likelihood and suciency, which are central
to statistical inference. It also highlights how these fundamental
concepts are used in the basic Bayesi
4 Concepts from last lecture
Likelihood Principle
Suciency
Principles of Bayesian Inference
( | x) ()p(x | )
1
4 Frequentist point estimation
Chapter 4 focuses on Frequentist point estimation and describes
the principles behind it. We discuss some desi
5 Likelihood and Bayesian methods of point estimation
Chapter 5 of the Lecture Notes describes the principles and basic
methods of Likelihood and Bayesian point estimation.
1
5.1 Maximum likelihood estimation
Recall:
The maximum likelihood estimate (mle)
Concepts from last lecture
Score function: V (X) = ln L(; X ), with E(V ) = 0 and var(V ) =
cfw_
2
I() = E 2 ln L(; X )
CRLB: If T is unbiased for m():
var(T )
[m()]2
I()
Exponential Family
p(x; ) = expcfw_a()T (x) + b() + c(x).
Extension to multipa
YP: STAT2001/3101, 2012-2013
21
Solutions to Exercise Sheet 9
1. Let X1 , . . . , Xn be iid from U (0, ) and consider X as well as X(n) as bases for
estimators for . Check whether they are unbiased:
E(X) = E(Xi ) = ,
2
so that T1 = 2X is unbiased for . Th
YP: STAT2001/3101, 2012-2013
19
Exercise Sheet 8
A: warm-up question
1. For application of the CLT we rst need the expectation and variance of X.
1
2x2 dx =
E(X) =
0
1
2
3
E(X 2 ) =
0
1
2x3 dx = .
2
Thus var(X) = 1/2 4/9 = 1/18. The CLT for sums now state
STAT2001/STAT3101:
Probability and Inference
Lecturer: Yvo Pokern, Room 144 Dept. of Statistical
Science
Lectures:
Christopher Ingold XLG2 Auditorium on Thursdays
1pm-2pm
Darwin B40 Lecture Theatre on Fridays 2pm-4pm
Ofce Hour: Thursdays 2:30pm-3:30pm,
YP: STAT2001/3101, 2012-2013
13
Exercise Sheet 6
A: Warm-up question
1. For geometrically distributed variables X and Y we have that P(X > n) = (1 p)n
(no successes in n trials, or sum the GP). Further, with Z = min(X, Y ) we have
P(Z > n) = P(X > n and Y
YP: STAT2001/3101, 2012-2013
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Exercise Sheet 4
A: Warm-up questions
1. From the specication of the bivariate normal distribution for (X, Y ), we can read
o: 1 = 0, 2 = 1, 1 = 1, 2 = 2, = 1 .
2
Therefore, the joint density is easy to write down using a fo