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Exercise Sheet 5
The exercise sheet is subdivided into three parts:
Part A contains relatively straightforward questions which you should attempt
in your own time full solutions for these will be available on Moodle after
YP: STAT2001/3101, 2012-2013
11
Exercise Sheet 4
The exercise sheet is subdivided into three parts:
Part A contains relatively straightforward questions which you should attempt
in your own time full solutions for these will be available on Moodle after
YP: STAT2001/3101, 2012-2013
3
Solutions to Exercise Sheet 2
A: Warm-up questions
1. (a) P (A) = P (cfw_1, 2) = P (cfw_1 cfw_2) = P (cfw_1) + P (cfw_2) = 1/2 + 1/4 = 3/4.
(b) P (X = 1|X A) = P (X = 1|A) =
P (X=1 and XA)
P (XA)
=
1/2
3/4
= 2/3
2. (a) Its g
YP: STAT2001/3101, 2012 2013
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
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
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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
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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,
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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
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)
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
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
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
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(
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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
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Example 1.5: An Application of the Iterated Conditional Expectation Formula
Question:
Suppose that R and N have joint distribution in which R|N is
Bin(N, ) and N is Poi(). Show that R is Poi(). Find the
mean of R using the i
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Exercise Sheet 8
The exercise sheet is subdivided into three parts:
Part A contains relatively straightforward questions which you should attempt
in your own time full solutions for these will be available on Moodle after
YP: STAT2001/3101, 2012-2013
18
Exercise Sheet 7
The exercise sheet is subdivided into three parts:
Part A contains relatively straightforward questions which you should attempt
in your own time full solutions for these will be available on Moodle after
YP: STAT2001/3101, 2012-2013
16
Exercise Sheet 6
The exercise sheet is subdivided into three parts:
Part A contains relatively straightforward questions which you should attempt
in your own time full solutions for these will be available on Moodle after
YP: STAT2001/3101, 2012-2013
8
Exercise Sheet 3
The exercise sheet is subdivided into three parts:
Part A contains relatively straightforward questions which you should attempt
in your own time full solutions for these will be available on Moodle after t
YP: STAT2001/3101, 2012 2013
1
Combinatorics of the Multinomial Distribution
As quite a few students seemed to have problems with the combinatorics of the multinomial distribution, I thought a simple reference to the STAT1005 lecture notes (p.12)
might no
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Exercise Sheet 2
The exercise sheet is subdivided into three parts:
Part A contains relatively straightforward questions which you should attempt
in your own time full solutions for these will be available on Moodle after t
YP: STAT2001/3101, 2012-2013
23
Exercise Sheet 9
These exercises cover sections 6.1 to 6.2.2. You should attempt all questions
but you do not need to hand in your answers. Full answers will be published
on moodle in due course.
1. Suppose that X1 , . . .