Department of Statistics and Actuarial Science
The University of Hong Kong
STAT 1302/2602
Probability and Statistics II
Semester 2, 2014/2015
Example Class 9
Solution:
1. (a) The test can be formed as
H0 : X Geo(p) vs H1 : otherwise, for some p (0, 1),
wh
m 1
7 Tests Based on Large Samples
Y
Y1 Y2
m1
pm1
p1 p2
Appendix
m
Proof of
2ln
Oi Ei 2
Ei
i 1
in Section
m 1
n Yi
i 1
m 1
1 pi
,
i 1
i 1, 2, , m 1.
of pi such that
Denote the value
ln L( p1 , p2 ,., pm1 )
0 as pi , i 1, 2, ,
pi
m 1. Then
PS7
1
L( p1,
f(x1, x2, , xn; 0)dx1dx2dxn,
6 Hypothesis Testing: Methods
C' D
Appendix
Proof of Theorem 6.1 (Neyman-Pearson
Lemma)
Suppose D is the rejection region of any other
test which has significance level * .
We consider first the continuous case. When
0,
f(
14/15 Fall
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT2602 Probability & Statistics II
Assignment 4 Solution
1. (a) Test H 0 : x y versus H 0 : x y at 0.05 .
X Y
Test statistic: T
S pool
1 1
m n
At 5% significance leve
14/15 Fall
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT2602 Probability & Statistics II
Assignment 4
Due Date: December 4, 2014
(Hand in your solutions for Questions 1, 4, 9, 11, 13, 14, 17, 20, 28)
1.
(a) The following
14/15 Fall
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT2602 Probability & Statistics II
Assignment 2 Solution
1. (a) E 1
1
1
1
1
1
1
EX 1 EX 2 E X 3
3
3
3
3
3
3
1
1
1
1
1
1
E 2 E X 1 E X 2 E X 3
4
2
4
4
2
4
Both estim
14/15 Fall
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT2602 Probability & Statistics II
Assignment 2
Due Date: October 20, 2014
(Hand in your solutions for Questions 1, 3, 4, 7, 9, 11, 15, 17, 18)
1. Based on a random sa
14/15 Fall
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT2602 Probability & Statistics II
Assignment 3 Solution
1. (a) A 95% confidence interval for the mean of all recent selling prices is
X t24,0.025
S
62000
148000 2.06
14/15 Fall
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT2602 Probability & Statistics II
Assignment 3
Due Date: November 12, 2014
(Hand in your solutions for Questions 1, 3, 7, 14, 17, 21, 22, 26, 30)
1.
A real estate age
14/15 Fall
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT2602 Probability & Statistics II
Assignment 1 Solution
1. Moment generating function of Gamm , : M X t
, t
t
The moment generating function of X is given by
n
14/15 Fall
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT2602 Probability & Statistics II
Assignment 1
Due Date: October 3, 2014
(Hand in your solutions for Questions 1, 4, 6, 8, 9, 14, 19, 22, 23)
1. Let X 1 , X 2 ,., X n
PointEstimation
population
unknown
parameters
, 2
estimation
sample
estimators
X
2 S 2
X ,S 2
observed
samplestatistics
StatisticalModel
Astatisticalmodelisasimplificationanddescriptionofthecomplexsystem
basedonmathematicalformulation.
RandomVector
X X
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT2602
PROBABILITY AND STATISTICS II (14-15, 1st)
TUTORIAL 1
Review of Stat2601
Combinatorial Analysis
(a) Selection of r from n distinct objects:
Ordered
Unordered
With replacem
8 Comparing Different
Populations
1
C
Appendix
Generalised likelihood ratio test in Case I of
Section 8.1
We want to find the generalised likelihood
ratio for testing
H0: 1 2 against H1: 1 2 .
Note that
0 cfw_(1, 2): 1 2 ,
cfw_(1, 2): 1 and 2 are real nu
Alternative hypothesis (H1 or Ha) a
hypothesis that will be accepted if the null
hypothesis is rejected
5 Hypothesis Testing: Basic
Concepts
1. Elements of hypothesis testing
2. P-value
Section 5.1 Elements of hypothesis testing
Hypothesis a statement (or
d (number of parameters to estimate when
determining L()
(number of parameters to estimate
when determining L(0).
[Proof] Omitted.
7 Tests Based on Large Samples
1. Approximate distributions of generalised
likelihood ratios
2. Goodness-of-fit tests
3. Te
Department of Statistics and Actuarial Science
The University of Hong Kong
STAT 1302/2602
Probability and Statistics II
Semester 2, 2014/2015
Example Class 9
Summary:
Main theorem:
Let be the generalized likelihood ratio for testing H0 : = 0 Rd vs H1 : o
1 as a lower 100(1 )% confidence
4 Interval Estimation: One
Population
limit (or confidence bound),
2 as an upper 100(1 )%
confidence limit (or confidence
bound),
1, 2 ) as a 100(1 )% confidence
(
interval.
1. Introduction
2. Estimation of means of norma
Note that MX(0) E(e0) 1.
1 Introduction
1.
2.
3.
4.
Moment generating functions
Simple random sampling
Sampling distributions of statistics
Order statistics
Section 1.1 Moment generating functions
Suppose r is a positive integer. The r-th
moment (about th
least on the average namely, that its
expected value should equal the parameter
which it is supposed to estimate.
3 Point Estimation: Concepts
1.
2.
3.
4.
Unbiasedness
Efficiency
Consistency
Sufficiency
Many different estimators may be obtained
for the sa
4 Interval Estimation: One
Population
Appendix
Proof of Theorem 4.2
X
X n
S
S n
where
X
n
and
X
n
,
(n 1) S 2 2
n 1
(n 1) S 2 2
are
X
follows N(0, 1)
n
(by Theorem 1.5) and (n 1) S 2 2 follows
X
2(n 1) (by Theorem 1.7). Therefore
S n
follows t(n 1
8 Comparing Different
Populations
1. Inferences about the difference between
two normal population means: independent
samples
2. Inferences about the difference between
two normal population means: paired
samples
3. Comparing two normal population
varianc
2 Point Estimation: Methods
1. Introduction
2. Method of moments
3. Method of maximum likelihood
Section 2.1 Introduction
Traditionally, problems of statistical inference
are divided into problems of estimation and
tests of hypotheses. The main difference
Since
3 Point Estimation: Concepts
Appendix
Proof of Lemma (Chebyshevs inequality)
[Proof] Suppose X is a continuous random
variable and its density function is f(x). Then
2 E([X ]2)
c
( x ) f( x) dx
2
c2
c
c
( x ) 2 f( x) dx
c
c
( x ) 2 f( x) dx
(
6 Hypothesis Testing: Methods
1.
2.
3.
4.
5.
Most powerful tests
Uniformly most powerful tests
Generalised likelihood ratio tests
One normal population cases
Tests and estimation
Section 6.1 Most powerful tests
When testing the null hypothesis 0
against t
n
1 Introduction
function of
X
2
i
i 1
is
n2
Appendix
Proof of Theorem 1.6
The moment generating function of X12 is
x2
1 2
tX12
tx 2
Ee
e
e dx
2
1
(1 2t ) x 2
exp
dx
2
2
1
(1 2t )
1
x2
exp
dx
1
2 (1 2t ) 1
2 (1 2t )
1
1
for t
(1 2t )
2
becaus
Chapter 8
where Eij
p.25
mj
ni is the expected frequency
N
corresponding to Oij when H0 is true, i 1,
2, , r, j 1, 2, , c. Note that
nm
m
E
Eij i j ij j .
ni
N
n
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT 2602 PROBABILITY AND STATISTICS II (2014-15)
EXAMPLE CLASS 10 SOLUTIONS
1. Test H0 : = 10 vs H1 : unrestricted, where df (H0 ) = 0 and df (H1 ) = 1.
The likelihood function is
Fall 14/15
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT2602 Probability and Statistics II
Class Test Solution
Date: Friday, October 31, 2014
Time: 11:30a.m. 12:20p.m.
1. [30 marks]
X 1 , X 2 ,., X n
Let
be a random samp
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT1302/2602 Probability and Statistics II (1st semester, 2016-2017)
Suggested solutions for Assignment 2
Q1. Let X1 , . . . , Xn be independent random sample from Poisson().
and
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT1302/2602 Probability and Statistics II (1st semester, 2016-2017)
Suggested solutions for Assignment 1
Q1. Let the p.d.f. of X be defined by f (x) = 2( 13 )x , x = 1, 2, 3, .
(
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT1302/2602 Probability and statistics II
Example Class 4 Solution
1
Exercise
1. Let X Poisson(), where is unknown. We would like to estimate T = P (X = 0)2 = e2 .
a. Let T0 be a
Moments and Generating Functions
September 24 and 29, 2009
Some choices of g yield a specific name for the value of Eg(X).
1
Moments, Factorial Moments, and Central Moments
For g(x) = x, we call EX the mean of X and often write X or simply if only the ra
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT1302/2602 Probability and Statistics II
Example Class 1 (2016-17 2nd semester)
Review
1. Discrete random variables:
(1) Univariate:
(a) The probability mass function (pmf): p(x