1
STAT 3460: Assignment 1
Due: Monday January 23, 2012
SOLUTIONS
(18 points total)
1. Suppose that diseased trees are distributed randomly and uniformly throughout a large
forest with an average of per four-acres. The number of diseased trees observed in

MATH/STAT 3460, Intermediate Statistical Theory
Winter 2014
Toby Kenney
Homework Sheet 3
Due: Monday 10th February: 3:30 PM
Basic Questions
1. X1 , X2 , . . . , Xn be distributed as an exponential distribution with paramn
eter plus some unknown constant a

MATH/STAT 3460, Intermediate Statistical Theory
Winter 2014
Toby Kenney
Homework Sheet 5
Due: Friday 21st March: 3:30 PM
Basic Questions
1. A certain region is claimed to be the home of exactly 1000 birds. A team of
researchers captures 77, marks them, an

MATH/STAT 3460, Intermediate Statistical Theory
Winter 2014
Toby Kenney
Homework Sheet 2
Due: Friday 31st January: 3:30 PM
Basic Questions
1. A certain genetic trait is controlled by a single gene with two versions A
and B. Each person has two copies of t

MATH/STAT 3460, Intermediate Statistical Theory
Winter 2014
Toby Kenney
Homework Sheet 1
Due: Friday 24th January: 3:30 PM
Basic Questions
1. The number of car accidents on a given working day is believed to follow a
Poisson distribution with parameter .

MATH/STAT 3460, Intermediate Statistical Theory
Winter 2014
Toby Kenney
Homework Sheet 4
Due: Friday 7th March: 3:30 PM
Basic Questions
1. A die has probability p1 of rolling six, and probability p2 of rolling ve.
We roll the die 100 times, and count the

MATH/STAT 3460, Intermediate Statistical Theory
Winter 2014
Toby Kenney
Homework Sheet 6
Due: Friday 28th March: 3:30 PM
Basic Questions
1. Two investment advisers make predictions about cetain stocks. For each
stock they predict whether its price will in

MATH/STAT 3460, Intermediate Statistical Theory
Winter 2014
Toby Kenney
Homework Sheet 7
Due: Friday 4th April: 3:30 PM
Basic Questions
1. (a) Suppose that X1 , . . . , Xn are independent samples from a distribution
with parameter , which is symmetric abo

STAT 3460 - Lecture 12
Calculating Contours - 10.7
February 13, 2012
Two Parameter Likelihoods
The 100p% Contour: Made up of (, ) pairs that satisfy:
r (, ) log (p) = 0
For computing, it is convenient to transform the (, )
coordinates:
Two Parameter Likel

STAT 3460 - Lecture 8
Two Parameter Likelihoods - 10.2
January 23, 2012
Two Parameter Likelihoods
Relative Likelihood
Joint Relative Likelihood Function (RLF) of and is:
R(, ) =
L(, )
L( , )
Where 0 R(, ) 1 and R( , ) = 1.
The joint RLF ranks pairs of

STAT 3460 - Lecture 11
Dose-Response Example - 10.5
January 23, 2012
Two Parameter Likelihoods
Dose-Response Modeling - 10.5
Dose: Suppose a drug is administered at k dierent doses,
d1 , d2 , ., dk , where each dose corresponds to the
log (Concentration)

STAT 3460 - Lecture 13
Frequency Properties: Sampling Distributions 11.1
February 15, 2012
Frequency Properties
Investigating Estimation Procedure Properties
To evaluate and compare statistical procedures: Examine how
they would behave in a series of hypo

STAT 3460 - Lecture 14
Frequency Properties: Coverage Probability 11.2
February 17, 2012
Frequency Properties
Coverage Probability
Imagine a series of repetitions of an experiment with single
unknown parameter xed at 0 .
Interval [A, B] is computed from t

STAT 3460 - Lecture 16 Continued
Condence Intervals - 11.4
March 5, 2012
Frequency Properties
Example 11.4.3 Here n = 10 independent observations from an
exponential distribution with mean, . We wish to nd an
approximate 95% Condence Interval for both wit

STAT 3460 - Lecture 16
Condence Intervals - 11.4
February 29, 2012
Frequency Properties
Condence Intervals
A random interval [A, B] is called a condence interval for if
its coverage probability is the same for all parameter values 0 .
CP(0 ) = P(A 0 B | =

STAT 3460 - Lecture 9
Maximum Relative Likelihood - 10.3
February 1, 2012
Two Parameter Likelihoods
Maximum Relative Likelihood
The joint Relative Likelihood Function (RLF) of and
ranks pairs of (, ) values according to their plausibilities in
light of t

STAT 3460 - Lecture 15
Chi-Square Approximation - 11.3
February 27, 2012
Frequency Properties
Coverage Probability for Likelihood Intervals
The 100p% LI for is the set of values where r () log (p).
0 belongs to the 100p% LI for if and only if r (0 ) log (

STAT 3460 - Lecture 23
Tests for Independence: Contingency Tables 12.6
March 23, 2012
Tests for Independence
Cross-Classied Data
Study to evaluate three cancer treatments classifed n patients
according to treatment and survival.
Resulting frequencies, fij

STAT 3460 - Lecture 17
Two Parameter Models - 11.5
March 17, 2011
Frequency Properties
Two Parameter Models
Here we consider probability models with two unknown
parameters, and .
As before, r (, ) denotes the joint log RLF of and .
The 100p% likelihood re

STAT 3460 - Lecture 21
Tests for Binomial Probabilities - 12.4
March 21, 2012
Tests of Signicance
Tests for Binomial Probabilities
Consider an experiment where k treatments are compared on the
basis of success/failure data with the results tabulated below

STAT 3460 - Lecture 25
Tests of Signicance: Signicance Regions - 12.9
March 30, 2012
Signicance Regions
Signicance Regions - 12.9
Condence Interval Construction based on Tests of Signicance.
For a test of H : = 0 , the signicance level will depend on
the

STAT 3460 - Lecture 24
Cause and Eect - 12.7
Tests for Marginal Homogeneity - 12.8
March 28, 2012
Cause and Eect
Association versus Cause and Eect
Earlier, when the hypothesis of independence was tested
and a small signicance level observed, we noted obse

STAT 3460 - Lecture 18
Tests of Signicance - 12.1
March 12, 2012
2 () and Exponential Distributions
Show that if X is Exponentially distributed
with mean , then 2X is 2 (2) distributed
Let Y 2X /, where f (x) = (1/) e x/
For variable transform: g (y ) = f

STAT 3460 - Lecture 22
Tests for Multinomial Probabilities - 12.5
March 23, 2012
Tests of Signicance
Tests for Multinomial Probabilities - 12.5
Suppose we have n repetitions of an experiment and that we wish
to assess how well the data agree with an hypot

STAT 3460 - Lecture 20
Likelihood Ratio Tests: Composite Hypotheses 12.3
March 19, 2012
Tests of Signicance
Likelihood Ratio Test: Composite Hypothesis
Hypothesized Model: Consists of both the basic probability
model for the experiment and a hypothesis re

STAT 3460 - Lecture 19
Likelihood Ratio Tests: Simple Hypotheses - 12.2
March 16, 2012
Tests of Signicance
Likelihood Ratio Tests
Likelihood Ratio Statistic: When hypotheses can be
formulated as an assertion of the value of unknown
parameters in a probabi

STAT 3460 - Lecture 10
Chapter 10.3 Cont. & Normal Approx. - 10.4
February 8, 2012
Two Parameter Likelihoods
Example 10.3.2: Consider again Example 10.1.2, where the joint
pdf of n independent measurements, X1 , X2 , ., Xn , from the
Weibull distribution

STAT 3460 - Lecture 7
Two Parameter Likelihoods - 10.1
January 25, 2012
Two Parameter Likelihoods
Maximum Likelihood Estimation
When the probability model for an experiment involves two
unknown parameters, and , the probability of the
data(observed event

STAT 3460: Assignment 6
Due: Friday March 23, 2012
1. Let Xl. X3. Xn be HD random variables having a gan'nna distribution with pdf.
at) = , e % for :7: > 0
Where 6 is a positive unknown parameter.
(a) . how that the likelihood ratio statistic is
E 2r(60)

Due: Wednesday April 4 2012
l
STAT 3460: Assignment 7 SOLUTIONS Q l a
1. Let Y1. ng. Y}; be independent Poisson variates with means. #1. [.03. - - - .nk.
(a) Show that the likelihood ratio statistic for testing H : p1 = #3 = - - - = M is given
bv:
D = 2