MATH-440: Bayesian Statistics
Mid-term Exam Sample Solutions
1. [14 points ]
The number of eggs per nest, ni , and the number of eggs hatched, xi , were observed for four
pairs of a rare specimen of birds that nested last season. The observed data were n1
MATH503:MathematicalStatistics
Readings: Casella & Berger Chapter 5
Lecture 1:
An Introduction to Statistical Inference
Properties of a Random sample
Outline
Course introduction, and logistics
Review of MATH-501 topics
Introduction to Statistical Infer
MATH-503
Homework Assignment 3 Solutions
Spring 2012
Instructor: Dr. Arab
1.
n
n
f ( xi | ) = n xi
i =1
i =1
1
= n xi
i =1
n
1
n
xi
i =1
,
n
1
g (T | ) = x i , h( x) = n
i =1
x i
i =1
Thus, T = xi is a sufficient statistic for by
MATH-503
Homework Assignment 4 Solutions
Spring 2012
Instructor: Dr. Arab
1.
This is a location-invariant family distributionand thus there are several choices for the
ancillary statistics including: R=X(n)-X(1).
2.
a)
b)
c)
3.
4.
d. yes, the MLE and the
MATH-503
Homework Assignment 5 Solutions
1.
a)
f ( x; ) = x 1
L( ; x) = n (xi ) 1
n
log L( ; x) = n log + ( 1) log xi
i =1
nn
log L( ; x) = + log xi = 0 (solve for ).
i =1
^
n
MLE is: = n
.
log xi
i =1
b.
f ( x / ) = x 1 ,0 < x < 1,0 < <
x ~ beta ( ,1
MATH-503
Homework Assignment 6 Solutions
1.
a) population mean not sample mean.
b) Null should be that there is no change.
c) A small P-value is needed for significance.
2.
a) P=0.9852
b) P=0.0418
c) P=0.0836
3.
a) decision rule: Reject the null if z1.645
MATH-503 Midterm Exam 1 Guide
Section 2:
Tuesday, February 28, 6:30pm-8:30pm (Dont be late!)
Section 1:
Wednesday, February 29, 6:30pm-8:30pm (Dont be late!)
Important: You MUST take the exam in the section you are enrolled in.
Exam 1 covers: Chapters 5,
MATH-503 Final Exam Guide
Sections 1 & 2:
Tuesday, May 8, 7pm-9pm, White-Gravenor 311
Dont be late!
Exam covers:
Chapters 5-11 from the course textbook (Casella & Berger) or comparable
topics (Sections that were skipped, not included in the exam!),
Offic
MATH-503
Homework Assignment 9
Spring 2012
Instructor: Dr. Arab
Due: Thursday April 26, 2012 (by or before 6pm)
Show all work to get full credit. You may turn in your assignment by or before the
due date, in class or drop it in the mailbox labeled Yun-Ju
Midterm Exam 2
MATH 503-Section 2
‘ Date: Tues., April 10, 2012
Name: M Time: 6:30pm—8:30pm
Instructor: Prof. Arab
Read all of the following information before starting the exam:
0 Show all work, clearly and in order, to get partial credit.
0 You may use
MATH-503: Mathematical Statistics
Readings: Casella & Berger Chapter 8
Hypothesis Testing
Hypothesis Testing
Setup: r.v. X with pdf/pmf f(x;), unknown.
Goal: to resolve a test of the form
H0 : 0 vs. H1 : 1
where 0,1 subsets of ; 0U1=
A hypothesis is a sta
MATH-503
Review
Prerequisite Concept Review
Readings: Casella & Berger, Chapters 1-5
Department of Mathematics and Statistics
Georgetown University
Spring 2012
Outline
Review of MATH-501 topics:
Probability, and Conditional Probability
Random variables
MATH-503: Mathematical Statistics
Readings: Casella & Berger Chapter 10
Asymptotic Evaluations
Consistency
The idea of Consistency is that the estimator converges to
the correct value as the sample size becomes infinite.
Let Wn=Wn(X1, Xn) be a sequence
MATH-503
Homework Assignment 2 Solutions
1.
iid
X i ~ Poisson( )
By CLT:
n ( X ) d N (0, )
Let g ( ) = e , g ' ( ) = e , [g ' ( )]2 = e 2 ,
By Delta Method,
n (e X e ) d N (0, e 2 )
or
e X d N (e ,
e 2
n
)
2.
X ~ Binomial (n, )
p = x/n
By CLT,
n ( p ) d
MATH-503
Homework Assignment 1 Solutions
Spring 2012
Instructor: Dr. Arab
1.
a)
n
n
i =1
i =1
M Y (t ) = E (eYt ) = E (e i ) = E (e txi ) = M xi (t ) = M xni (t )
t
x
, because xi 's i.i.d.
where
M x1 (t ) = pe t + (1 p)
n
M Y (t ) = M x1 (t ) = ( pe t +
MATH503:MathematicalStatistics
Readings: Casella & Berger Chapter 7
Methods of Point Estimation
Outline
Two common approaches to estimating
Method of moments (MOM)
Maximum likelihood estimation (MLE)
MethodofMoments(MoM)
Perhaps the oldest method of p
MATH503:MathematicalStatistics
Readings: Casella & Berger Chapter 8
Hypothesis Testing
HypothesisTesting
Setup: r.v. X with pdf/pmf f(x;), unknown.
Goal: to resolve a test of the form
H0 : 0 vs. H1 : 1
where 0,1 subsets of ; 0U1=
A hypothesis is a stateme
MATH503:MathematicalStatistics
Readings: Casella & Berger Chapter 9
Interval Estimation
ConfidenceIntervals
Point estimates do not provide identification of an estimators
precision. We use confidence intervals to quantify the amount
of uncertainty in an
MATH503
Review
Prerequisite Concept Review
Readings: Casella & Berger, Chapters 1-5
Department of Mathematics and Statistics
Georgetown University
Spring 2012
Outline
Review of MATH-501 topics:
Probability, and Conditional Probability
Random variables
Course Information and Syllabus
MATH 503: MATHEMATICAL STATISTICS (Sections 1 & 2)
Spring 2012
Time and Location:
Section 1: St. Marys 110; Wednesdays 6:30pm-9:00pm
Section 2: WGR 311; Tuesdays 6:30pm-9:00pm
Instructor:
Office:
Voice:
Email:
Dr. Ali Arab
MATH-503
Homework Assignment 1
Spring 2012
Instructor: Dr. Arab
Due: Thursday January 26, 2012 (by or before 6pm)
Show all work to get full credit. You may turn in your assignment by or before the
due date, in class or drop it in the mailbox labeled Yun-J
MATH-503
Homework Assignment 2
Spring 2012
Instructor: Dr. Arab
Due: Thursday February 2, 2012 (by or before 6pm)
Show all work to get full credit. You may turn in your assignment by or before the
due date, in class or drop it in the mailbox labeled Yun-J
MATH-503
Homework Assignment 3
Spring 2012
Instructor: Dr. Arab
Due: Thursday February 9, 2012 (by or before 6pm)
Show all work to get full credit. You may turn in your assignment by or before the
due date, in class or drop it in the mailbox labeled Yun-J
MATH-503
Homework Assignment 4
Spring 2012
Instructor: Dr. Arab
Due: Thursday February 16, 2012 (by or before 6pm)
Show all work to get full credit. You may turn in your assignment by or before the
due date, in class or drop it in the mailbox labeled Yun-
MATH-503
Homework Assignment 5
Spring 2012
Instructor: Dr. Arab
Due: Monday February 27, 2012 (by or before 6pm)
Show all work to get full credit. You may turn in your assignment by or before the
due date, in class or drop it in the mailbox labeled Yun-Ju
MATH-503
Homework Assignment 6
Spring 2012
Instructor: Dr. Arab
Due: Thursday March 22, 2012 (by or before 6pm)
Show all work to get full credit. You may turn in your assignment by or before the
due date, in class or drop it in the mailbox labeled Yun-Ju
MATH-503
Homework Assignment 7
Spring 2012
Instructor: Dr. Arab
Due: Thursday March 29, 2012 (by or before 6pm)
Show all work to get full credit. You may turn in your assignment by or before the
due date, in class or drop it in the mailbox labeled Yun-Ju