Final Exam April 21, 2006
Problem 1. Suppose that Y1 , . . . , Yn are independent and identically distributed random
variables with each Yi having density function
y2
exp cfw_y/ , y > 0,
2 3
where > 0 is a parameter. It is known that E(Yi ) = 3 and Var(Yi
Statistics 252Mathematical Statistics
Winter 2007 (200710)
Final Exam Solutions
Instructor: Michael Kozdron
1. (a) By denition, the likelihood function L() is given by
n
n
fY (yi |) =
L() =
i=1
n
2
2
expcfw_2 yi
2 y
n 2n
n
yi exp
=2
i=1
i=1
2
2
yi
.
i=
Statistics 252 Practice Midterm (Solutions) Winter 2007
1.
Let U = Y / so that for u > 0,
u
2y
y2
exp 2
2
P (U u) = P (Y u) =
0
2
dy = 1 eu .
Thus, we must nd a and b so that
a
2
2ueu du =
0
and
2
2
2ueu du =
b
log(1 /2) and b =
Computing the integrals w
Statistics 252 Midterm #1 January 30, 2006
This exam has 5 problems and 6 numbered pages.
You have 50 minutes to complete this exam. Please read all instructions carefully, and check your
answers. Show all work neatly and in order, and clearly indicate yo
Final Exam Solutions
1. (a) To nd the method of moments estimator, we equate the rst population moment
with the rst sample moment. Since E(Yi ) = 3, we conclude MOM = Y .
3
1. (b) By denition, the likelihood function L() is given by
n
n
fY (yi |) =
L() =
Using SPSS
Note: The use of another statistical package
such as Minitab is similar to using SPSS
After starting the SSPS program the following dialogue
box appears:
If you select Opening an existing file and press OK the
following dialogue box appears
The
Measures of Variability
Variability
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
0
5
10
15
20
25
Measure of Variability
(Dispersion, Spread)
Variance, standard deviation
Range
Inter-Quartile Range
Pseudo-standard deviation
Range
Range
Definition
Let min = the smal
Statistical Inference
Making decisions regarding the
population base on a sample
Decision Types
Estimation
Deciding on the value of an unknown parameter
Hypothesis Testing
Deciding a statement regarding an unknown parameter
is true of false
Predictio
CMPT 280
Intermediate Data Structures and Algorithms
Introduction
Mark G. Eramian
University of Saskatchewan
Mark G. Eramian
CMPT 280
1/17
Course Website
CMPT 280 course on Moodle:
http:/moodle.cs.usask.ca
Features:
Download lecture notes, assignments,
CMPT 280
Topic 1: Review of Lists
Mark G. Eramian
University of Saskatchewan
Mark G. Eramian
CMPT 280
1/12
References
Textbook, Chapter 1
Mark G. Eramian
CMPT 280
2/12
Array-based List Implementations
Implementation 1: Ordered first-to-last at beginning
Statistics 252 Midterm #2 March 16, 2007
This exam has 5 problems and is worth 40 points.
Instructor: Michael Kozdron
You must answer all of the questions in the exam booklet provided.
You have 50 minutes to complete this exam. Please read all instruction
Statistics 252 Winter 2007 Midterm #2 Solutions
1. If Y Uniform(, 2), then fY (y |) = 1 , y 2. Let U = Y / so that if 1 u 2, then
u
P (U u) = P (Y u) =
1
dy = u 1,
2
a
and so fU (u) = 1, 1 u 2. We must now nd a and b such that 1 du = and b du =
2
Solving
Statistics 252 Practice Midterm Winter 2007
1.
(8 points ) Consider a random variable Y with density function
fY (y ) =
y2
2y
exp 2
2
,
y > 0,
where > 0 is a parameter. Use the pivotal method to verify that if 0 < < 1, then
Y
log(/2)
,
Y
log(1 /2)
is a
Statistics 252 Winter 2006 Midterm #1 Solutions
1. (a) Since Y1 , Y2 , Y3 are independent and identically distributed we nd
E (Y ) = E
Y1 + Y2 + Y3
3
E (Y1 ) + E (Y2 ) + E (Y3 )
3E (Y1 )
=
= E (Y1 ).
3
3
=
We also compute that
y fY (y ) dy =
E (Y1 ) =
0
2
Statistics 252 Winter 2007 Midterm #1 Solutions
1. (a) Since B () = E () , we must rst compute E (). To determine E (), we need to nd
the density function of , which requires us rst to nd the distribution function of . Since
Y1 , . . . , Yn are i.i.d., we
Make sure that this examination has 11 numbered pages
University of Regina
Department of Mathematics & Statistics
Final Examination
200710
(April 20, 2007)
Statistics 252-001
Mathematical Statistics
Name:
Student Number:
Instructor: Michael Kozdron
Time:
Statistics 252 Midterm #2 March 3, 2006
1.
(10 points )
(a) Suppose that a random variable Y has density function fY (y |) = ( + 1)y , 0 y 1.
Determine the Fisher information I () for this random variable.
(b) Let Y1 , . . . , Yn be independent and identi
Statistics 252 Midterm #1 February 5, 2007
This exam has 5 problems and is worth 40 points.
Instructor: Michael Kozdron
You must answer all of the questions in the exam booklet provided.
You have 50 minutes to complete this exam. Please read all instructi