HW1
Solutions
Problem 1
2
1
Note that X1 X = 3 X1 3 X2 1 X3 and similarly for other deviations. Thus
3
2/3 1/3 1/3
A = 1/3 2/3 1/3 .
1/3 1/3 2/3
The covariance matrix is ACov(X)A = 2 AA = 2 A. Hence we get B = A.
Problem 2
(a) Note rst
MX (t) = E[etX ] =
THE UNIVERSITY OF ILLINOIS
Department of Statistics
STATISTICS 510
Mathematical Statistics I, Fall 2014
Homework 1: Due on 3pm, Tuesday, Sep. 2, 2014
Put your solution in the drop box STAT510 in the Illini Hall
1. Suppose X is a 3 1 random vector with cov
THE UNIVERSITY OF ILLINOIS
Department of Statistics
STATISTICS 510
Mathematical Statistics I, Fall 2014
Homework 4: Due on 3pm, Monday, Oct. 13, 2014
Put your solution in the drop box STAT510 in the Illini Hall
1. Suppose X1 , , Xn iid N (, 2 ), where bot
THE UNIVERSITY OF ILLINOIS
Department of Statistics
STATISTICS 510
Mathematical Statistics I, Fall 2014
Homework 5: Due on 3pm, Friday, Oct. 31, 2014
Put your solution in the drop box STAT510 in the Illini Hall
1. CB7.38
2. Suppose that X1 , , Xn are iid
THE UNIVERSITY OF ILLINOIS
Department of Statistics
STATISTICS 510
Mathematical Statistics I, Fall 2014
Homework 3: Due on 3pm, Wednesday, Sep. 24, 2014
Put your solution in the drop box STAT510 in the Illini Hall
1. Suppose that X1 , X2 are iid N (, 1),
THE UNIVERSITY OF ILLINOIS
Department of Statistics
STAT 510 Mathematical Statistics I
Homework 2 Solutions
Fall 2014
Problem 1: [6 pts] Let (F ) to be the variance of the distribution F . (a) Write out the expression
of (F ); (2pts) (b) Find the expressi
THE UNIVERSITY OF ILLINOIS
Department of Statistics
STAT 510 Mathematical Statistics I
Homework 3 Solutions
Fall 2014
Problem 1: Suppose that X1 , X2 are iid N (, 1), where is unknown. Let T = X1 + 2X2 be a
statistic. Show that T is not a sucient statisti
THE UNIVERSITY OF ILLINOIS
Department of Statistics
STATISTICS 510
Mathematical Statistics I, Fall 2014
Homework 2: Due on 3pm, Monday, Sep. 15, 2014
Put your solution in the drop box STAT510 in the Illini Hall
1. Let (F ) to be the variance of the distri
THE UNIVERSITY OF ILLINOIS
Department of Statistics
STATISTICS 510
Mathematical Statistics I
Fall 2016 (Homework 1: Due Aug 31th 2016, in class)
1. Suppose X is a 3 1 random vector with cov(X) = 2 I3 . Find the matrix A so that
X1 X
AX = X2 X
,
X3 X
=
THE UNIVERSITY OF ILLINOIS
Department of Statistics
STATISTICS 510
Mathematical Statistics I
Fall 2016
Instructor information:
Xiaofeng Shao, Ph.D.
Professor
104D Illini Hall
Webpage: publish.illinois.edu/xshao
Oce hours: Monday 1:45pm-2.45pm and Thursday
THE UNIVERSITY OF ILLINOIS
Department of Statistics
STAT 510 Mathematical Statistics I
Homework 4 Solutions
Fall 2014
Problem 1: [5 pts] Suppose X1 , , Xn iid N (, 2 ), where both and 2 are unknown
parameters. Let S 2 = (n 1)1 n (Xi X)2 be the sample vari
THE UNIVERSITY OF ILLINOIS
Department of Statistics
STAT 510 Mathematical Statistics I
Homework 5 Solutions
Fall 2014
Problem 1: [6 pts] (CB 7.38)
Solution:
For both part (a) and (b), it can be shown that the distributions belong to the exponential family
Some useful equalities!
Sum of geometric series: 1 + p+ p^2 + . + p^m = (1-p^cfw_m+1)/(1-p), when p<1. !
Gamma(n) = (n-1)!, !
Gamma(t+1) = t Gamma(t)!
!
The mean and variance of a linear combination of independent random variables!
HW2: 7; PP1: 5!
!
S
STAT510 Fall 2014
Mathematical Statistics I
Midterm Solutions
October 8, 2014, Wednesday, 10:00am11:50am
Student ID:
Name:
1. Please print your name and student ID number in the above space and circle the
discussion section number.
2. This is a closed-boo
THE UNIVERSITY OF ILLINOIS
Department of Statistics
STATISTICS 510
Mathematical Statistics I, Fall 2014
Homework 8: Due on 3pm, Monday, Dec. 15, 2014
Put your solution in the drop box STAT510 in the Illini Hall
1. For the random sample X1 , , Xn from N (,
THE UNIVERSITY OF ILLINOIS
Department of Statistics
STAT 510 Mathematical Statistics I
Homework 8 Solutions
Fall 2014
Problem 1: [12 pts] For the random sample X1 , , Xn from N (, 2 ), nd the asymptotic
distribution of mn = n1 n (Xj Xn )3 , where Xn is th
THE UNIVERSITY OF ILLINOIS
Department of Statistics
STAT 510 Mathematical Statistics I
Homework 7 Solutions
Fall 2014
Problem 1: [10 pts] (CB 5.42)
Solution:
(a)
x
)
n
x
= 1 [P (X1 < 1 )]n
n
x n
= 1 (1 )
n
P (n (1 X(n) ) x) = P (X(n) 1
Let = 1/. Then we
THE UNIVERSITY OF ILLINOIS
Department of Statistics
STATISTICS 510
Mathematical Statistics I, Fall 2014
Homework 7: Due on 3pm, Monday, Dec. 1, 2014
Put your solution in the drop box STAT510 in the Illini Hall
1. CB5.42
2. CB5.38(a,b)
3. Consider a sequen
THE UNIVERSITY OF ILLINOIS
Department of Statistics
STATISTICS 510
Mathematical Statistics I, Fall 2014
Homework 6: Due on 3pm, Wednesday, Nov. 12, 2014
Put your solution in the drop box STAT510 in the Illini Hall
1. CB6.42
2. CB6.43
3. CB7.65
4. Assume t
THE UNIVERSITY OF ILLINOIS
Department of Statistics
STAT 510 Mathematical Statistics I
Homework 6 Solutions
Fall 2014
Problem 1: [10 pts] (CB 6.42)
Solution:
x
(a) hg1 (x1 , x2 , ., xn ) = (cx1 + a, cx2 + a, ., cxn + a) W (hg1 (x1 , x2 , ., xn ) = c + a +
Chapter
14
More on Maximum Likelihood Estimation
In the previous chapter we have seen some situations where maximum likelihood
has yielded fairly reasonable estimators. In fact, under certain conditions, MLEs are
consistent, asymptotically normal as n ! ,
Chapter
15
Hypothesis Testing
Estimation addresses the question, What is q? Hypothesis testing addresses questions like, Is q = 0? Confidence intervals do both. It will give a range of plausible
values, and if you wonder whether q = 0 is plausible. you ju
Chapter
9
Asymptotics: Mapping and the D-method
The law of large numbers and central limit theorem are useful on their own, but they
can be combined in order to find convergence for many more interesting situations.
In this chapter we look at mapping and
Chapter
10
Statistical Models and Inference
Most, although not all, of the material so far has been straight probability calculations,
that is, we are given a probability distribution, and try to figure out the implications
(what X is likely to be, margin
Chapter
4
Transformations: DFs and MGFs
A major task of mathematical statistics is finding, or approximating, the distributions of random variables that are functions of other random variables. For example,
given X1 , . . . , Xn iid observations, we would
NA M E _
STAT 510 Exam #1 Answers
Friday, October 1, 2004
Closed book & Notes. Calculators are ok.
30 points 1. Suppose (X,Y) is uniform over the unit disk, so that the space is cfw_Cay l 2:2 +112 < 1
and the pdf is f(x,y) : 1/77. Also, let R : x/X2 + Y2.
STAT 510 Practice Problems
The exam is Friday, March 18, 2016, in the usual room at the usual time
The exam is closed book & notes. It will have four or five problems, although
each will have several parts. It covers everything up through Chapter 7 and Ho
NAME _
STAT 510 Exam #1: Answers
Friday, October 13. 2006
Closed book 85 Notes.
30 points 1. Suppose (X1,X2) has space
X :cfw_($1,3:2) $1 > O,$2 > 0 and $1+zg <1
and pdf f(x1,x2) = 2.
(a) Sketch the space X.
0.8
X2
0.4
0.0
0.0 0.4 0.8
X1
(b) Let Y1 = x/X1
Midterm Solutions
1. Suppose X = (X1 , X2 ) has space
cfw_(x1 , x2 ) | x1 > 0, x2 > 0, x1 + x2 < 1
and pdf fX (x1 , x2 ) = 2.
(a) (5 points) Sketch the space of X.
The space of X is the triangle in the xy-plane with vertices (0, 0), (0, 1) and (1, 0).
(b)
Chapter
12
Linear Regression
12.1
Regression
How is height related to weight? How are sex and age related to heart disease? What
factors influence crime rate? Questions such as these have one dependent variable
of interest, and one or more explanatory or
Chapter
13
Likelihood, Suciency, and MLEs
13.1
The likelihood function
If we know q, then density tells us what X is likely to be. In statistics, we do not
know q, but we do observe X = x, and wish to know what values q is likely to
be. The analog to the
Chapter
8
Asymptotics: Convergence in Probability and
Distribution
So far we have been concerned with finding the exact distribution of random variables
and functions of random variables. Especially in estimation or hypothesis testing,
functions of data c
Chapter
1
Distributions and Densities
1.1
Probability
This section quickly reviews the basic definition of a probability distribution. Starting
with the very general, suppose X is a random object. It could be a single variable, a
vector, a matrix, or some
Chapter
16
Model selection: AIC and BIC
We often have a number of models we wish to consider, rather than just two as in
hypothesis testing. (Note also that hypothesis testing may not be appropriate even
when choosing between two models, e.g., when there