Stat 553: Asymptotic Tools
Midterm
Fall 2011
SOLUTIONS
Problem 1. For n = 1, 2, . . ., suppose that Xn is distributed as a Bernoulli(pn ) random variable. That is,
P (Xn = 1) = 1 P (Xn = 0) = pn .
P
n
(a) [3 points] Suppose that pn = 1/n. Prove that X n 0
Stat 553: Asymptotic Tools
Final Exam
Fall 2011
SOLUTIONS
This closed-book nal is worth 20 points. You have 110 minutes. You are allowed to use three
sheets (double-sided) of your own notes. Write your answers on separate pages, and be sure that
your name
Stat 553: Asymptotic Tools
Midterm
Fall 2011
SOLUTIONS
Problem 1. For n = 1, 2, . . ., suppose that Xn is distributed as a Bernoulli(pn ) random variable. That is,
P (Xn = 1) = 1 P (Xn = 0) = pn .
P
n
(a) [3 points] Suppose that pn = 1/n. Prove that X n 0
Stat 597D: Asymptotic Tools
Midterm 2
Fall 2000
November 2, 2000
Name:
This closed-book midterm is worth 15 points. You have 1 hour. Note that not all questions are weighted
equally; allot your time accordingly.
Problem 1. Let X1 , X2 , . . . be iid Poiss
Stat 597A: Asymptotic Tools
Midterm 2
Fall 2002
November 20, 2002
Name:
This closed-book midterm is worth 15 points. You have 75 minutes. Note that not all questions are weighted
equally; allot your time accordingly.
Problem 1. [2 points] Suppose X (n) is
Stat 597D: Asymptotic Tools
Midterm 2
Fall 2000
Solutions
Name:
This closed-book midterm is worth 15 points. You have 1 hour. Note that not all questions are weighted
equally; allot your time accordingly.
Problem 1. Let X1 , X2 , . . . be iid Poisson() ra
Stat 553: Asymptotic Tools
Midterm 2
Fall 2011
SOLUTIONS
This midterm is worth 15 points. You have 60 minutes.
Problem 1. Let X1 , X2 , . . . be independent and identically distributed from a beta(1, ) distribution for
some > 0.
The beta(1, ) distribution
Stat 553: Asymptotic Tools
Midterm 1 sample questions
Problem 1. For 0 < p < 1 and q = 1 p, let X1 , X2 , . . . be iid random variables such that
P (Xk = 0) = P (Xk = 2p)
P (Xk = p)
= p/2,
= q.
(a) Dene
n
=
1
n
n
2
Xk Xk 3(X n )3 .
k=1
Show that n is a co
Stat 553: Asymptotic Tools
Midterm 2
Fall 2011
Sample problems
Problem 1. Let X1 , X2 , . . . be iid Poisson() random variables. For each i, let Yi = I cfw_Xi = 0. Find the
asymptotic distribution of 2X n + log Y n . Show all steps, including nding the jo
Assignment 1
Exercise 1.1 Assume that an a and bn b, where a and b are real numbers.
(a) Prove that an bn ab
Hint: Show that |an bn ab| |(an a)(bn b)| + |a(bn b)| + |b(an a)| using
the triangle inequality.
(b) Prove that if b = 0, an /bn a/b.
Sketch of so
Assignment 2
Exercise 1.20 According to the result of Exercise 1.16, the limit (1.21) implies that
the relative dierence between n=1 (1/i) and log n goes to zero. But this does
i
not imply that the dierence itself goes to zero (in general, the dierence ma
Assignment 3
Exercise 1.43 The complex plane C consists of all points x + iy , where x and y are
real numbers and i = 1. The elegant result known as Eulers formula relates
the points on the unit circle to the complex exponential function:
expcfw_it = cos
Assignment 4
Exercise 2.10 The goal of this Exercise is to construct an example of an independent
P
sequence X1 , X2 , . . . with E Xi = such that X n but Var X n does not
converge to 0. There are numerous ways we could proceed, but let us suppose
that fo
Assignment 5
a.s.
Exercise 3.2 The diagram at the end of this section suggests that neither Xn X
qm
nor Xn X implies the other. Construct two counterexamples, one to show that
qm
qm
a.s.
Xn X does not imply Xn X and the other to show that Xn X does not
a.
Assignment 6
Exercise 1.41 Kolmogorovs inequality is a strengthening of Chebyshevs inequality
for a sum of independent random variables: If X1 , . . . , Xn are independent random
variables, dene
k
(Xi E Xi )
Sk =
i=1
to be the centered k th partial sum fo
Assignment 7
Exercise 4.3 Use the Continuity Theorem to prove the Cramr-Wold Theorem, Thee
orem 4.12.
d
Hint: a Xn a X implies that a
Xn (1)
a
X (1).
Sketch of solution: As we pointed out in class, the only tricky part of the
d
d
Cramr-Wold Theorem is sh
Assignment 8
Exercise 4.7 Use the Cramr-Wold Theorem along with the univariate Central Limit
e
Theorem (from Example 2.12) to prove Theorem 4.9.
Sketch of solution: This proof is actually given in the course notes, just
before Exercise 4.8: Let X Nk (0, )
Assignment 9
Exercise 5.5 Let Xn binomial(n, p), where p (0, 1) is unknown. Obtain condence intervals for p in two dierent ways:
d
(a) Since n(Xn /n p) N [0, p(1 p)], the variance of the limiting distribution
P
depends only on p. Use the fact that Xn /n p
Stat 553: Asymptotic Tools
Final Exam WITH SOLUTIONS
Fall 2004
December 13, 2004
This closed-book nal is worth 20 points. You have 120 minutes. You are allowed to use two sheets
(double-sided) of your own notes. Write your answers on separate pages, and b
Stat 553: Asymptotic Tools
Final Exam
Fall 2004
December 13, 2004
This closed-book nal is worth 20 points. You have 120 minutes. You are allowed to use two sheets
(double-sided) of your own notes. Write your answers on separate pages, and be sure that you
Stat 597A: Asymptotic Tools
Final Exam WITH SOLUTIONS
Fall 2003
December 16, 2003
Problem 1. [3 points] Let X1 , . . . , Xn be an iid sequence with E Xi = and Var Xi = 2 < . Prove
d
the central limit theorem: n(X n ) N (0, 2 ).
You may use the fact that t
Stat 553: Asymptotic Tools
Midterm 2
Fall 2011
SOLUTIONS
This midterm is worth 15 points. You have 60 minutes.
Problem 1. Let X1 , X2 , . . . be independent and identically distributed from a beta(1, ) distribution for
some > 0.
The beta(1, ) distribution
Assignment 1
Exercise 1.1 Assume that an a and bn b, where a and b are real numbers.
(a) Prove that an bn ab
Hint: Show that |an bn ab| |(an a)(bn b)| + |a(bn b)| + |b(an a)| using
the triangle inequality.
(b) Prove that if b = 0, an /bn a/b.
Sketch of so
Assignment 2
Exercise 1.20 According to the result of Exercise 1.16, the limit (1.21) implies that
the relative dierence between n=1 (1/i) and log n goes to zero. But this does
i
not imply that the dierence itself goes to zero (in general, the dierence ma
Assignment 3
Exercise 1.43 The complex plane C consists of all points x + iy , where x and y are
real numbers and i = 1. The elegant result known as Eulers formula relates
the points on the unit circle to the complex exponential function:
expcfw_it = cos
Assignment 4
Exercise 2.10 The goal of this Exercise is to construct an example of an independent
P
sequence X1 , X2 , . . . with E Xi = such that X n but Var X n does not
converge to 0. There are numerous ways we could proceed, but let us suppose
that fo
Assignment 5
a.s.
Exercise 3.2 The diagram at the end of this section suggests that neither Xn X
qm
nor Xn X implies the other. Construct two counterexamples, one to show that
qm
qm
a.s.
Xn X does not imply Xn X and the other to show that Xn X does not
a.
Assignment 6
Exercise 1.41 Kolmogorovs inequality is a strengthening of Chebyshevs inequality
for a sum of independent random variables: If X1 , . . . , Xn are independent random
variables, dene
k
(Xi E Xi )
Sk =
i=1
to be the centered k th partial sum fo
Assignment 7
Exercise 4.3 Use the Continuity Theorem to prove the Cramr-Wold Theorem, Thee
orem 4.12.
d
Hint: a Xn a X implies that a
Xn (1)
a
X (1).
Sketch of solution: As we pointed out in class, the only tricky part of the
d
d
Cramr-Wold Theorem is sh
Assignment 8
Exercise 4.7 Use the Cramr-Wold Theorem along with the univariate Central Limit
e
Theorem (from Example 2.12) to prove Theorem 4.9.
Sketch of solution: This proof is actually given in the course notes, just
before Exercise 4.8: Let X Nk (0, )