Solutions Homework 2
September 9, 2009
Solution to Exercise 2.1:
x f 1 (Ac ) f (x) Ac
f (x) A.
/
Now if x f 1 (A), then of course f (x) A, which cant happen if x f 1 (Ac ), i.e.,
by contraposition x f 1 (Ac ) implies x f 1 (A)c , which means f 1 (Ac ) f
STAT 582 Homework 7
Due date: In class on Wednesday, April 13, 2005
Instructor: Dr. Rudolf Riedi
2
21. Let Xn be a sequence of Gaussian random variables with means n and variances n . Assume that
D
2
Xn X. Show that necessarily n for some IR and n 2 for s
STAT 582 Homework 6
Due date: In class on Friday, March 25, 2005
Instructor: Dr. Rudolf Riedi
17. Let Xn be a sequence of random variables such that
P [Xn = n] = 1/n
P [Xn = 0] = 1 1/n.
(a) Does the sequence cfw_Xn n converge in probability? If so to what
STAT 582 Homework 5
Due date: In class on Friday, March 18, 2005
Instructor: Dr. Rudolf Riedi
14. (a) Let Xn be independent, Gaussian r.v. with
IE[Xn ] = 0
Show that
n Xn
converges a.s. i
2
n n
2
var(Xn ) = n .
< .
Hint: Use the Three Series Theorem only
STAT 582 Homework 4
Due date: In class on Friday, March 4, 2005
Instructor: Dr. Rudolf Riedi
11. Here, we establish an extension to the SLLN.
+
Assume that cfw_Xn are iid with IE[X1 ] < and IE[X1 ] = .
(a) Using Kolmogorovs SLLN show that for any c > 0 w
STAT 582 Homework 3
Due date: In class on Friday, February 25, 2005
Instructor: Dr. Rudolf Riedi
9. Recall that
1
n n
= , but
n
(1)n
n
converges. Let Xn be iid with
P [Xn = 1] =
Does
Xn
n n
1
2
converge in probability? Does it converge almost surely? In L
STAT 582 Homework 1
Due date: In class on Friday, February 4, 2005
Instructor: Dr. Rudolf Riedi
P
a.s.
1. Let Xn be a monotone sequence of random variables. Assume that Xn X. Show that Xn X.
Hint: Consider subsequences.
2. Let Xn be any sequence of random
STAT 582 Homework 2
Due date: In class on Friday, February 11, 2005
Instructor: Dr. Rudolf Riedi
2
5. Let Xn have normal distribution with mean 0 and variance n . When is the family cfw_Xn n u.i.?
6. Suppose cfw_Xn n and cfw_Yn n are two u.i. families den
Solutions Homework 8
December 10, 2011
Solution to Exercise 2.4.4
(a) This is trivial by properties of the determinant: the determinant of
a product is the product of the determinant, and taking transpose doesnt
change the determinant. Thus, if V = AAt ,
Solutions Homework 9
December 10, 2011
Solution to Exercise 3.2.1: Suppose that > 0, P [ X n X >
] 0. Then, given > 0, we can nd an N such that for all n N,
P [ X n X > ] < . Simply take = to get the desired conclusion.
Conversely, suppose > 0, N such tha
1
Solution to Exercise 2.1.5 If x 1 and p < q, then xp xq . This
follows since the exponential function is monotone increasing, so xp = exp[p log x]
exp[q log x] = xq since p log x q log x when p < q and log x 0, i.e. x 1.
Thus
E[ X p ]
=
E[ X p I[0,1] (
Solutions Homework 6
November 17, 2011
Solution to Exercise 5.1: Lets try guring out E[X|Y = y] using intuition
about conditional distributions. If we observe Y = 1/2, then we know 0 1/2
and nothing more. Using elementary conditional probability:
P [X = 1
Solutions Homework 5
October 4, 2010
Solution to Exercise 3.17: Since the measure of a set is the integral of its
indicator, we have
(1 2 )(A)
=
1 2
=
1
IA (1 , 2 ) d(1 2 )(1 , 2 )
2
IA (1 , 2 ) d2(2 ) d1(1 ) ,
where of course Fubinis theorem was applied
Solutions Homework 1
September 7, 2009
Solution to Exercise 1.1: Lets see if we can use logical equivalence (if and
only if, symbolized by , and abbreviated i) to cut the steps in half.
c
x
A
x
/
AA
A
AA
it is not true that A A, x A
A A, x A
/
x
Ac .
AA
I
Solutions Homework 4
September 29, 2010
Solution to Supplement Exercise SupEx 2: Assume 0 fn f .
Then f = limn fn = lim inf n fn . Thus, by Fatous Lemma, we have f lim inf n fn .
Now fn is an increasing sequence of extended real numbers by the monotonicit
Solutions Homework 2
September 20, 2010
Solution to Exercise 2.12 The denition given is that f is integrable i
both f are nite. If f is integrable (statement (i), then f = f+ f , and since
both terms in the latter expression are nite, we have that f is de
STAT 582 Homework 8
Due date: In class on Monday, April 25, 2005
Instructor: Dr. Rudolf Riedi
24. Let U be a discrete random variable with values cfw_ui iI . Let Ai := cfw_ : U () = ui .
Let Y be in L1 . Recall that
IE[Y |U ] = IE[Y | (Ai , i I)] =
ai IIA