HOMEWORK 4 SOLUTIONS
Problem 1 First observe that (1) n1 ESn = n1
1kn
EXn,k =
1kn
n 1 u
k n
which in particular is the Riemann sum of u for the partition
1
k : 1 k n . Since u is n integrable, we know that that the Riemann sum converges to the integral of
205B homework, week 4; due Thursday February 19
Durrett Chapter 5 Exercises 5.7, 5.8, 5.11 1. Let (Xn ) be an irreducible Markov chain on S with transition matrix (p(x, y ). Let B be a nite subset of S such that the chain a.s. visits B innitely often. Let
831 Theory of Probability Fall 2009 Homework 4
Due Tuesday, October 13
1. Monte Carlo integration. Assume given a continuous function u on [0, 1] such that 0 u(x) 1. We use u to create an array of cfw_0, 1-valued random variables as follows: let cfw_Xn,k
831 Theory of Probability Fall 2009 Homework 3
Due Thursday, October 1
1. Let > 0. Show that, for a > 0, lim en (n)k k! k : 0kna = 0, if a < 1, if a > .
n
Hint: apply the WLLN to Poisson random variables. 2. For each n N let cfw_Xn,k : 1 k n be IID random
205B homework, week 5; due Thursday February 26 These are miscellaneous questions on Markov chains, not necessarily closely connected to this weeks class material.
1. Let Xn be the Markov chain on states 0, 1, . . . , K with transition matrix p(i, i + 1)
831 Theory of Probability Fall 2009 Homework 2
Due Tuesday, September 22
1. Let cfw_Xn,i : n N, i I be real-valued random variables on (, F, P ). Assume that for each n N, the variables cfw_Xn,i : i I are independent. Assume that for each i I there is a r
205B homework, week 3; due Thursday February 12 Durrett Chapter 5 Exercises 4.3, 4.4, 4.8, 4.10 1. Give an example to show that, if Xn is a Markov chain, then f (Xn ) need not be a Markov chain. 2. Let A and B be disjoint subsets of a nite state space S .
Lecture 1 : Introduction
We will start with a simple combinatorial problem. Consider cfw_1, 11000 . How many elements x cfw_1, 11000 satisfy
1000
xi 50?
i=1
More generally, for any n N and > 0 how many elements x cfw_1, 1n satisfy
n
xi n?
i=1
The answer i
Lecture 2 : Ideas from measure theory
2.1
Probability spaces
This lecture introduces some ideas from measure theory which are the foundation of the modern theory of probability. The notion of a probability space is dened, and Dynkins form of the monotone
Lecture 3 : Random variables and their distributions
3.1
Random variables
Let (, F ) and (S, S ) be two measurable spaces. A map X : S is measurable or a random variable (denoted r.v.) if X 1 (A) cfw_ : X ( ) A F for all A S One can write cfw_X A or (X A)
Lecture 4: Expected Value
4-1
Lecture 4 : Expected Value
References: Durrett [Section 1.3]
4.5
Expected Value
Denote by (, F , P) a probability space. Denition 4.5.1 Let X : R be a F\B -measurable random variable. The expected value of X is dened by E(X )
Lecture 5: Inequalities
5-1
Lecture 5 : Inequalities
5.7
Inequalities
Let X, Y etc. be real r.v.s dened on (, F , P). Theorem 5.7.1 (Jensens Inequality) Let be convex, E(|X |) < , E(|(X )|) < . Then (E(X ) E(X ) (5.11) Proof Sketch: As is convex, is the s
Lecture 6 : Distributions
Theorem 6.0.1 (Hlders Inequality) If p, q [1, ] with 1/p + 1/q = 1 then o E(|XY |) |X |p |Y |q (6.1)
Here |X |r = (E(|X |r )1/r for x [1, ); and |X | = inf cfw_M : P(|X | > M ) = 0. Proof: See the proof of (5.2) in the Appendix o
Lecture 7 : Product Spaces
7.3
Product spaces and Fubinis Theorem
i
Denition 7.3.1 If (i , Fi ) are measurable spaces, i I (index set), form For simplicity, i = 1 .
i
i .
i (write for this) is the space of all maps: I 1 . For i i , = (i : i I, i i ). is e
Lecture 8: Weak Law of Large Numbers
8-1
Lecture 8 : Weak Law of Large Numbers
References: Durrett [Sections 1.4, 1.5] The Weak Law of Large Numbers is a statement about sums of independent random variables. Before we state the WLLN, it is necessary to de
205B homework, week 8; due Thursday March 19 1. Prove the following slight extension of Azumas inequality. Let (Mn ) be a martingales such that |Mn Mn1 | Kn for constants Kn . Then for x > 0
n 1 P (|Mn M0 | x) 2 exp 2 x2 / i=1
Ki2 .
2. Suppose you have n
Mathematics 241, Problem Set #8, due October 21, 2009
Write clear proofs that have good grammar and consist of complete sentences. Royden, page 122, #7
Royden, page 126, # 9, 12, 14, 15, 16, 17
Royden, page 135, #22
Mathematics 241, Problem Set #7, due October 12, 2009
Write clear proofs that have good grammar and consist of complete sentences. Royden, page 101, #1,4.
Royden, page 104, # 7, 10.
Royden, page 110, #12, 14, 15
Mathematics 241, Problem Set #6, due October 7, 2009
Write clear proofs that have good grammar and consist of complete sentences. Royden, page 40, #22 Royden, page 46, #37 Royden, page 50, #48 Royden, page 64, #14
Hint: follow the Beale notes given out in
Mathematics 241, Problem Set #5, due September 28, 2009 (dont turn in)
Write clear proofs that have good grammar and consist of complete sentences. Royden, page 96, #20,21,24 Royden, page 94, #16, 17
Mathematics 241, Problem Set #3, due September 14, 2009
Write clear proofs that have good grammar and consist of complete sentences.
Royden, page 70, #19,21,23,24.
Royden, page 73, #29,30.
HOMEWORK 2 SOLUTIONS
Problem 1 Fix i1 , i2 , ., ik I. We shall show that Yi1 , .Yik are independent by showing that
k
Pcfw_Yi1 ti , ., Yik tk =
l=1
P(Yil tl ),
t1 , t2 , .tk R.
Step1: For any bounded continuous functions f1 , f2 , ., fk , we have
k k
E
l
Mathematics 241, Problem Set #2, due September 7, 2009
Write clear proofs that have good grammar and consist of complete sentences.
Royden, page 55, #4.
Royden, page 58, #6,7,8.
Royden, page 64, #10,13.
Note. Number 13 is long but rewarding. The previous
Mathematics 241, Problem Set #9, due October 28, 2009
Write clear proofs that have good grammar and consist of complete sentences. Royden, page 258, #5a,b,c
Royden, page 267, # 19, 21, 22
Royden, page 275, #27, 28
Royden, page 279, #33, 35