Probability Theory: STAT310/MATH230; May 1, 2010 Amir Dembo
E-mail address: [email protected] Department of Mathematics, Stanford University, Stanford, CA 94305.
Contents
Preface Chapter 1. Probability, measure and integration 1.1. Probability spaces
Stat 310B/Math 230B Theory of Probability
Homework 2 Solutions
Andrea Montanari
Due on 1/22/2014
Exercise [4.3.11]
In the proof of Theorem 4.1.2 we use the Radon-Nikodym theorem only for showing the existence of C.E. in
case X L1 (, F, P) and X 0.
So, ass
Stat 310A/Math 230A Theory of Probability
Homework 6 Solutions
Andrea Montanari
Due on November 14, 2012
Exercises on the strong law of large numbers and the central limit
theorem
Exercise [2.3.13]
Clearly, |Xn | = |Xn1 |Un |, resulting with
n
log |Xn | =
Stat 310A/Math 230A Theory of Probability
Homework 7 Solutions
Andrea Montanari
Due on November 28, 2012
Exercises on weak convergence of distributions
Exercise [3.2.8]
p
D
(a). It is easy to see that Yn c if and only if Yn c (for example, this is a conse
Stat 310A/Math 230A Theory of Probability
Homework 1 Solutions
Andrea Montanari
Due on 10/3/2012
Option 1: Exercises on measure spaces
Exercise [1.1.4]
1. A and B \ A are disjoint with B = A (B \ A) so P(A) + P(B \ A) = P(B) and rearranging gives the
desi
Stat 310B/Math 230B Theory of Probability
Homework 5 Solutions
Andrea Montanari
Due on 2/19/2014
Exercise [5.3.20]
1. We claim that
2n
E[h|Fn ] = 2
n
h(u)du IAi,n (t) .
i=1
(1)
Ai,n
Indeed, integrability and Fn -measurability of the RHS of (1) are obvious
Stat 310A/Math 230A Theory of Probability
Homework 8 Solutions
Andrea Montanari
Due on December 5, 2012
Exercises on characteristic functions
Exercise [3.3.10]
1. Denoting by X () the ch.f. of X, since X and X are i.i.d., the ch.f. of X is X () = X ().
He
Stat 310A/Math 230A Theory of Probability
Homework 5 Solutions
Andrea Montanari
Due on November 7, 2012
Exercises on the law of large numbers and Borel-Cantelli
Exercise [2.1.5]
Let > 0 and pick K = K( ) nite such that if k K then r(k) . Applying the Cauc
Stat 310A/Math 230A Theory of Probability
Homework 2 Solutions
Andrea Montanari
Due on 10/10/2012
Exercises on measurable functions and Lebesgue integration
Exercise [1.2.14]
The same method works for all four parts.
1. Since B = (cfw_(, ] : R), it follow
Stat 310A/Math 230A Theory of Probability
Homework 3 Solutions
Andrea Montanari
Due on
Exercises on inequalities and convergence
Exercise [1.3.21]
(a). Cauchy-Schwarz implies
2
(EY I(Y >a) ) EY 2 P(Y > a)
For EY > a 0 the left hand side is larger than (EY
Stat 310B/Math 230B Theory of Probability
Homework 3 Solutions
Andrea Montanari
Due on 1/22/2014
Exercise [5.1.8]
(a). Recall Proposition 5.1.5 that for all k 1 a.s. E[Dk |Fk1 ] = 0. Obviously this a.s. zero random
variable has zero expectation, which amo
Stat 310B/Math 230B Theory of Probability
Homework 4 Solutions
Andrea Montanari
Due on 2/5/2014
Exercise [5.2.10]
2
It is easy to check that cfw_Sn s2 is a martingale (but you should do it). Let A = cfw_maxn |Sk | > x and
n
k=1
= infcfw_k : |Sk | > x n.
Stat 310B/Math 230B Theory of Probability
Homework 1 Solutions
Andrea Montanari
Due on 1/15/2014
Exercise [4.1.3]
We need to show that E(XIA ) = E(Y IA ), for all A G = (P). Let L = cfw_A F : E(XIA ) = E(Y IA ).
The assumption implies P L. By Dynkins theo
Stat 310B/Math 230B Theory of Probability
Homework 7 Solutions
Andrea Montanari
Due on 3/5/2014
Exercise [6.1.12]
1. Since cfw_N n = cfw_Tn Gn for any non-negative integers , n, it follows that each N is a stopping
time for Gn . Further, Tk k for all k (
Stat 310A/Math 230A Theory of Probability
Homework 4 Solutions
Andrea Montanari
Due on October 24, 2012
Exercises on independent random variables and product measures
Exercise [1.4.18]
We will use the following fact repeatedly: the probability that X is d
Stat 310B/Math 230B Theory of Probability
Homework 6 Solutions
Andrea Montanari
Due on 2/26/2014
Exercise [5.4.14]
Recall that k are i.i.d. random variables taking values in cfw_A, B, . . . , Z such that P(k = x) = 1/26 if
x cfw_A, B, . . . , Z and 0 othe
310c: Homework Solutions 2010
Homework 1 Solution. 7.1.12 Fixing h > 0 and t 0, let Y = Xt+h Xt and L = cfw_A F : P(A C ) = P(A)P(C ) for all C (Y ). Note that L is a -system. Also, for s = (s1 , . . . , sm ) such that 0 s1 < < sm = t considering our assu
310C: HOMEWORK SETS 2010
363
310c: Homework Sets 2010 Homework rules: Arguments should be precise, concise and correct. We expect rigorous and complete proofs. Answers should be clean and legible. Please use a separate sheet of paper for each problem (eas
Stat 310B/Math 230B Theory of Probability
Homework 8 Solutions
Andrea Montanari
Due on 3/12/2014
Exercise [6.2.5]
1. For any x S\C, our assumption that Px (C < ) > 0 implies that Px (TC Nx ) = Px (C Nx ) > 0
for nite non-random constants Nx . Since S \ C