Probability Theory : STAT 310 / MATH 230 ; Nov 27 , 2012

Probability Theory : STAT 310 / MATH 230 ; August 27 , 2013

Probability Theory : STAT 310 / MATH 230 ; September 12 , 2010 Amir

Contents Preface 5 Chapter 1. Probability, measure and integration 7 1.1. Probability spaces, measures and σ-algebras 7 1.2. Random variables and their distribution 18 1.3. Integration and the (mathematical) expectation 30 1.4. Independence and product measures 54 Chapter 2. Asymptotics: the law of large numbers 71 2.1. Weak laws of large numbers 71 2.2. The Borel-Cantelli lemmas 77 2.3. Strong...

Stat 310A/Math 230A Theory of Probability Midterm Solutions

with a1, a2 > 0. (Indeed any rank 2 matrix can be written as RA with A as above, and R orthogonal. Hence any affine transformation can be written as h ◦ f with f as claimed and g a rigid motion.) Hence, by Caratheodory uniqueness theorem it is sufficient to show that λ2(ASu(α1, α2)) = |det(A)|λ2(Su(α1, α2)) for all u = (u1, u2) ∈ R, αi ∈ R+ (3) Su(α, β) ≡ { x = (x1, x2) ∈ R : x1 ∈ (u1, u1 + α1]...

SYLLABUS – MCS 494: Probability on Graphs

OFFICE HOURS: 1-2 MWF. TEXT: None. I will try to point out the sections of some relevant books, which will address the corresponding material we cover. Some of these books are listed below in bibliography. PREREQUISITE: Math 310 (first course in Linear Algebra), and Stat 401 (Introduction to Probability) or equivalent courses. Basic knowledge of graphs is welcome, (some parts of MCS 261), but n...

Stat 310A/Math 230A Theory of Probability Practice Final Solutions

Solution : Assume, without loss of generality |x2 − x1| ≥ 2−n+1. Then there exists an integer k ∈ {1, . . . , 2 − 1}, such that x1 < k · 2−n < (k + 1)2−n < x2. Of course PX((x1, x2)) ≥ P(k · 2−n ≤ X(ω) ≤ (k + 1)2−n) . (4) The integer k admits the unique binary expansion k = ∑n i=1 ki2 n−i. Then P(k · 2−n ≤ X(ω) ≤ (k + 1)2−n) = P(Cn,(k1...,kn)) = p 1(1− p)0 , (5) with n0(k) and n1(k) the number ...