Transcription of Chapter 1
1 RS Chapter 1 Random Variables6/14/20191 Chapter 1 Probability Theory: IntroductionBasic Probability General In a probability space ( , , P), the set is the set of all possible outcomesof a probability experiment . Mathematically, is just a set, with elements . It is called the sample space. An eventis the answer to a Yes/No question. Equivalently, an event is a subset of the probability space: A . Think of A as the set of outcomes where the answer is Yes , and Acis the complementary set where the answer is No . A -algebra is a mathematical model of a state of partial knowledge about the outcome.
2 Informally, if is a -algebra and A , we say that A if we know whether A or Chapter 1 Random Variables6/14/20192 Definitions AlgebraDefinitions: Semiring(of sets)A collection of sets Fis called a semiringif it satisfies: F. If A, B F, then A B F. If A, B F, then there exists a collection of sets C1, C2, .., Cn F, such that A \ B = .(A\B all elements of A not in B)Definitions: AlgebraA collection of sets F is called an algebraif it satisfies: F. If 1 F, then 1C F.(Fis closed under complementation) If 1 F & 2 F,then 1 2 F. (Fis closed under finite unions).
3 Definitions: sigma -algebraDefinition: sigma -algebraA sigma -algebra( -algebra or -field) Fis a set of subsets of : F. If F, then C F.( C= complement of ) If 1, 2,.., n,.. F, then, F( 1i s are countable)Note: The set E = {{ },{a},{b},{c},{a,b},{b,c},{a,c},{a,b,c}} is an algebra and a -algebra. -algebras are a subset of algebras in the sense that all -algebras are algebras, but not vice versa. Algebras only require that they be closed under pairwise unions while -algebras must be closed under countablyinfinite Chapter 1 Random Variables6/14/20193 Theorem: All -algebras are algebras, and all algebras are , if we require a set to be a semiring, it is sufficient to show instead that it is a -algebra or algebra.
4 sigma algebras can be generated from arbitrary sets. This will be useful in developing the probability :For some set X, the intersection of all -algebras, Ai, containingX that is, x X x Aifor all i is itself a -algebra, denoted (X). This is called the -algebra generatedby Space, Definition:Sample SpaceThe sample space is the set of all possible unique outcomes of the experiment at : If we roll a die, = {1; 2; 3; 4; 5; 6}.In the probability space, the -algebra we use is ( ), the -algebra generated by . Thus, take the elements of and generate the "extended set" consisting of all unions, compliments, compliments of unions, unions of compliments, etc.
5 Include ; with this "extended set" and the result is ( ), which we denote as .DefinitionThe -algebra generated by , denoted , is the collection of possible events from the experiment at Chapter 1 Random Variables6/14/20194 DefinitionThe -algebra generated by , denoted , is the collection of possible events from the experiment at : We have an experiment with = {1, 2}. Then, = {{ },{1},{2},{1,2}}. Each of the elements of is an event. Think of events as descriptions of experiment outcomes ( : the nothing occurs event).Note that -algebras can be defined over the real line as well as over abstract sets.
6 To develop this notion, we need the concept of a : There are many definitions of topology based on the concepts of neighborhoods, open sets, closed set, etc. We present the definition based on open sets.( , )Definition (via open sets): A topological space is an ordered pair (X, ), whereXis a set and is a collection of subsets ofX, satisfying:1. ; X 2. is closed under finite is closed under arbitrary element of a topology is known as an open set. The collection is called : We have an experiment with = {1, 2, 3}. Then, = {{ },{1,2,3}} is a (trivial) topology on.
7 = {{ },{1},{1,2,3}} is also a topology on . = {{ },{1,2},{2,3},{1,2,3}} is NOT a topology on . Topological SpaceRS Chapter 1 Random Variables6/14/20195 Definition: Borel -algebra (Emile Borel (1871-1956), France.)The Borel -algebra(or, Borel field) denoted B, of the topological space (X; ) is the -algebra generated by the family of open sets. Its elements are called Borel : Let C = {(a; b): a < b}. Then (C) = BRis the Borel field generated by the family of all open intervals do elements of BRlook like? Take all possible open intervals. Take their compliments.
8 Take arbitrary unions. Include and R. BRcontains a wide range of intervals including open, closed, and half-open intervals. It also contains disjoint intervals such as {(2; 7] U (19; 32)}. It contains (nearly) every possible collection of intervals that are -algebraDefinition:Measurable Space A pair (X, ) is a measurable spaceif Xis a set and is a nonempty -algebra of subsets of measurable space allows us to define a function that assigns real-numbered values to the abstract elements of .Definition: Measure Let (X, ) be a measurable space. Aset function defined on is called a measureiff it has the following 0 (A) for any A.)
9 2. ( ) = ( -additivity). For any sequence of pairwise disjoint sets {An} such that Un=1An , we have MeasuresRS Chapter 1 Random Variables6/14/20196 Intuition: A measure on a set, S, is a systematic way to assign a positive number to each suitable subset of that set, intuitively interpreted as its size. In this sense, it generalizes the concepts of length, area, Examples of measures:- Counting measure: (S) = number of elements in Lebesgue measure on R: (S) = conventional length of is, if S = [a,b] (S) = [a,b] = b Note: A measure may take as its value.
10 Rules:(1) For any x R, + x= , x * = if x> 0, x * = if x< 0, and 0 * = 0;(2) + = ;(3) * a= for any a> 0;(4) or / are not :Measure Space A triplet (X, , ) is a measure space if (X, ) is a measurable space and : [0; ) is a measure. If (X) = 1, then is a probability measure, which we usually use notation P, and the measure space is a probability & Measure SpaceRS Chapter 1 Random Variables6/14/20197 There is a unique measure on (R, BR) that satisfies ([a, b]) = b afor every finite interval [a, b], < a b< . This is called the Lebesgue measure.]
