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Probability Theory Probability Theory "A random variable is neither random nor variable." Gian-Carlo Rota, M.I.T.. Florian Herzog 2013 Probability space Probability space A probability space W is a unique triple W = fΩ; F;P g: • Ω is its sample space •F its σ-algebra of events • P its probability measure Remarks: (1) The sample space Ω is the set of all possible samples or elementary events !: Ω = f! j ! 2 Ωg. (2)The σ-algebra F is the set of all of the considered events A, i.e., subsets of Ω: F = fA j A ⊆ Ω;A 2 Fg. (3) The probability measure P assigns a probability P (A) to every event A 2 F: P : F! [0; 1]. Stochastic Systems, 2013 2 Sample space The sample space Ω is sometimes called the universe of all samples or possible outcomes !. Example 1. Sample space • Toss of a coin (with head and tail): Ω = fH;T g. • Two tosses of a coin: Ω = fHH;HT;TH;TT g. • A cubic die: Ω = f!1;!2;!3;!4;!5;!6g. • The positive integers: Ω = f1; 2; 3;::: g. • The reals: Ω = f! j ! 2 Rg. Note that the !s are a mathematical construct and have per se no real or scientific meaning. The !s in the die example refer to the numbers of dots observed when the die is thrown. Stochastic Systems, 2013 3 Event An event A is a subset of Ω. If the outcome ! of the experiment is in the subset A, then the event A is said to have occurred. The set of all subsets of the sample space are denoted by 2Ω. Example 2. Events • Head in the coin toss: A = fHg. • Odd number in the roll of a die: A = f!1;!3;!5g. • An integer smaller than 5: A = f1; 2; 3; 4g, where Ω = f1; 2; 3;::: g: • A real number between 0 and 1: A = [0; 1], where Ω = f! j ! 2 Rg. We denote the complementary event of A by Ac = ΩnA. When it is possible to determine whether an event A has occurred or not, we must also be able to determine whether Ac has occurred or not. Stochastic Systems, 2013 4 Probability Measure I Definition 1. Probability measure A probability measure P on the countable sample space Ω is a set function P : F! [0; 1]; satisfying the following conditions • P(Ω) = 1. • P(!i) = pi. • If A1;A2;A3; ::: 2 F are mutually disjoint, then 1 1 [ X P Ai = P (Ai): i=1 i=1 Stochastic Systems, 2013 5 Probability The story so far: • Sample space: Ω = f!1;:::;!ng, finite! • Events: F = 2Ω: All subsets of Ω P • Probability: P (!i) = pi ) P (A 2 Ω) = pi wi2A Probability axioms of Kolmogorov (1931) for elementary probability: • P(Ω) = 1. • If A 2 Ω then P (A) ≥ 0. • If A1;A2;A3; ::: 2 Ω are mutually disjoint, then 1 1 [ X P Ai = P (Ai): i=1 i=1 Stochastic Systems, 2013 6 Uncountable sample spaces n Most important uncountable sample space for engineering: R, resp. R . Consider the example Ω = [0; 1], every ! is equally "likely". • Obviously, P(!) = 0. • Intuitively, P([0; a]) = a, basic concept: length! Question: Has every subset of [0; 1] a determinable length? Answer: No! (e.g. Vitali sets, Banach-Tarski paradox) Question: Is this of importance in practice? Answer: No! Question: Does it matter for the underlying theory? Answer: A lot! Stochastic Systems, 2013 7 Fundamental mathematical tools Not every subset of [0; 1] has a determinable length ) collect the ones with a determinable length in F. Such a mathematical construct, which has additional, desirable properties, is called σ-algebra. Definition 2. σ-algebra A collection F of subsets of Ω is called a σ-algebra on Ω if • Ω 2 F and ; 2 F (; denotes the empty set) • If A 2 F then ΩnA = Ac 2 F: The complementary subset of A is also in Ω S1 • For all Ai 2 F: i=1 Ai 2 F The intuition behind it: collect all events in the σ-algebra F, make sure that by performing countably many elementary set operation ([; \;c ) on elements of F yields again an element in F (closeness). The pair fΩ; Fg is called measure space. Stochastic Systems, 2013 8 Example of σ-algebra Example 3. σ-algebra of two coin-tosses • Ω = fHH;HT;TH;TT g = f!1;!2;!3;!4g •F min = f;; Ωg = f;; f!1;!2;!3;!4gg. •F 1 = f;; f!1;!2g; f!3;!4g; f!1;!2;!3;!4gg. •F max = f;; f!1g; f!2g; f!3g; f!4g; f!1;!2g; f!1;!3g; f!1;!4g; f!2;!3g; f!2;!4g; f!3;!4g; f!1;!2;!3g; f!1;!2;!4g; f!1;!3;!4g; f!2;!3;!4g; Ωg: Stochastic Systems, 2013 9 Generated σ-algebras The concept of generated σ-algebras is important in probability theory. Definition 3. σ(C): σ-algebra generated by a class C of subsets Let C be a class of subsets of Ω. The σ-algebra generated by C, denoted by σ(C), is the smallest σ-algebra F which includes all elements of C, i.e., C 2 F. Identify the different events we can measure of an experiment (denoted by A), we then just work with the σ-algebra generated by A and have avoided all the measure theoretic technicalities. Stochastic Systems, 2013 10 Borel σ-algebra The Borel σ-algebra includes all subsets of R which are of interest in practical applications (scientific or engineering). Definition 4. Borel σ-algebra B(R)) The Borel σ-algebra B(R) is the smallest σ-algebra containing all open intervals in R. The sets in B(R) are called Borel sets. The extension to the n multi-dimensional case, B(R ), is straightforward. • (−∞; a), (b; 1), (−∞; a) [ (b; 1) • [a; b] = (−∞; a) [ (b; 1), S1 S1 • (−∞; a] = n=1[a − n; a] and [b; 1) = n=1[b; b + n], • (a; b] = (−∞; b] \ (a; 1), T1 1 1 •f ag = n=1(a − n; a + n), Sn •f a1; ··· ; ang = k=1 ak. Stochastic Systems, 2013 11 Measure Definition 5. Measure Let F be the σ-algebra of Ω and therefore (Ω; F) be a measurable space. The map µ : F! [0; 1] is called a measure on (Ω; F) if µ is countably additive. The measure µ is countably additive (or σ-additive) if µ(;) = 0 and for every sequence of disjoint sets (F : i 2 ) in F with F = S F we have i N i2N i X µ(F ) = µ(Fi): i2N If µ is countably additive, it is also additive, meaning for every F; G 2 F we have µ(F [ G) = µ(F ) + µ(G) if and only if F \ G = ; The triple (Ω; F; µ) is called a measure space. Stochastic Systems, 2013 12 Lebesgue Measure The measure of length on the straight line is known as the Lebesgue measure. Definition 6. Lebesgue measure on B(R) The Lebesgue measure on B(R), denoted by λ, is defined as the measure on (R; B(R)) which assigns the measure of each interval to be its length. Examples: • Lebesgue measure of one point: λ(fag) = 0. P1 • Lebesgue measure of countably many points: λ(A) = i=1 λ(faig) = 0. • The Lebesgue measure of a set containing uncountably many points: { zero { positive and finite { infinite Stochastic Systems, 2013 13 Probability Measure Definition 7. Probability measure A probability measure P on the sample space Ω with σ-algebra F is a set function P : F! [0; 1]; satisfying the following conditions • P(Ω) = 1. • If A 2 F then P (A) ≥ 0. • If A1;A2;A3; ::: 2 F are mutually disjoint, then 1 1 [ X P Ai = P (Ai): i=1 i=1 The triple (Ω; F;P ) is called a probability space. Stochastic Systems, 2013 14 F-measurable functions Definition 8. F-measurable function The function f :Ω ! R defined on (Ω; F;P ) is called F-measurable if −1 f (B) = f! 2 Ω: f(!) 2 Bg 2 F for all B 2 B(R); −1 i.e., the inverse f maps all of the Borel sets B ⊂ R to F. Sometimes it is easier to work with following equivalent condition: y 2 R ) f! 2 Ω: f(!) ≤ yg 2 F This means that once we know the (random) value X(!) we know which of the events in F have happened. •F = f;; Ωg: only constant functions are measurable •F = 2Ω: all functions are measurable Stochastic Systems, 2013 15 F-measurable functions Ω: Sample space, Ai: Event, F = σ(A1;:::;An): σ-algebra of events X(!):Ω 7! R R 6 j X(!) 2 ! R A2 A1 t (−∞; a] 2 B Y A3 −1 Ω A4 X : B 7! F X: random variable, B: Borel σ-algebra Stochastic Systems, 2013 16 F-measurable functions - Example Roll of a die: Ω = f!1;!2;!3;!4;!5;!6g We only know, whether an even or and odd number has shown up: F = f;; f!1;!3;!5g; f!2;!4;!6g; Ωg = σ(f!1;!3;!5g) Consider the following random variable: 1; if ! = ! ;! ;! ; f(!) = 1 2 3 −1; if ! = !4;!5;!6. Check measurability with the condition y 2 R ) f! 2 Ω: f(!) ≤ yg 2 F f! 2 Ω: f(!) ≤ 0g = f!4;!5;!6g 2= F ) f is not F-measurable. Stochastic Systems, 2013 17 Lebesgue integral I Definition 9.
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