introduction-probability.pdf

We take the system of subsets a a b such that a can

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]) we take the system of subsets A ( a, b ] such that A can be written as A = ( a 1 , b 1 ] ( a 2 , b 2 ] ∪ · · · ∪ ( a n , b n ] where a a 1 b 1 ≤ · · · ≤ a n b n b . For such a set A we let P 0 ( A ) := 1 b - a n i =1 ( b i - a i ) . Proposition 1.3.1 The system G is an algebra. The map P 0 : G → [0 , 1] is correctly defined and satisfies the assumptions of Carath´ eodory’s extension theorem Proposition 1.2.17. Proof . For notational simplicity we let a = 0 and b = 1. After a standard reduction (check!) we have to show the following: given 0 a b 1 and pair-wise disjoint intervals ( a n , b n ] with ( a, b ] = n =1 ( a n , b n ] we have that b - a = n =1 ( b n - a n ). Let ε (0 , b - a ) and observe that [ a + ε, b ] n =1 a n , b n + ε 2 n . Hence we have an open covering of a compact set and there is a finite sub- cover: [ a + ε, b ] n I ( ε ) ( a n , b n ] b n , b n + ε 2 n for some finite set I ( ε ). The total length of the intervals ( b n , b n + ε 2 n ) is at most ε > 0, so that b - a - ε n I ( ε ) ( b n - a n ) + ε n =1 ( b n - a n ) + ε.
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1.3. EXAMPLES OF DISTRIBUTIONS 27 Letting ε 0 we arrive at b - a n =1 ( b n - a n ) . Since b - a N n =1 ( b n - a n ) for all N 1, the opposite inequality is obvious. Definition 1.3.2 [Uniform distribution] The unique extension P of P 0 to B (( a, b ]) according to Proposition 1.2.17 is called uniform distribution on ( a, b ]. Hence P is the unique measure on B (( a, b ]) such that P (( c, d ]) = d - c b - a for a c < d b . To get the Lebesgue measure on R we observe that B ∈ B ( R ) implies that B ( a, b ] ∈ B (( a, b ]) (check!). Then we can proceed as follows: Definition 1.3.3 [Lebesgue measure] 8 Given B ∈ B ( R ) we define the Lebesgue measure on R by λ ( B ) := n = -∞ P n ( B ( n - 1 , n ]) where P n is the uniform distribution on ( n - 1 , n ]. Accordingly, λ is the unique σ -finite measure on B ( R ) such that λ (( c, d ]) = d - c for all -∞ < c < d < . We can also write λ ( B ) = B ( x ). Now we can go backwards: In order to obtain the Lebesgue measure on a set I R with I ∈ B ( R ) we let B I := { B I : B ∈ B ( R ) } and λ I ( B ) := λ ( B ) for B ∈ B I . Given that λ ( I ) > 0, then λ I ( I ) is the uniform distribution on I . Impor- tant cases for I are the closed intervals [ a, b ]. Furthermore, for -∞ < a < b < we have that B ( a,b ] = B (( a, b ]) (check!). 8 Henri L´ eon Lebesgue, 28/06/1875-26/07/1941, French mathematician (generalized the Riemann integral by the Lebesgue integral; continuation of work of Emile Borel and Camille Jordan).
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28 CHAPTER 1. PROBABILITY SPACES 1.3.5 Gaussian distribution on R with mean m R and variance σ 2 > 0 (1) Ω := R . (2) F := B ( R ) Borel σ -algebra. (3) We take the algebra G considered in Example 1.1.4 and define P 0 ( A ) := n i =1 b i a i 1 2 πσ 2 e - ( x - m ) 2 2 σ 2 dx for A := ( a 1 , b 1 ] ( a 2 , b 2 ] ∪· · ·∪ ( a n , b n ] where we consider the Riemann - integral 9 on the right-hand side. One can show (we do not do this here, but compare with Proposition 3.5.8 below) that P 0 satisfies the assump- tions of Proposition 1.2.17, so that we can extend P 0 to a probability measure N m,σ 2 on B ( R ).
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  • Spring '17
  • Probability, Probability theory, Probability space, measure, lim P, Probability Spaces

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