ECONS
introduction-probability.pdf

Now we define the the borel σ algebra on r n

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Now we define the the Borel σ -algebra on R n . Definition 1.2.20 For n ∈ { 1 , 2 , ... } we let B ( R n ) := σ (( a 1 , b 1 ) × · · · × ( a n , b n ) : a 1 < b 1 , ..., a n < b n ) . Letting | x - y | := ( n k =1 | x k - y k | 2 ) 1 2 for x = ( x 1 , ..., x n ) R n and y = ( y 1 , ..., y n ) R n , we say that a set A R n is open provided that for all x A there is an ε > 0 such that { y R n : | x - y | < ε } ⊆ A. As in the case n = 1 one can show that B ( R n ) is the smallest σ -algebra which contains all open subsets of R n . Regarding the above product spaces there is Proposition 1.2.21 B ( R n ) = B ( R ) ⊗ · · · ⊗ B ( R ) . If one is only interested in the uniqueness of measures one can also use the following approach as a replacement of Carath´eodory ’s extension theo- rem: Definition 1.2.22 [ π -system] A system G of subsets A Ω is called π - system , provided that A B ∈ G for all A, B ∈ G . Any algebra is a π -system but a π -system is not an algebra in general, take for example the π -system { ( a, b ) : -∞ < a < b < ∞} ∪ {∅} . Proposition 1.2.23 Let , F ) be a measurable space with F = σ ( G ) , where G is a π -system. Assume two probability measures P 1 and P 2 on F such that P 1 ( A ) = P 2 ( A ) for all A ∈ G . Then P 1 ( B ) = P 2 ( B ) for all B ∈ F .
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1.3. EXAMPLES OF DISTRIBUTIONS 25 1.3 Examples of distributions 1.3.1 Binomial distribution with parameter 0 < p < 1 (1) Ω := { 0 , 1 , ..., n } . (2) F := 2 Ω (system of all subsets of Ω). (3) P ( B ) = μ n,p ( B ) := n k =0 ( n k ) p k (1 - p ) n - k δ k ( B ), where δ k is the Dirac measure introduced in Example 1.2.4. Interpretation: Coin-tossing with one coin, such that one has heads with probability p and tails with probability 1 - p . Then μ n,p ( { k } ) equals the probability, that within n trials one has k -times heads . 1.3.2 Poisson distribution with parameter λ > 0 (1) Ω := { 0 , 1 , 2 , 3 , ... } . (2) F := 2 Ω (system of all subsets of Ω). (3) P ( B ) = π λ ( B ) := k =0 e - λ λ k k ! δ k ( B ). The Poisson distribution 7 is used, for example, to model stochastic processes with a continuous time parameter and jumps: the probability that the process jumps k times between the time-points s and t with 0 s < t < is equal to π λ ( t - s ) ( { k } ). 1.3.3 Geometric distribution with parameter 0 < p < 1 (1) Ω := { 0 , 1 , 2 , 3 , ... } . (2) F := 2 Ω (system of all subsets of Ω). (3) P ( B ) = μ p ( B ) := k =0 (1 - p ) k k ( B ). Interpretation: The probability that an electric light bulb breaks down is p (0 , 1). The bulb does not have a ”memory”, that means the break down is independent of the time the bulb has been already switched on. So, we get the following model: at day 0 the probability of breaking down is p . If the bulb survives day 0, it breaks down again with probability p at the first day so that the total probability of a break down at day 1 is (1 - p ) p . If we continue in this way we get that breaking down at day k has the probability (1 - p ) k p . 7 Sim´ eon Denis Poisson, 21/06/1781 (Pithiviers, France) - 25/04/1840 (Sceaux, France).
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26 CHAPTER 1. PROBABILITY SPACES 1.3.4 Lebesgue measure and uniform distribution Using Carath´eodory’s extension theorem, we first construct the Lebesgue measure on the intervals ( a, b ] with -∞ < a < b < . For this purpose we let (1) Ω := ( a, b ], (2) F = B (( a, b ]) := σ (( c, d ] : a c < d b ), (3) As generating algebra G for B (( a, b
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  • Spring '17
  • Probability, Probability theory, Probability space, measure, lim P, Probability Spaces

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