lecture2_2

# lecture2_2 - Notes 2 for Honors Probability and Statistics...

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Unformatted text preview: Notes 2 for Honors Probability and Statistics Ernie Croot August 24, 2010 1 Examples of σ-algebras and Probability Mea- sures So far, the only examples of σ-algebras we have seen are ones where the sample space is finite. Let us begin with a FALSE example of a σ-algebra where S is infinite: False example of a σ-algbra. We say that a subset A ⊆ N has density ρ if and only if lim n →∞ # { a ∈ A : a ≤ n } n = ρ. Another, more compact way of writing the numerator in the limit is | A ( n ) | , which is the cardinality of the set A ( n ), which in turn is the number of elements of A that are ≤ n . Not every subset of N has a density. For example, the set B with the property that b ∈ B if and only if b is an integer lying in an interval [2 3 j , 2 3 j +1 ] for some integer j ≥ 0, does not have a density. Another way of writing B is as the infinite union B = { 1 , 2 } ∪ { 8 , 9 ,..., 16 } ∪ { 64 , 65 ,..., 128 } ∪ ··· To see that B does not have a density, consider the size of | B (2 3 k − 1) | , for some integer k ≥ 1. This set will have (1 + 1) + (8 + 1) + (64 + 1) + ··· + (2 3( k- 1) + 1) = k + 2 3 k − 1 7 elements . 1 So, | B (2 3 k − 1) | / (2 3 k − 1) will tend to the limit 1 / 7 as k → ∞ . But now consider the size of B (2 3 k +1 ). This set is exactly the same as B (2 3( k +1) − 1), and so has size ( k + 1) + (2 3( k +1) − 1) / 7 elements; and so, | B (2 3 k +1 ) | / (2 3 k +1 ) tends to the limit 4 / 7 as k → ∞ . So, the limit | B ( n ) | /n does not exist, meaning that B does not have a density. Now, it seems intuitively obvious that if we let S = N , and let Σ be the set of all subsets A ⊂ S that have a density ρ , and then let P ( A ) = ρ , then ( S, Σ ,P ) is a probability space. It turns out that this is not true! The problem here is that Σ is NOT a σ-algebra, and, in particular, is not closed under intersections. An example of a pair of sets C,D ∈ Σ whose intersection is not in Σ is given as follows: Let C denote the set of even integers, and let D = D ∪ D 1 , where D is the set of even integers lying in an intervals of the form [2 3 j , 2 3 j +1 ] (where j ≥ 0 is an integer), and where D 1 is the set of odd integers that lie in intervals of the form (2 3 j +1 , 2 3( j +1) ) (again,...
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lecture2_2 - Notes 2 for Honors Probability and Statistics...

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