Download PDF by Jordan M. Stoyanov: Counterexamples in probability

By Jordan M. Stoyanov

ISBN-10: 0471965383

ISBN-13: 9780471965381

Counterexamples (in the mathematical feel) are robust instruments of mathematical idea. This publication covers counterexamples from chance conception and stochastic procedures. This new increased version contains many examples and the newest examine effects. the writer is considered one of many leading specialists within the box. comprises numbers examples.

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Extra resources for Counterexamples in probability

Example text

A1,b1)(a2,b1)(a1,b2)(a3,b1)(a2,b2)(a1,b3)... Each element of the direct product appears somewhere on the list. We can make this formal as follows. After traversing k − 1 diagonals, the number of elements enumerated is 1 + 2 + ... + k − 1 = k(k − 1)/2. , k. This is the n = k(k − 1)/2 + jth element on the list. Substituting i for k − j + 1, we see that the pair (ai, bj) is the n = (i + j − 1)(i+j − 2)/2 + jth element on the list. We have established that the direct product of two countably infinite sets is in 1 − 1 correspondence with the natural numbers.

Because we want to be able to consider countable unions, we require the allowable sets ℱ to be a sigma-field. Therefore, it is understood in probability theory and throughout the remainder of this book that we are allowed to consider only the events belonging to the sigma-field ℱ. We will see that fields also play an important role in probability because a common technique is first to define a probability measure on a field and then extend it to a sigma-field. 4. In 5-card poker, each simple event ω is a specific collection of 5 cards, such as ω = {5 of Diamonds, 8 of Hearts, 2 of Clubs, 3 of Spades, King of Spades}.

4. In 5-card poker, each simple event ω is a specific collection of 5 cards, such as ω = {5 of Diamonds, 8 of Hearts, 2 of Clubs, 3 of Spades, King of Spades}. The sample space Ω is the set of all such collections of 5 cards. , {Ace of Diamonds, Ace of Hearts, Ace of Clubs, Ace of Spades, King of Spades}}. The natural sigma-field to use here is ℱ =the set of all possible unions of simple events; that way we can talk about any possible event. If instead we foolishly choose ℱ = {{full house}, {all hands other than a full house}, ∅, Ω}, then we would be allowed to consider only whether or not a full house occurred; we could not talk about events like a pair, a flush, etc.

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Counterexamples in probability by Jordan M. Stoyanov

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