Download An Introduction to Probability Theory and Its Applications, by William Feller PDF

By William Feller

Significant alterations during this variation contain the substitution of probabilistic arguments for combinatorial artifices, and the addition of latest sections on branching strategies, Markov chains, and the De Moivre-Laplace theorem.

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Additional info for An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd Edition

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In ≤ n, ∀1 ≤ j < k ≤ n : ij = ik , so that Γ(ci1 ) ⊆ Γ(ci2 ) ⊆ . . ⊆ Γ(cin ). Intuitively, it must be possible to arrange the focal sets so that they form a ‘‘(stair) pyramid’’ or a ‘‘(stair) cone’’ of ‘‘possibility mass’’ on Ω (with the focal sets corresponding to horizontal ‘‘slices’’, the thickness of which represents their probability). With this picture in mind it is easy to see that requiring consonant focal sets is sufficient for ∀E ⊆ Ω : Π(E) = maxω∈E π(ω). In addition, it is immediately clear that a random set with consonant nonempty focal sets must be consistent, because all elements of the first focal set in the inclusion sequence are possible in all contexts.

2 on page 36) loses information. 15. It shows two random sets over Ω = {1, 2, 3, 4, 5}, both of which lead to the same basic possibility assignment. However, with the left random 42 CHAPTER 2. 15 Two random sets that induce the same basic possibility assignment. The numbers marked with a • are possible in the contexts. 1 c1 : c2 : c3 : π 1 4 1 4 1 2 • 1 4 2 3 4 5 • • • • c1 : • c2 : • 1 2 • • • 1 3 4 c3 : 1 2 π 1 4 1 4 1 2 1 2 3 • • • • • • • • • 1 3 4 1 2 1 4 1 2 4 5 set, Π({1, 5}) = 12 (maximum of the degrees of possibility of the elementary events), but with the right random set, Π({1, 5}) = 34 (sum of the degrees of possibility of the elementary events).

In our opinion this drawback disqualifies this approach, because it practically eliminates the capability of possibility theory to handle situations with imprecise, that is, set-valued information. Nevertheless, there is a (surprisingly simple) way out of this dilemma, which we discuss in the next but one section. It involves a reinterpretation of a degree of possibility for general, non-elementary events while keeping the adopted interpretation for elementary events. 8 Mass Assignment Theory Although we already indicated above that we reject the assumption of consonant focal sets, we have to admit that a basic possibility assignment induced by a random set with consonant nonempty focal sets has an important advantage, namely that from it we can recover all relevant properties of the inducing random set by computing a so-called mass assignment.

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