By Tim Bedford, Roger Cooke

ISBN-10: 0521773202

ISBN-13: 9780521773201

Probabilistic probability research goals to quantify the danger brought on by excessive know-how installations in occasions the place classical statistical research is hard or very unlikely. This publication discusses the basic proposal of uncertainty, its courting with likelihood, and the bounds to the quantification of uncertainty. Drawing on large adventure within the conception and purposes of danger research, the authors specialize in the conceptual and mathematical foundations underlying the quantification, interpretation and administration of chance. They conceal common issues in addition to vital new matters equivalent to using professional judgment and uncertainty propagation.

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**Sample text**

These lead to very diﬀerent probabilities (see Exercise 1), yet angles and x-coordinates seem equally ‘equally possible’. Frequency interpretation This interpretation was championed by R. von Mises [von Mises, 1919, von Mises, 1981] and Reichenbach [Reichenbach, 1932]. Here Σ = ﬁnite set of outcomes of an experiment that can be repeated indeﬁnitely under ‘identical conditions’. Now, P (A) is deﬁned as limn→∞ (1/n) 1Ai ; where 1Ai is 1 if the outcome of experiment i is in A and is 0 otherwise. The ﬁrst concern of the frequentist is to demonstrate that P (A) is well deﬁned, that is, that the limit limn→∞ (1/n) 1Ai exists.

They then introduce the notion of uncertainty about the frequency (the ‘probability of a frequency’). This more sophisticated notion of risk will be discussed further in Chapter 2. 3 Scope of probabilistic risk analyses 11 For the moment the reader should bear in mind that the basic method of risk analysis is the identiﬁcation and quantiﬁcation of scenarios, probabilities and consequences. The tools that will be described in this book are dedicated to these tasks. 3 Scope of probabilistic risk analyses The goals and sizes of probabilistic risk analyses vary widely.

F implies Ai ∈ F. If A ⊂ PΩ, then F(A) denotes the smallest (σ-) ﬁeld which contains all elements of A. F(A) is called the (σ-) ﬁeld generated by A: F(A) = ∩{G|G is a (σ-) ﬁeld and G ⊇ A}. 2 P : F → [0, 1] is a (ﬁnitely additive) positive normalized measure on (Ω, F) if 39 40 3 Probabilistic methods (i) ∀A ∈ F, 0 ≤ P (A) ≤ 1, (ii) P (Ω) = 1, (iii) ∀A, B ∈ F, A ∩ B = ∅ implies P (A ∪ B) = P (A) + P (B). Furthermore, P is countably additive or σ-additive if (iv) whenever A1 , A2 , . . ∈ F, Ai ∩ Aj = ∅ for i = j, P ( Ai ) = P (Ai ).

### Probabilistic Risk Analysis: Foundations and Methods by Tim Bedford, Roger Cooke

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