Interval Probabilities and Enclosures
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Interval Probabilities and Enclosures Marcilia A. Campos · M. das Gracas dos Santos
Received: 13 March 2012 / Accepted: 6 January 2013 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2013
Abstract This paper deals with the computation of interval probabilities and enclosures for probabilities. The theoretical support for getting all the results are interval analysis and high-accuracy arithmetic. The enclosures for Standard Normal are from INTLAB. Keywords
High accuracy arithmetic · Interval analysis · Probability · Validated numerics
Mathematics Subject Classification (2000)
Primary 65G20; Secondary 60A05
1 Introduction In this paper, we present results related to the use of interval mathematics (Alefeld and Herzberger 1983; Moore 1966, 1979; Moore et al. 2009; Sunaga 1958) and high-accuracy arithmetic (Kulisch and Miranker 1981; Rump 1999) applied for proposing a new methodology to compute probabilities and enclosures for probabilities. Starting with a definition of interval probability (Campos 1997) and interval probabilities for discrete random variables (Campos 2000), we present (1) a definition of interval conditional probability (Santos 2010) and (2) computation of enclosures for the Uniform, Exponential and Standard Normal continuous random variables (Santos 2010), the last two based on Caprani et al. (2002). This paper is organized as follows. In Sect. 2, the proposal of interval probability and interval conditional probability is shown. In Sect. 3, we compute interval probabilities for discrete random variables. The computation of enclosures are detailed in Sect. 4. Finally, conclusions are presented in Sect. 5.
Communicated by Renata Hax Sander Reiser. M. A. Campos (B) · M. das Gracas dos Santos Centro de Informática-UFPE, Recife, Pernambuco, Brazil e-mail: [email protected] M. das Gracas dos Santos e-mail: [email protected]
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M. A. Campos, M. das Gracas dos Santos
2 An interval probability The proposed-interval probability uses interval analysis to accomplish self-validating numerical computation of probabilities. However, it must be consistent with the semantics of the probability function, as it is known in mathematics (Burrill 1972; Feller 1968). Therefore, Campos (1997) proved that the interval probability has properties similar to the ones of the real probability. A difference between interval probability approach and the real probability (the probability as it is known) is that the former provides additional information about the reliability of the computed results (Forsythe 1977; Goldberg 1991). Definition 2.1 Given X , Y intervals and ∗ ∈ {+, −, ·, /}, the functions FI and F∗ are defined as follows: F I : R → IR FI (x) = X, x ∈ X.
F∗ : R × R → IR × IR F∗ (x, y) = X ∗ Y, x ∈ X, y ∈ Y.
Definition 2.2 Let FI (1) = 1ε = [1 − ε, 1 + ε], the smallest machine interval such that 1 ∈ 1ε and FI (0) = 0ε = [−ε, +ε], the smallest machine interval such that 0 ∈ 0ε . Lemma 2.3 Given the sample space, , a σ -field of subsets of , A, the probability function defined on A, P, a
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