# Formalized Probability Theory and Applications Using Theorem by Osman Hasan, Sofiène Tahar

By Osman Hasan, Sofiène Tahar

Scientists and engineers frequently need to care for platforms that show random or unpredictable parts and needs to successfully assessment possibilities in every one state of affairs. desktop simulations, whereas the conventional device used to resolve such difficulties, are restricted within the scale and complexity of the issues they could solve.

Formalized likelihood idea and purposes utilizing Theorem Proving discusses the various obstacles inherent in computers whilst utilized to difficulties of probabilistic research, and offers a singular way to those obstacles, combining higher-order common sense with computer-based theorem proving. Combining sensible software with theoretical dialogue, this booklet is a vital reference device for mathematicians, scientists, engineers, and researchers in all STEM fields.

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Extra resources for Formalized Probability Theory and Applications Using Theorem Proving

Example text

24 ⊢ ∀ m1 m2. prod_measure m1 m2 = ( λ a. integral m1 ( λ s1. measure m2 (PREIMAGE g(s1) a))) The integral in this definition is the Lebesgue integral for which we present the formalization in the next section. Now it can be verified in HOL4 that the product measure can be reduced to μ (a1 × a2) = μ1 (a1) × μ2 (a2) for finite measure spaces. 9 ⊢ prod_measure m1 m2 (a1 x a2) = measure m1 a1 x measure m2 a2 The above definition can be used to define products of more than two measure spaces as follows.

22 Probabilistic Analysis Using Theorem Proving Figure 1. Formal Probabilistic Analysis Methodology The theory of probability can be formalized in higher-order logic according to the Kolmogorov axiomatic definition of probability. This definition provides a mathematically consistent way for assigning and deducing probabilities of events. This approach consists in defining a set of all possible outcomes, Ω, called the sample space, a set F of events which are subsets of Ω and a probability measure p such that (Ω; F; p) is a measure space with p (Ω) = 1.

F x + g x) ∈ measurable a Borel) (( λ x. f x * g x) ∈ measurable a Borel) (( λ x. max (f x) (g x)) ∈ measurable a ˄ ˄ ˄ ˄ Borel) The notation (λx. f x) s the lambda notation of f, used to represent the function f: x → f (x). If (fn) is a monotonically increasing sequence of extended-real-valued measurable functions with respect to (A, B (R)), such that ∀x, f (x) = supn∈ N fn (x) then f is also (A, B (R)) measurable. 6 ⊢ ∀ a f fi. sigma_algebra a ˄ ∀ i. fi i ∈ measurable a Borel ˄ ∀ x. mono_increasing (λi.