2 implies that p t is equal to the sum of the areas

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Unformatted text preview: ace (Ω, F , P ) such that any rectangular region that is a subset of Ω of the form R = {(u, v ) : a ≤ u < b, c ≤ v < d} is an event, and P (R) = area of R = (b − a)(d − c). Let T be the triangular region T = {(x, y ) : x ≥ 0, y ≥ 0, x + y < 1}. Since T is not rectangular, it is not immediately clear whether T is an event. Show that T is an event, and find P (T ), using the axioms. Solution Consider the infinite sequence of square regions shown in Figure 1.2. Square 1 has area 1/4, squares 2 and 3 have area 1/16 each, squares 4,5,6, and 7 have area (1/4)3 each, and so on. The set of squares is countably infinite, and their union is T , so T is an event by Axiom E.3. Since 12 CHAPTER 1. FOUNDATIONS 8 4 9 10 2 5 11 12 6 1 13 3 14 7 15 Figure 1.2: Approximation of a triangular region. the square regions are mutually exclusive, Axiom P.2 implies that P (T ) is equal to the sum of the areas of the squares: P (T ) = 1/4 + 2(1/4)2 + 22 (1/4)3 + 23 (1/4)4 + · · · = (1/4)(1 + 2−1 + 2−2 + 2−3 + · · · ) = (1/4) · 2 = 1/2. Of...
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This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Institute of Technology.

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