Integration

J. C. Taylor
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J. C. Taylor: McGill University, Department of Mathematics and Statistics

Chapter Chapter II in An Introduction to Measure and Probability, 1997, pp 29-85 from Springer

Abstract: Abstract In elementary probability, if Ω is finite, say Ω = {1, 2,…, n}, F is the collection of all subsets of Ω, and if P gives weight a i ≥ 0 to {i} with Σn i=1 a i = 1, then the (mathematical) expectation E[X] of a random variable X on Ω (i.e., a function X : Ω → ℝ) is defined to be Σ n i=1 X(i) a i , = Σn i=1 X(i)P({i{). When a i = 1/n for each i, this expectation is the usual average value of X. Heuristically, this number is what we expect as the average of a large number of “observations” of X — see the weak law of large numbers in Chapter IV. Also, if X is non-negative, then E[X] can be conceived as the “area” under the graph of X: over each i one may imagine a rectangle of width P({a i }) and height X(i); then Σn i=1 X(i)P({i}) is the sum of the “areas” of the rectangles that make up the set under the graph.

Keywords: Lebesgue Measure; Finite Interval; Borel Function; Finite Measure; Lebesgue Measure Zero (search for similar items in EconPapers)
Date: 1997
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DOI: 10.1007/978-1-4612-0659-0_2

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