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Integration

In: An Introduction to Measure and Probability

Author

Listed:
  • J. C. Taylor

    (McGill University, Department of Mathematics and Statistics)

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.

Suggested Citation

  • J. C. Taylor, 1997. "Integration," Springer Books, in: An Introduction to Measure and Probability, chapter 0, pages 29-85, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4612-0659-0_2
    DOI: 10.1007/978-1-4612-0659-0_2
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