Integration

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 Σⁿ _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.