Expectation and Integrals
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In the previous chapter, the mathematical expectation of a real random variable X is characterized as the probability-weighted average of the possible values of X. In the case of a simple random variable, which has a finite set of possible values, the expectation is a sum. The idea of expectation as an average is extended by intuitive and heuristic arguments to the case of random variables with absolutely continuous distributions. The sum becomes an integral. For mixed distributions, we may use the Stieltjes integral, or a mixture of a sum (for the discrete part) and an integral (for the absolutely continuous part).
KeywordsMathematical Expectation Finite Measure Monotone Convergence Nondecreasing Sequence Fundamental Lemma
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