Continuous Random Variables
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چکیده
Math 394 1 (Almost bullet-proof) Definition of Expectation Assume we have a sample space Ω, with a σ−algebra of subsets F , and a probability P , satisfying our axioms. Define a random variable as a a function X : Ω → R, such that all subsets of Ω of the form {ω |a < X(ω) ≤ b}, for any real a ≤ b are events (belong to F). Assume at first that the range of X is bounded, say it is contained in the interval [A,B]. We work with X by approximating it with a sequence of discrete random variables X, defined as
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