Prediction and Conditional Expectation
Let (Ω, F, P) be a probability space, and let L2 be the vector space of random variables Zwith E[Z2<∞. Let X∈L2 be an unobservable random variable, whose value we wish to predict from observation of other random variables Y1,..., Yn. (For example, Xmay be the value of a stochastic process at some time in the future, or a spatial process over a region where it cannot be observed). In order to use knowledge of Y1,..., Yn to predict X, the predictor must be function of Y1,..., Yn, and, in particular, is a random variable
KeywordsMean Square Error Orthogonal Projection Conditional Distribution Conditional Expectation Minimum Mean Square Error
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