STAT_5205_fall_2019_SLR.pdf - 1 Statistical Models and...

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1Statistical Models and Conditional Expectation“essentially, all models are wrong, but some are useful"George E.P. BoxMathematical modelAmathematical modelis a description of a system using mathematical concepts and language.Statistical modelAstatistical modelembodies a set of assumptions concerning the generation of the observeddata, and similar data from a larger population.Relations between variablesAfunctional relationbetween two variables is expressed by a mathematical formula. Ifxisthe independent variable andyis the dependent variable, then a function relation is of the form:y=f(x).Astatistical relation, unlike a functional relation, is not a perfect one. In general, the observa-tions for a statistical relation do not fall directly on the curve of relationship. This is commonlyexpressed as a functional relation coupled with a random error . Ifxis the independent variableandYis the dependent variable, then a statistical relationoftentakes the form:Y=f(x) +.A statistical relation is also commonly expressed in terms ofconditional expectation. That is,for random variablesYandX,Y=E[Y|X=x] +.1
1.1Conditional ExpectationGoal:The goal of this section is to motivate conditional expectation:E[Y|X=x]Consider an example from probability theory.Example 1Consider rolling two fair six sided dice (D1&D2) and recording the sum of the faces and themaximum of the faces. Define two random variables
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12345624681012Sum and Max of Two DiceXYCriticize this motivating example:4

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