Stat 312: Lecture 17 Simple Linear Model Moo K. Chung firstname.lastname@example.org March 27, 2003 Concepts 1. Deterministic model: y = β0 + β 1 x. For example, x is the speed of a car and y is the distance the car traveled in 1 hour. This is an unrealistic model because the car can not maintain the absolute constant speed. So we introduce a noise term ± in the above equa-tion. 2. Stochastic model: let ± be a random noise with ± = 0 and Var ± = σ 2 . Instead of the deterministic model we will have y = β0 + β 1 x + ± . Since ± is a random variable, we use Y instead of y : Y = β0 + β 1 x + ±. Note that Y = β0 + β 1 x and Var Y = σ 2 . 3. Given paired data ( x 1 , y 1 ) , ··· , ( x n , y n ) , we have a lin-ear stochastic relationship Y j = β0 + β 1 x j + ± j , where y j is the observed value of a random variable Y j and ± j ∼ ± . Note that Y j = β0 + β 1 x j . Let ˆ β0 , ˆ β 1 be estimators of β0 , β 1 . Then the predicted values or fitted values ˆ y j = ˆ β0 + ˆ β 1 x j are estimators of Y j
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