Stat 312: Lecture 19 Linear Regression Moo K. Chung email@example.com Nov 30, 2004 Concepts 1. Let x be the speed of a car and y be the distance the car traveled in an hour hour. Then we have model y = β0 + β 1 x. Suppose we have n paired measurements ( x i ,y i ) ,i = 1 , ··· ,n . Since all measurement are supposed to be noisy, we introduce a noise term ² in the above equation. Our modiﬁed stochastic model is y = β0 + β 1 x + ², where ² ∼ N (0 ,σ 2 ) . Since ² is a random variable, we use Y instead of y for convenience: Y = β0 + β 1 x + ². Note that E Y = β0 + β 1 x and V Y = σ 2 . 2. Equivalently we can write the above linear model for each paired measurement ( x i ,y j ) : Y j = β0 + β 1 x j + ² j , where y j is the observed value of random variable Y j and ² j ∼ ² . Note that E 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 are given by ˆ y j = ˆ β0 + ˆ β 1 x j . The differences between the observations
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