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   The simple linear regression model consists of the mean function and the variance function. SLR, mean function E(Y|X=x) = β0 + β1x The value of parameters  are usually unknown and must be estimated using data. SLR, variance function Var(Y|X=x) = σ2 In SLR, the variance function is assumed to be constant, with a positive value of σ2 that is usually unknown. yi is observed value of i the response y, and will typically not equal its expected value E(Y|X=xi) because σ2 > 0. yi = E(Y|X=x) + ei, where ei is statistical error (implicit equation for ei) ei can be defined explicitly as, ei = yi - E(Y|X=x) = yi - (β0 + β1x) The errors, ei, depend on the unknown parameters in the mean function and so are not observable quantities. They are random variables. Assumption of ei, E(ei|xi) = 0 (mean of statistical errors is 0). So, if you draw a scatterplot of the ei vs the xi, we would have a null scatterplot, with no patterns. Assumption of ei, they are independent. Assumption of ...

  Regression is study of dependence. How Y changes on the average as the value of X is varied . Linear regression is important instance of regression methodology and is most commonly used. Virtually all other regression methods build upon an understanding of how linear regression works. The goal of regression is to understand how the values of Y change as X is varied over  its range of possible values. One important function of the scatterplot is to decide if we might  reasonably assume  that the response on the vertical axis is independent of the predictor on the horizontal axis. The extreme values on the left and right of the horizontal axis are points that are likely to be important in fitting  regression models  and are called  leverage points . The separated points on the vertical axis are potentially  outlier . Outliers are more easily discovered in residual plots. Residual plot gain resolution in the plot. Residual plot is obtained by remov...