The difference between sample regression function and total regression function

The overall regression function also becomes the theoretical regression function, the
model for E (y | x) = a + B x
where the parameter AB exists but unknown, is an expectation, the
sample regression function also becomes the empirical regression function
model for y^ = a^ + b^ x
a^, b^ In order to estimate the value based on the sample data, y^ is also the value predicted by the estimated equation.
non-actual model, knowledge is used to fit the actual model.

The overall regression line is unknown, only one. The sample regression line is fitted according to the sample data, and a sample regression line can be fitted for each set of samples. The
β1 and β2 in the overall regression function are unknown parameters, which are expressed as constants. and the sample regression function         is a random variable, whose specific value varies with the observed values of the sampled samples.
UT in the overall regression function is the longitudinal distance between the YT and the unknown total regression line, and it cannot be observed directly. In the sample regression function, et is the longitudinal distance between YT and sample regression line, and the specific value of ET can be calculated after the sample regression is fitted out according to the sample observations.