# Estimator of $Y$ in the simple linear regression model

In a statistics textbook I saw that the linear simple regression model is defined as $$$$Y = \alpha + \beta x + e$$$$ where $$x$$ is a value of the independent variable, $$Y$$ is the response, $$\alpha$$ and $$\beta$$ are the parameters, and $$e$$ is a random error with a mean of 0.

Then this textbook talks about estimators. It says $$A$$ is an estimator of $$\alpha$$, and $$B$$ is an estimator of $$\beta$$. Also, $$A+Bx$$ is an estimator of $$\alpha+\beta x$$. However, according to the textbook, $$A+Bx$$ is an estimator of $$Y$$. I don't understand that $$A+Bx$$ can be an estimator of $$Y$$, since $$Y$$ is a random variable but not a parameter. Am I wrong?

To me it makes sense only to say that $$A+Bx$$ is an estimator of the 'mean' of the random variable $$Y$$. Then, given that the mean of $$e$$ is zero, $$A+Bx$$ is an estimator of the mean of $$\alpha + \beta x + e$$.

Then, given that the mean of $$e$$ is zero, $$A+Bx$$ is an estimator of $$\alpha + \beta x + e$$.
Note that $$\alpha+\beta x+e$$ is a random variable and it equals $$Y$$. If you oppose to calling $$A+Bx$$ and estimator of $$Y$$, you should oppose calling it an estimator of $$\alpha+\beta x+e$$ for the sake of internal consistency.
When we say $$A$$ is an estimator of $$\alpha$$ it means that $$\mathbb{E}(A)=\alpha$$. Since $$A+Bx$$ is a random variable(because of $$\alpha$$ and $$\beta$$) we can write: $$\mathbb{E}(A+Bx)=\mathbb{E}(A)+\mathbb{E}(Bx)+0=\mathbb{E}(A)+\mathbb{E}(B)x+\mathbb{E}(e)=\mathbb{E}(Y)$$ Hope it helps.