# Why does the $E(y_i) = E(\beta x_i + e_i) = \beta x_i$?

We have this information :

We have this regression model $$Y = \beta x + e$$ with ($$e\sim N(0, \sigma^2)$$) . We also have the estimate of $$\beta$$ which is $$\hat{\beta} = \dfrac{1}{\sum_{i=1}^n x_i^2}\sum_{i=1}^n x_i y_i$$. Consider $$\hat{\beta}$$ as if = $$\beta$$.

Why does the $$E(y_i) = E(\beta x_i + e_i) = \beta x_i$$?

Update: Ok, I get that $$E(e_i) = 0$$ but then I don't understand why does $$E(\beta x_i) = \beta x_i$$ ? I thought that the $$E(x) = \sum x f(x)$$. Where is our $$f(x)$$ in this case?

• $E(e_i)=0$ and sum rule for expectation. Dec 12, 2019 at 17:29
• but why is the expectation of $\beta x_i$ equal to itself? Dec 12, 2019 at 17:35
• I am not familiar with the details of the example. However, if these quantities are not random, then the expectations are the quantities. Specifically $\beta x_i$ is not random. Dec 12, 2019 at 19:13

I don't understand why does $$E(\beta x_i) = \beta x_i$$ ?
Because $$\beta$$ and $$x_i$$ are deterministic quantities.