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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?

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  • $\begingroup$ $E(e_i)=0$ and sum rule for expectation. $\endgroup$ Dec 12, 2019 at 17:29
  • $\begingroup$ but why is the expectation of $\beta x_i$ equal to itself? $\endgroup$
    – WindBreeze
    Dec 12, 2019 at 17:35
  • $\begingroup$ 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. $\endgroup$ Dec 12, 2019 at 19:13

1 Answer 1

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I don't understand why does $E(\beta x_i) = \beta x_i$ ?

Because $\beta$ and $x_i$ are deterministic quantities.

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