- #1
twoflower
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Homework Statement
Consider model of linear regression:
[tex]
Y_i = \beta_0 + x_i \beta_1 + \epsilon_i
[/tex]
i = 1, ..., 5, where [itex]\epsilon_i \sim \mathcal{N}(0, \sigma^2)[/itex] are independent. Find expected value and variance of predicted values [tex]\widehat{Y}_i[/tex] considering that observations are obtained in points 1, 2, 3, 4, 5 (ie. [tex]x_i = i[/tex] for i = 1, ..., 5) and [tex]\sigma^2 = 1[/tex]. Hint: remember that
[tex]
\widehat{Y} = HY
[/tex]
Homework Equations
[tex]
H = X\left(X^T X\right)^{-1}X^T Y
[/tex]
The Attempt at a Solution
My attempt is
[tex]
E \widehat{Y} = \beta_0 + X\beta_1 = (\beta_0 + \beta_1, \beta_0 + 2\beta_1, \beta_0 + 3\beta_1, \beta_0 + 4\beta_1, \beta_0 + 5\beta_1)
[/tex]
Is it correct?
Anyway, even if it is, how do I find the variance and how do I use the hint? :)
Thank you.