In my lecture note, it states that $\hat{y} \sim\mathcal{N}(X\beta, \sigma^{2}X(X^{T}X)^{-1}X^{T}$)
but isn't $y \sim\mathcal{N}(X\beta, \sigma^{2}I_n$) ?
Which one is accurate or are they the same thing but with different representation?
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Sign up to join this communityIn my lecture note, it states that $\hat{y} \sim\mathcal{N}(X\beta, \sigma^{2}X(X^{T}X)^{-1}X^{T}$)
but isn't $y \sim\mathcal{N}(X\beta, \sigma^{2}I_n$) ?
Which one is accurate or are they the same thing but with different representation?
Then $X Cov(\hat\beta) X^T$ is the covariance $\Sigma$ for the error of the estimated values $\hat y = X \hat \beta$
(that is different from the true sample values $y$).
Imagine a linear regression line which is always less accurate at the ends due to the uncertainty in the slope.
Possibly it may become more clear when we use different notation (use the $\hat \mu$ instead of $\hat y$)
In the case of prediction, if you would like to estimate an error for the estimate of a new value, then you would actually use the sum of the two variances expressed above (the estimate of the mean plus the 'error' of a sampled value with respect to the mean).