Why is $y\sim\mathcal{N}(Xβ,\sigma^{2} I_n)$? (i.e. $\varepsilon\sim \mathcal{N}(0, \sigma^{2} I_n)$)
How can I obtain $y\sim\mathcal{N}(Xβ,\sigma^{2} I_n)$?
Adding a constant $X\beta$ to a normally distributed random variable is again normal.
The mean of the $y = X\beta + \epsilon$ is: $$E(y) = E(X\beta + \epsilon) = E(X\beta) + E(\epsilon) = X\beta + 0 = X\beta$$ and the variance is $$\operatorname{var} (y) = E(y -X\beta)(y -X\beta)^T = E(\epsilon \epsilon^T) = \operatorname{var} (\epsilon) = \sigma^2 I$$ We conclude that $$y \sim N(X \beta, \sigma^2 I)$$
The variance will not change if you add a constant, i.e $Var(X+c)=Var(X)$, here $X\beta$ is a constant
The formula is :
$Y=X\beta+\varepsilon$ where $\varepsilon\sim \mathcal{N}(0, \sigma^{2} I_n)$