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Given the simple linear regression y = Beta0 + Beta1x + e (epsilon) and e~N(0,sigma^2), to prove y is a multivariate normal I must show that it is a linear combination of multivariate normals correct?

So since I was given e~N(0,sigma^2), I let zi = ei/sigma and then e = sigma * I * zi. Therefore e = (e1, ..., en) is a MVN. Now do I simply set e = y - Beta0 + Beta1x and then state simply it must be a MVN since e is? I am lost at this help and any help or suggestion would be greatly appreciated! Thanks

user272760
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  • The formula for $\hat\beta$ is a linear combination of the $y_i$ and therefore is an affine linear combination of the $e_i,$ *QED.* Re "therefore:" this conclusion requires an assumption you haven't stated: namely, that the $e_i$ have a *multivariate* Normal distribution. (It isn't enough that their marginals are Normal). – whuber Feb 05 '20 at 23:47
  • I should add that I re-interpreted your post to be asking about the distribution of the parameter estimates, which might not be what you intended. According to some definitions of multivariate normality, if you assume the $e_i$ are *independent,* there's nothing you need to show: it's immediate that $y$ is MVN. But according to other definitions, there is a (little bit) to show. For instance, some people define the MVN in terms of its characteristic function, so you have to relate the cf of $e$ to that of $y.$ Thus, assuming this is what you need, please tell us how you define the MVN. – whuber Feb 06 '20 at 00:02

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