Independence relation between observation and error [duplicate]

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Having a simple model like: Y = a + bX + u where u represent the error term, if I know that the observation of X and Y are i.i.d, also the error term is i.i.d?

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• stats.stackexchange.com/questions/94872 answers the question about independence, because $u = Y-a-bX$ is a function of the iid variables $(X,Y).$ As far as whether the $u_i$ have identical distributions, perhaps you can answer that part of the question yourself? – whuber Dec 29 '18 at 18:34
• If $X$ and $Y$ are both standard Normal, then perforce $a=0$ and $b=1$ and you must still conclude the $u_i$ are identically distributed. – whuber Dec 29 '18 at 22:57
• You haven't supplied assumptions that imply $u$ is the sum of independent standard Normal variables. – whuber Dec 30 '18 at 16:07