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how can I find the distribution of Yi = β1Xi + εi, when we know Xi is normally distributed and mean and standard deviation are unknown and εi is normally distributed with (0,1)? Can we assume it is also normally distrubuted?

jonny16
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  • Linear combination of normally distributed random variables I think is also normally distributed. – Fiodor1234 May 09 '21 at 20:26
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    @Fiodor Only when you also assume all the variables have a multivariate normal distribution. See https://stats.stackexchange.com/questions/30159. – whuber May 09 '21 at 20:53
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    @whuber Really englihting post! So, as I understand it $Y_{i}$ could be assumed normal if we it already holds that $(b_{1}X_{i},\epsilon_{i})$ is multivariate normal, based on the second anwser of the post? – Fiodor1234 May 10 '21 at 09:24
  • @Fiodor1234 Yes, that's right. And since we usually assume all $\varepsilon_i$ are independent of all $X_j,$ that implies the entire collection of $X_j$ and $\varepsilon_i$ have a multivariate Normal distribution when the $X_j$ are multivariate Normal and the $\varepsilon_i$ are multivariate Normal. – whuber May 10 '21 at 14:32

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