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I am reading a textbook that mentions the following linear model:

$$y \sim \sum_i x_i b_i + \sigma z,$$

where $y,b_i,z \in \mathbb{R}^n$, $x_i,\sigma \in \mathbb{R}$, and $z \sim \mathcal N(0,I)$. Now, the text then goes on to talk about properties of the sample and population covariance without defining these terms:

  • could someone give a definition of the sample and population covariances?
  • Is the population covariance independent of the distribution from which the noise term ($z$) is sampled?
  • For example, could I have samples of the same population where the noise term is taken from a different distribution for each sample?
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