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In a text i read

We shall further assume that the distribution of the target data is Gaussian. More specifically, we assume that the target variable $t_k$ is given by some deterministic function of $\mathbf{x}$ with added Gaussian noise $\varepsilon$, so that $$t_k=h_k(\mathbf{x})+\varepsilon_k.$$

Now I'm wondering if the two concepts are equivalent i.e. having a normal distribution is the same of being sum of a deterministic function and a term with normal distribution?

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  • $\begingroup$ No, they are not. Only the term $\epsilon_k$ denotes a Gaussian distribution. The author is describing a deterministic model with added noise, and the expression "the target data is Gaussian" is a shortcut not to be taken literally. $\endgroup$
    – user65203
    Commented Sep 30, 2016 at 20:05

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If you add a constant to a Gaussian-distributed variable, you get another Gaussian-distributed variable, with a different mean. A Gaussian noise is understood as a Gaussian-distributed random variable with zero mean. So the variable $t_k$ indeed follows a Gaussian distribution, with mean $h_k(\mathbb x)$.

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