# For a Gaussian process, how does $\mathcal{N}(\mathbf{f}, {\sigma_n}^2\mathbf{I})$ relate to the Normal PDF?

I am told during the Gaussian Process class that the following notations are equivalent as convention: \begin{align*} \mathbf{y}|\mathbf{f} &\sim \mathcal{N}(\mathbf{f}, {\sigma_n}^2\mathbf{I}) \\ p(\mathbf{y}|\mathbf{f}) &= \mathcal{N}(\mathbf{f}, {\sigma_n}^2\mathbf{I}) \end{align*}

How does it relate to the typical definition of pdf of a Gaussian distribution:

Normal Probability Density Function $$F(x) = \frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}$$

• @max I have flagged my question to move to stats exchange site since it is more relevant and has a gaussian process tag. Could you help to move it? – paradite Sep 9 '16 at 6:14
• Sorry, I can't help. Flagging might have been the right thing to do. I'm not sure. I think it would be ok for you to copy-and-paste this question to a new question on Cross Validated SE and then delete this one. But I'm not sure. I do not think it should be a big problem though. – Em. Sep 9 '16 at 6:19
• Also, this kind of question is relevant here too. If your question is not getting enough attention, then consider looking at this post: How to grab users' attention on an old question? – Em. Sep 9 '16 at 6:23
• @Max thanks for the info! – paradite Sep 9 '16 at 6:24