TITLE: normal distribution unknown notation
QUESTION [1 upvotes]: I just stumbled upon some notation that I don't understand, a variable is defined as having a distribution of the following form: 
$\mathcal{N}(a | b, c)$ 
What's what here? Pretty sure $c$ is the covariance, but what does the rest mean? Could it be an error?
In my particular case, it's even weirder, the distribution is actually: 
$\mathcal{N}(a | \mathbf{0}, c)$

REPLY [1 votes]: As you said, $c$ represents the covariance. In your case, $b$ represents the mean. In short, we have:
$$
\mathcal{N}(a|b,c)=\frac{1}{\sqrt{2\pi c}} \exp \left\{ -\frac{(a-b)^2}{2c} \right\},
$$
or, in case $b$ represents a vector and $c$ represents a covariance matrix, we have:
$$
\mathcal{N}(a|b,c)=\frac{1}{\sqrt{2\pi |c|}} \exp \left\{ -(a-b)^Tc^{-1}(a-b) \right\},
$$
where $|c|$ represents the determinant of the covariance matrix.
Sometimes, the $a$ is omitted, such that the normal distribution is written as $\mathcal{N}(b,c)$, see, e.g., wikipedia.
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