The Gaussian distribution has many interesting properties, many of which make it useful in various different applications. Before moving further, let us just define the univariate PDF with a mean $\mu$ and variance $\sigma^2$
$$ \mathcal{N}(x | \mu, \sigma^2) = \frac{1}{\sqrt{2 \pi \sigma^2}} \exp \left( -\frac{(x - \mu)^2}{2 \sigma^2} \right). $$
In the general multi-dimensional case, the mean becomes a mean vector, and the variance turns into a $D \times D$ covariance matrix.
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