In this video, I would like to remind you key properties of normal distribution.

This will be needed in the next videos.

The normal or Gaussian distribution is a continuous probability distribution.

The probability density in

one dimensional case is defined by the equation shown on the slide.

In this equation, x is the random value,

Mu is the mean of the distribution,

and sigma squared is the variance or just sigma is the standard deviation.

Examples of normal distribution with zero mean and

different sigmas are shown in the figure on the slide.

Sigma defines width of the distribution.

The larger the sigma value the more wider the distribution.

These examples are with the same sigma and three different mean values.

So, mean of the distribution defines position of it.

Let's complicate the distribution.

The multivariate normal or Gaussian distribution is

a generalization of the one-dimensional normal distribution to higher dimensions.

The probability density in N dimensions is defined by

the equation of the slide where vector x is a N-dimensional random vector,

Mu is a N-dimensional mean vector,

and sigma is a covariance matrix of the distribution with size N to N.

This is the same distribution as on

the previous slide but with more details about its parameters.

Mean vector Mu is the mean of input vector x.

And each element of the covariance matrix represents dependency between

two points x_i and x_j as it's shown on the slide.

Consider the 2D example with the following mean vector and the covariance matrix.

The mean vector defines the position of the center of

the distribution and the covariance matrix defines its width.

Covariance matrix also defines shape of the distribution

and its orientation as it's shown on this slide, in this example.

In the next videos,

we will use properties of conditional distribution.

Consider the multivariate normal distribution shown on the slide.

Suppose that the input vector consists of the two vectors,

x_a and x_b, with size k and N minus k, respectively.

Also, suppose that the Mu vector also consists of the two vectors,

and the covariance matrix has the following structure shown on the slide where

each element of these matrix is also a matrix.

In this case, the conditional distribution is normal that

it defines by its mean vector and the covariance matrix.

They are estimated by the equations shown on the slide.

One more property we need is that

the marginal distribution of the normal distribution is also normal.

And its parameters estimated as it's shown on the slide.

All these properties of the normal distribution,

we will use in the next video about Gaussian processes for regression.