And then to denote a Gaussian distribution,

sometimes I'm going to write script N parentheses mu comma sigma script.

So this script N stands for normal since Gaussian and

normal they mean the thing are synonyms.

And the Gaussian distribution is parametarized by two parameters,

by a mean parameter which we denote mu and

a variance parameter which we denote via sigma squared.

If we plot the Gaussian distribution or Gaussian probability density.

It'll look like the bell shaped curve which you may have seen before.

And so

this bell shaped curve is paramafied by those two parameters, mu and sequel.

And the location of the center of this bell shaped curve is the mean mu.

And the width of this bell shaped curve, roughly that,

is this parameter, sigma, is also called one standard deviation,

and so this specifies the probability of x taking on different values.

So, x taking on values here in the middle here it's pretty high,

since the Gaussian density here is pretty high, whereas x taking on values further,

and further away will be diminishing in probability.

Finally just for completeness let me write out the formula for

the Gaussian distribution.

So the probability of x, and I'll sometimes write this as the p (x)

when we write this as P ( x ; mu, sigma squared), and so this denotes that

the probability of X is parameterized by the two parameters mu and sigma squared.

And the formula for the Gaussian density is this 1/ root 2 pi,

sigma e (-(x-mu/g) squared/2 sigma squared.

So there's no need to memorize this formula.

This is just the formula for the bell-shaped curve over here on the left.

There's no need to memorize it,

and if you ever need to use this, you can always look this up.

And so that figure on the left, that is what you get if you take a fixed value of

mu and take a fixed value of sigma, and you plot P(x) so this curve here.

This is really p(x) plotted as a function of X for

a fixed value of Mu and of sigma squared.

And by the way sometimes it's easier to think in terms of sigma squared that's

called the variance.

And sometimes is easier to think in terms of sigma.

So sigma is called the standard deviation, and

so it specifies the width of this Gaussian probability density,

where as the square sigma, or sigma squared, is called the variance.