In a past video, we've described the Gaussian distribution for a single random variable so we can have
a normal or Gaussian distribution that has a mean of you and a standard deviation of sigma squared.
And this gives us a probability density function as shown on this equation here.
So this describes the gasoline distribution for a single random variable.
We can extend the Gaussian distribution from a one dimensional example using a single random variable
into a higher order, Gaussian distribution using a random vector.
So now if we use a random vector, we can describe the distribution.
So the Gaussian Egyptian based on our main vector X bar and covariance matrix C of X.
And when we do this, we get this equation here for the Gaussian distribution.
So for a two dimensional Gaussian distribution, we'll end up with something that looks like this.
So this is an example, distribution for a third Gaussian.
So basically, instead of having a single dimension, we now have multiple dimensions and we can work
out a Gaussian distribution of any dimension that we want.
So it could be a three dimensional, four dimensional, five dimensional Gaussian.
It could be any number.
It just becomes very difficult to visualize higher order Gaussian density functions.
So in a third case, it's fairly simple.
In the three case is a 3D ellipsoid, but in the higher order terms, it's very difficult to visualize
it in a multi dimensional Gaussian distribution.
The main shifts the center of the distribution.
The variance controls the spread in the different axes, while the cross covariance is control the orientation
of the distribution.
So if we look at a two dimensional example here, so we have a variance in the X and the variance in
the Y, we have the main being.
This case can be zero zero.
So that shifts where the origin is.
We also have the covariance and the covariance controls the angle in this case for the 2D example of
the Ellipse here.
So this ellipse drawing here is going to be the one sigma uncertainty.
If we look at the two dimensional Gaussian distribution and trace a line where sigma equals one, we
end up with an ellipse.
So in this case, we're going to have a non-zero across variances because it's Ellipse is not aligned
with the X or Y axis, it is shifted or is being rotated.
So there's some cross correlations between the X and Y axis of this Gaussian distribution.
The more dimensional Gaussian distribution becomes very important when we start looking at data fusion
so we can look at a Gaussian distribution to describe the uncertainty of our estimate, where each state
is basically a random value inside a random variable vector.
This means we can describe the accuracy of the estimation process using a multidimensional Gaussian
distribution.