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In a past video, we've described the Gaussian distribution for a single random variable so we can have

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a normal or Gaussian distribution that has a mean of you and a standard deviation of sigma squared.

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And this gives us a probability density function as shown on this equation here.

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So this describes the gasoline distribution for a single random variable.

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We can extend the Gaussian distribution from a one dimensional example using a single random variable

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into a higher order, Gaussian distribution using a random vector.

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So now if we use a random vector, we can describe the distribution.

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So the Gaussian Egyptian based on our main vector X bar and covariance matrix C of X.

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And when we do this, we get this equation here for the Gaussian distribution.

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So for a two dimensional Gaussian distribution, we'll end up with something that looks like this.

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So this is an example, distribution for a third Gaussian.

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So basically, instead of having a single dimension, we now have multiple dimensions and we can work

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out a Gaussian distribution of any dimension that we want.

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So it could be a three dimensional, four dimensional, five dimensional Gaussian.

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It could be any number.

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It just becomes very difficult to visualize higher order Gaussian density functions.

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So in a third case, it's fairly simple.

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In the three case is a 3D ellipsoid, but in the higher order terms, it's very difficult to visualize

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it in a multi dimensional Gaussian distribution.

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The main shifts the center of the distribution.

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The variance controls the spread in the different axes, while the cross covariance is control the orientation

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of the distribution.

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So if we look at a two dimensional example here, so we have a variance in the X and the variance in

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the Y, we have the main being.

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This case can be zero zero.

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So that shifts where the origin is.

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We also have the covariance and the covariance controls the angle in this case for the 2D example of

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the Ellipse here.

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So this ellipse drawing here is going to be the one sigma uncertainty.

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If we look at the two dimensional Gaussian distribution and trace a line where sigma equals one, we

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end up with an ellipse.

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So in this case, we're going to have a non-zero across variances because it's Ellipse is not aligned

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with the X or Y axis, it is shifted or is being rotated.

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So there's some cross correlations between the X and Y axis of this Gaussian distribution.

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The more dimensional Gaussian distribution becomes very important when we start looking at data fusion

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so we can look at a Gaussian distribution to describe the uncertainty of our estimate, where each state

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is basically a random value inside a random variable vector.

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This means we can describe the accuracy of the estimation process using a multidimensional Gaussian

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distribution.