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The normal gasoline distribution is one of the most important probability density functions when it

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comes to data fusion, the distribution is a very natural way of expressing an estimated value and the

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associated uncertainty of the estimate.

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The whole distribution is continuous, which makes it a mathematically nice function to operate with.

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The shape of the distribution is shown here along with its function.

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The complete distribution is fully described by two parameters.

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It is described by the main and the variance, and it's usually notated up here.

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So we have the capital N for the normal distribution has the main and has a variance.

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Now the fact that it's the food distribution is described by only these two parameters to get this complete

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shape makes it a very compact way of describing the distribution.

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As usual, the complete area under the curve under the distribution has to equal one for it to be a

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valid PDF.

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This is why there's a normalization factor out of this distribution as variance of the distribution

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shrinks.

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So as the edges get closer, the likelihood or value of the distribution has to increase to maintain

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the unit area under the curve.

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The majority of the random values produced by the distribution are clustered around the main or the

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peak value, and they become less probable as they move further out either side of the main.

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As the main position shifts the take shifts as a variance, Frinks regrows the spread of the distribution

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shrinks or growers are calling a very useful property of the Gaussian distribution.

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Is the Three Sigma rule the probability that the random value is within one signal of the main.

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So if the value lies in this area, here is 68 percent.

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So if we integrate the PADF between around the main so the main minus one sigma all the way up to the

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main plus one sigma, if we integrate the probability density function, we get point six eight.

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So sixty eight percent.

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Now if we do the same thing for two sigma, so if we integrate the area under the curve between these

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two bands here, we then get a probability of point nine, five or 95 percent.

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And then again, if we do it with three sigma, so we integrate the complete area between these two

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elements here we end up with a probability of point nine, nine or 99 percent.

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This gives us a likely bound on the random number that might be produced by the random variable, it

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conversely also gives us a measure of how likely any value might be consistent with the random variable

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or distribution.

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We call it the three sigma rule, because pretty much any value produced by the distribution is most

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likely to be contained within plus or minus three sigma of the main value.

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And we can be 99 percent confident that this value is.

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One of the most important properties of the Gaussian distribution is the fact that any linear transformation

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of the Gaussian distribution is another Gaussian distribution.

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This will be explored later on in depth as it is a key property for many of the estimation processes

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later on.

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But basically this allows the computationally expensive operations involving convolutions or integrations

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of the probability density functions to be simplified down to a transformation of the main and various

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parameters themselves, rather than carrying out the complete transformation of the Paideia function.

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And again, we will look at this in more detail.

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But this is a very important concept of the Gaussian distribution.

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So now let's have a look at an estimation example to highlight the usefulness of this distribution.

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Imagine that we're trying to estimate a position of a car.

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So the estimation process gives us a Gaussian distribution for the estimate of the car described by

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a mean and variance.

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So the distribution for the position estimate looks something like this and it can be described using

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these numbers here.

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So imagine we have an estimate of the precision.

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It has a main value of one hundred and twenty five meters and a variance of four square meters squared.

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So we know the most likely position of the car.

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And we also know how good the estimate is.

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So using the three sigma rule, we know that the composition has to be within plus or minus three sigma

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of the main and we can be ninety nine point eight percent sure of this confidence.

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This gives us a bounce on a position estimate.

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So therefore the estimated position must be one hundred and twenty five meters, plus or minus three

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sigma or 12 meters.

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So this is a very useful way of expressing an estimated position and it's uncertainty.

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So hopefully you can see why the Gaussian distribution is a very nice way of expressing an estimated

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value and its associated uncertainty, and this is going to be very useful later on when we look in-depth

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at different estimation processes.

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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 or the Gaussian distribution based

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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 is 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.

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In a multidimensional Gaussian distribution, the main shifts the center of the distribution, the variance

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controls the spread in the different axes, while the crosscourt variances control the orientation of

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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.

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We have the main being, this case going to be zero zero, so that shifts where the origin is.

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We also have the covariance and that 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 ellipse.

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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 convergences 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.