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We have seen in the past that the linear transform of a one dimensional Gaussian distribution is just

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another Gaussian distribution, with the mean and variance transformed.

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So if we have a random variable X that belongs to a Gaussian distribution with I mean, of X bar and

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a variance of sigma squared X, if we multiply this Gaussian distribution and transform it by a linear

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transform of A, X plus B, we end up with a new Gaussian distribution with the main shifted based on

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the transformation.

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And the variance is just going to be scaled by the square of a scaling factor.

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Now we can extend this concept to a multidimensional Gaussian as well.

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So let's suppose we have a random vector X and this random vector is pulled from a multidimensional

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Gaussian distribution with Amane of our exposure and covariance of C of X.

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Let's suppose that we want to transform this Gaussian distribution using the linear transform of X plus

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B, so A here is going to be a transformation matrix multiplied by our random vector plus an offset

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vector.

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So this function here, we're going to call it G.

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This is the transformation that we want to apply to this Gaussian distribution up here.

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So the first step is to work out our inversed mapping function, so g inverse, and that's just going

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to be our inverse, a Thom's Y minus our inverse A time is B.

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So this gives us our inverse mapping function and then we can differentiate this mapping function with

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respect to Y just to end up without inverse A..

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Now we can use the relationship that we've covered in the past, which allows us to perform a mathematical

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transformation of a probability density function from one to the other.

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So it allows us to apply a function to the density function.

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So filling in the terms here, we know the probability density function is going to be our equation

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for our Gaussian multidimensional distribution.

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We have our derivative of our investment function.

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So if we fill all this in, we end up with this equation here.

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So we find out this is a probability of the multidimensional Gaussian distribution after is undergone

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our transformation of X plus B, so now inspecting these terms here, we can have a look.

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You can say now this is a function of why we know our Y bar is now just going to be X bar plus B, we

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also know that the covariance of this relationship, of this multidimensional Gaussian distribution

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is now going to be this term here.

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So this new covariance of the covariance y is just going to be a times variance matrix times our transpose.

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So by looking at these terms here, we can see that the linear transform of a multidimensional Gaussian

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distribution is just again a linear transform of the main and covariance parameters.

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So the linear transform of Gaussian PDF is just another Gaussian UPDF with the mean and variance transformed.

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So if we have this random variable vector X here and it comes from a multidimensional Gaussian distribution

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described by these parameters here, and we want to apply a linear transformation of X plus B, we end

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up with this Gaussian distribution described by these parameters here.

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So.

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So we just transform the main and we just transform the covariance using this relationship.

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So this is pretty important.

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It tells us if C X represents the uncertainty covariance, then it can be transformed into a number

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of frame using the linear transformation Y is equal to X..

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So if we have the covariance in frame X and we want to transform it to frame Y, and we know the transformation

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matrix, we can transform the covariance using this relationship here.

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So covariance and why is this going to be eight times covariance in X times a transpose?

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So we're going to use this transformation relationship quite a lot later on inside our common fiodor.

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So this is an important relationship and an important outcome of this process here.

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So if we have a random variable vector X and it undergoes a transformation and we know the original

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covariance before the transformation, we can work out the uncertainty after the transformation so we

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can transform the uncertainty from one frame together.

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Just like that, we can transform a vector from one frame to another.