Random variables can be transformed through a function, suppose that we have a random variable X and
it's associated PDF.
It is possible to apply a mathematical function to the BBF to find the transform PDF for the transform
random variable.
So if we have a random variable X and we have the associated PDF, if X, it is possible to apply a
mathematical function so Y easily equals G of X to the PDF to find the PDF of the transform variable
Y, which is the fly.
So let's assume the transfer functions are monotonic so they can be monotonically increasing or decreasing.
It doesn't matter as long as this transformation function, this why of G of X is monotonic.
So we have the transformation function.
Why is equal G of X..
Now we want to find the inverse mapping, so we want to find the function, the inverse G that maps
from the Y variable to back to the X variable.
And we're going to rename this inverse mapping the inverse G into H.
We then can express the transformation of the PDF using this function here, so we won't go through
the derivation of this function, but this is the function that will transform the PDF effects into
the PDF if y so this takes the random variable X and works out the random variable Y and the associated
the PDF of it.
So you can say it's pretty much the derivative of the inverse function, the absolute value of it.
Thom's pi the PDF for the X variable with the function substituted into it.
So using this relationship here, we then can look into the example of a linear transformation of a
Gaussian PDF and calculate the result.
So let's suppose we have a random variable X and we know that it is the normal distribution or Gaussian
distribution and it has a mean of X bar and a variant of Sigma X squared.
We know from the previous video that this here is a function of the PDF for this normal distribution.
So we have the main here and the variance here.
So now suppose that we have the transformation and we want to do a linear transformation.
So we have A X plus B equals Y.
So this is going to be a linear transformation function that we're going to apply to this random variable.
And in this of course, this coefficient A and B have to be real numbers and A cannot be equal to zero.
Otherwise, it's not a transformation.
So the first thing we want to do is to work out the inverse G mapping function.
So that's just going to be Y minus B divided by A.
So this is going to be our investment in function and we're going to define it to be hech.
Now we know that we need to get the derivative of its derivative have to exist to be able to do this
mapping and the derivative of the hedge function is simply just one over a that's fairly straightforward
from first order of differentiation.
So using the relationship that we presented on the previous slide, we can now fill in all this other
information.
So first we know that the derivative is just one Ivereigh, so we can fill that in here.
Next, we want to work out the second time here so the PDF for X with the function substituted in.
So what we can do is we can take our hedge function substituted in for X into the X PDF.
Once we've done that substitution we can then take the whole function and fill it in for this term here.
So if we do this, we end up with this equation here.
So this is the transformation and this gives us the PDF for the Y random variable after we do the transformation.
So you can see that this function here looks very similar to this function over here.
So in fact, they're both Gaussian distributions.
The only difference you can see is that we've changed the meaning of the Gaussian distribution and we've
also changed the variance of the Gaussian distribution.
So equating the two parameters, we can work out the new meaning of this random variable y.
It's just a time to previous mean plus B, so this is basically just doing the transformation on the
main.
So it's fairly straightforward.
Also, we can see the variance here is just being scaled by the scale factor, eh.
So the new variance for the Y pdf is just going to be a squared time.
The variance.
So we can see that the linear transformation of a Gaussian PDF is just another Gaussian PDF with the
mean and variance transformed.
So here's the original PDF of of main exposure and variance Sigma X, we apply the mapping function
of A, X plus B, so a linear transformation to get the new wife the random variable.
All we have to do is we just have to transform the main using this equation here and transform the variance
using this equation here.
So this is a major take away from a Bayesian probabilistic estimation.
A linear transformation of Gaussian PDF is just another Gaussian PDAF.
If we can express the PDF as a Gaussian, then we can dramatically simplify the mathematics involved
in estimation when we're working with probabilities in particular, we won't we will not have to carry
out integrations or payday's to a probability which may or may not have close form solutions.
So this is why when we're doing probabilistic data fusion, that ideally we'd like to use Gaussian distributions
because we can apply these simple transformations and find the results.
So let's suppose we have a random vector X and this random vector is pulled from a multidimensional
Gaussian distribution with a of our exposure and covariance of C of X, let's suppose that we want to
transform this Gaussian distribution using the linear transform of X plus B, so A, here is going to
be a transformation matrix multiplied by our random vector plus an offset vector.
So this function here, we're going to call it G.
This is the transformation that we want to apply to this Gaussian distribution up here.
So the first step is to work out our inversed mapping function, so g inverse, and that's just going
to be our inverse, a Thom's Y minus our inverse A times B, so this gives us our inverse mapping function
and then we can differentiate this mapping function with respect to Y just to end up with our inverse.
A Now we can use the relationship that we've covered in the past, which allows us to perform a mathematical
transformation of a probability density function from one to the other.
So it allows us to apply a function to the density function.
So filling in the terms here, we know the probability density function is going to be our equation
for our gorshin multidimensional distribution, we have a derivative of our investment function G.
So if we fill all this in, we end up with this equation here.
So we find out this is a probability of the multidimensional Gaussian distribution after is undergone
our transformation of X plus B..
So now inspecting these terms here, we can have a look.
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
also know that the covariance of this relationship, of this multidimensional Gaussian distribution
is now going to be this term here.
So this new covariance of the covariance of Y is just going to be a times OK, variance, matrix times
array transpose.
So by looking at these terms, we can see that the linear transform of a multidimensional Gaussian distribution
is just again a linear transform of the main and covariance parameters.
So the linear transform of Gaussian pdf is just another Gaussian UPDF with the mean and variance transformed.
So if we have this random variable vector X here and it comes from a multidimensional Gaussian distribution
described by these parameters here, and we want to apply a linear transformation of X plus B, we end
up with this Gaussian distribution described by these parameters here.
So.
So we just transform the main and we just transform the covariance using this relationship.
So this is pretty important, it tells us if Sea X represents the uncertainty covariance, then it can
be transformed into another frame using the linear transformation Y is equal to X..
So if we have the covariance in frame X and we want to transform it to frame Y, and we know the transformation
matrix, we can transform the covariance using this relationship here.
So covariance and why is this going to be a times covariance in X times transpose?
So we're going to use this transformation relationship quite a lot later on inside our computer.
So this is an important relationship and an important outcome of this process here.
So if we have a random variable vector X and it undergoes a transformation and we know the original
covariance before the transformation, we can work out the uncertainty after the transformation so we
can transform the uncertainty from one frame to other.
Just like that, we can transform a vector from one frame to another.