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 F.
Why.
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 they call 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.
You can look at the associated notes if you want to go into more detail, 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 had 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 name 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 function is simply just one over a that's fairly straightforward from
first order 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 everyday, 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 Hej function substituted in for X into the Expedia.
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 all 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 for a Bayesian probabilistic estimation, a linear transformation of a Gaussian
PDF is just another Gaussian PDF.
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.