I don't think the Probability and Random Variables class I followed really touched on this topic at all.
I wrote a Mathematica script to compute moments of a univariate probability distribution via the MGF (it's just the left half of the image)...
I'm aware that there is higher level functionality in Mathematica; the idea was half to make a tool to quickly find moments and half to review using the MGF. I started a parallel version for bivariate (to later be extended to general multivariate) probability distributions (the right half of the image). It has a small bug (not really the point of this thread, but I have to run the code twice to get the correct output after resetting the Mathematica Kernel), but otherwise finds the MGF.
My question is what to do next, in parallel to the univariate case. The space of moments of bivariate distributions seems to be richer than of univariate distributions. I have found a notion of cross-moments apparently generalizing univariate moments, each of which seem to be computed by taking a (non-strictly) mixed partial derivative of the MGF and setting all its variables to 0. But I take it this generalization is not the whole picture, as each cross-moment is scalar-valued, and the mean and variance of a bivariate distribution are a vector and a matrix, respectively. Since the MGF is scalar-valued after ironing out all the dot products and matrix-vector multiplication, the intuitive idea would be to take the gradient of the MGF and set all its variable to 0 to find the mean. If so, perhaps the pattern is that the cross-moments (including non-mixed univariate moments as a special case) are the elements of the bivariate distribution moments. Looking at the second bivariate distribution moment then (the covariance matrix), obviously its variance elements are univariance moments, and perhaps it is the case that its covariance elements are also cross-moments.
How close am I to the right track? How can I compute bivariate moments using the MGF? Is this going to require knowing tensor calculus to go beyond the second moment? What about if I stick to two moments, but extend to the general multivariate distribution case?
I wrote a Mathematica script to compute moments of a univariate probability distribution via the MGF (it's just the left half of the image)...
I'm aware that there is higher level functionality in Mathematica; the idea was half to make a tool to quickly find moments and half to review using the MGF. I started a parallel version for bivariate (to later be extended to general multivariate) probability distributions (the right half of the image). It has a small bug (not really the point of this thread, but I have to run the code twice to get the correct output after resetting the Mathematica Kernel), but otherwise finds the MGF.
My question is what to do next, in parallel to the univariate case. The space of moments of bivariate distributions seems to be richer than of univariate distributions. I have found a notion of cross-moments apparently generalizing univariate moments, each of which seem to be computed by taking a (non-strictly) mixed partial derivative of the MGF and setting all its variables to 0. But I take it this generalization is not the whole picture, as each cross-moment is scalar-valued, and the mean and variance of a bivariate distribution are a vector and a matrix, respectively. Since the MGF is scalar-valued after ironing out all the dot products and matrix-vector multiplication, the intuitive idea would be to take the gradient of the MGF and set all its variable to 0 to find the mean. If so, perhaps the pattern is that the cross-moments (including non-mixed univariate moments as a special case) are the elements of the bivariate distribution moments. Looking at the second bivariate distribution moment then (the covariance matrix), obviously its variance elements are univariance moments, and perhaps it is the case that its covariance elements are also cross-moments.
How close am I to the right track? How can I compute bivariate moments using the MGF? Is this going to require knowing tensor calculus to go beyond the second moment? What about if I stick to two moments, but extend to the general multivariate distribution case?