A question I couldn't find a reference for. Any hint or (book)reference would be very appreciated.The problem reads:I have a few random variables, say, x_1, x_2, x_3, x_4, and a funcion that maps them into R i.e.: f: (x_1, x_2, x_3, x_4) --> R. The question is: is it possible to derive the density of the resulting random variable f(x_1, x_2, x_3, x_4) based on the densities of the "input" random variables. Does it simplify the whole matter if the X_i are iid? I guess it does. How to proceed in this case then?Cheers,JPBD

Finding a joint distribution function given marginal distributions is not a simple problem. Of course if you have them to be iid then the joint distribution is just the product of the individual marginal distributions. If they are not then there are two problems to solve. Problem 1: In what way are they dependent? An assumption most people make to begin with is that there is some sort of a correlation matrix describing the dependency.Problem 2: Given the dependency, does a joint distribution even exist for the combination of marginals and the correlations? IF yes, do we have a closed form solution ? One way to tackle this problem is using families of joint distribution functions such as the Farlie Gumbel Morgernstern(FGM) family of joint distributions. The trick however is that the alphas which are used as parameters can capture pairwise correlations only in the range -1/3 to 1/3 I found someones masters thesis that deals with bivariate distribution functions. it has some useful ideas. Construction of Bivariate DistributionsHope this helps

the problem is not so difficult ifthe marginals are normalor uniform ( atleastfor the bivariate case but maybe beyond bivariate , it is also difficult ) but, for other distributions besides those two, i am unawareof a general technique.

Thanks for your reactions. In the meantime I have realized that the problem at hand is not trivial at all. I might have missexplained it: actually the resulting rv is not a multivariate one, but a scalar one. For expample: 1) given x1, x2, x3 iid, what is the distribution of X1+x2+x3?2) given x1, x2, x3 iid, what is the distribution of (X1)^2 + (X2)^2 + (X3)^2?3) given x1, x2, x3 iid, what is the distribution of (X1)^2*(X2)^2*(X3)^2?4) given x1, x2, x3 iid, what is the distribution of log[ (X1)^2*(X2)^2 +(X3)^2] ?...etcWhat is behind 1-4 is a function mapping R*R*R into R i.e. the new/generated random is scalar.Solutions for 1-2 are well-known if, for example, Xi are (standard) gaussian. But...a) how about 1-2 when the density is known but arbitrary?b) What if the underlying function gets more complicated / is arbitrary?c) and, even worse, can we say anything if both the density is known but arbitrary and the mapping R*R*R -> R is arbitrary (say, like in 3-4) as well.So far I have found nothing on textbook so as to 'solve' this problem....is this an "unsolved" problem in general terms?Cheers,Jpbd

there is no "general technique"for arbitrary distriibutionsbut techniques can be appliedonce the distribution isgiven, depending on what the distribution is.the sum can be simple forcertain distributions ( normal, exponential )and the sum of the squared can alsobe simple for certain distributions ( normaland maybe others, for example, the t distributioni think ) but there is no general 'technique",i don't think, particularly when you start doingstrange things like 3) and 4).it's not clear to where 3) and 4 ) would arisein practice so the problem probably hasn't beenworked on that vigorously for something like3) or 4).

I think you have to use copulas in here. Copula 'joins' together marginals.the joint density f_{1234} looks like:f_{1234}=f_1*f_2*f_3*f_4*c(F_1,F_2,F_3,F_4)where f_i are densities of X_i and F_i are the distributions and c a copula!

MissWawrzyniak - you are correct!Given the marginal distriutions there is not enough information given to determine the joint distribution, which is what you need to answer questions about arbitrary multivariate functions.You have to add information to the problem in the form of an assumed copula function, which you can then use to create the joint density.Sklar's theorem guarantees that if a joint distribution exists and the marginal distributions exist, then a unique copula exists. I think Sklar may have actually called it a subcopula ...In practise, there tends to be fairly synchronous historical data available, and there are methods you can use to decide upon a copula function. So the info you need to add back in can come from historical data.A correlation matrix is also not enough information to add in. You still need the copula function assumption.I have worked in finance since 1989 (mostly as a derivs trader) and I first heard of copulas in 2002. Amazing it is not more common knowledge. It should be taught in Quantitative Finance 101! (Well, maybe 102).