Suppose that X1, X2 are random variables with given probability distributions fx1(x), fx2(x).
Let fy(x) = fy( fx1(x) , fx2(x) ) be a known probability distribution of "unknown" random variable Y. Is it possible to determine how the variable Y is related to X1 and X2?
Excellent question. Usually we know how Y is related to X1 and X2, we know fx1(x1) and fx2(x2) and the question what is fy(y). Even, this is found for simple cases. For example when Y=X1+X2 the pdf of Y or fy(y) will be the convolution of the two pdfs. Also, when we have Y=X1/X2 the resulting fy(y) is some integral utilizing the Jacobian matrix (includes derivatives of x1 and x2 w.r.to Y) (see http://en.wikipedia.org/wiki/Probability_density_function#Sums_of_independent_random_variables). This is the usual problem. It means that we know the system and our inputs (their pdfs) hence we can find out the output(s). Knowing the system means that we know how X1 and X2 interact to produce Y (whether it X1+X2 or X1/X2 or something else).
In your question, which is again a good one, you want to reverse the problem, i.e. let us assume that we know the output (its pdf) and we would like to know how the inputs X1 and X2 are related. In other words you want to know whether it is X1+X2 or X1/X2 or something else. In other words, again, you want to know the system (in a black box) by measuring its output (assuming also you know the inputs, or their pdfs - you didn't say that, but I would assume that or otherwise it gets too far). If we know how to do that it would be wonderful. To find out if that possible with random variables let us first check if that possible with deterministic variables. Assume that we have a black box with 2 inputs and one output, we measured the output and we found it 4 and measured the inputs and we found them both equal 2. Hence, we can assume that Y=X1+X2 or Y=X1*X2 or we can find myriads of such relations (say Y=X1^3/X2, Y=2X1-X2, ...etc). One might argue that continuous monitoring of inputs and outputs could lead to a clue on the relationship. For example if two (deterministic) sinusoidal signals are imputed we may tell from monitoring the output what the relationship might be (e.g. is it a summation or multiplication) but this can get very complicated or impossible if we have all options valid. Hence, generally, no clue or very complicated. I would say if the problem is such in a deterministic world, how would it be if the inputs were random variables?. It must be near to impossible. I would say again, if you managed to reverse the problem, it would be something really cool, you would know what should be the system inside to satisfy a certain output pdf.
On the other hand if you relaxed your problem more so that whether knowing a Pdf (a required one) of an output can be used to tune a partially already known system, e.g to find optimum "a" value when Y=a.X1+X2 so as to attain the pdf of y (e.g. to set its mean or variance) then this can be negotiable and it is done (Feedback systems to control gain for example so as the output has certain value). Analysis of such problems assume deterministic values (or deterministic values with some additive noise/ripple signals). I haven't come across such analysis for a more general pdf cases but would think that it can be done in special cases. Thanks. @AlDmour.
Dear Dr. Adam Rudziński, I can add the following link, may be of your help.
http://en.wikipedia.org/wiki/Random_variable
Dear Adam,
You may get all necessary details and codes from a well-known book "Numerical Recipes" (in C or Fortran) by William H. Press et al. Cambridge University Press. First edition has been published in 1988; second in 1992. There is section number 7 - Random Numbers.
can you show me a program on matlab entering the probability law and give me a random variable following this law?
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