Joint Distribution of Complicated Variables

[raging]

Simple joint distributions such as X+Y are usually worked out in textbooks, but how would we approach a general case. For example, let X any Y be independent variables each defined on the interval [0,infinity], and having densities f(x) and f(y) respectively. How do we find, for example, the joint distributions of (X/Y+1)? Anyone have any lead into?

 

[Stephen Tashi]

but how would we approach a general case.

The general idea for computing the cumulative distribution F(r) = probability that g(x,y) <= r for a function g(x,y) of two random variables would be to examine the implications of of the inequality g(x,y) < = r. For a particular r, you must determine what areas contain points (x,y) such that g(x,y) <= r. Then you must compute the probability that x and y both fall in these areas by integrating the joint density of (x,y) over those areas.

It is not always possible to do this calculation in symbols and get a concise formula for the answer. It may be necessary to use some algorithm to compute a numerical approximation to the answer.

There may be more specialized ways of doing the work when g(x,y) is a special kind of function - like g(x,y) = x + y, which you mentioned.

For example, let X any Y be independent variables each defined on the interval [0,infinity], and having densities f(x) and f(y) respectively. How do we find, for example, the joint distributions of (X/Y+1)? Anyone have any lead into?

Do you mean X/(Y+1) ?

The general method would say to look at the solution set to
X/(Y+1) <= r

Perhaps you can visualize this set by pretending that Y is a constant and looking at the curve of the x's that satisfy X = rY + r. Then decide if all the x's on one side of this curve satsify the inequality. Then visualize the area swept out by the curve as you vary r.

Presumably you get some boundary curve or curves for the area and these curves are a function of r. You must integrate the joint density over the area. The curves would be incorporated into your limits of integration, so the integral woud be a function of r.

I'm not going to tackle those details myself unless there some interesting question about that particular g(x,y)!

 

[raging]

I see. Thanks for the comment! I guess something like X/(Y+1) this just becomes a vector calculus problem. But would you strategy work for non-continuous functions as well? For example, the step function or the min and max functions? How would we decide what g(x,y) is less than if that quantity is always changing?

 

[Stephen Tashi]

How would we decide what g(x,y) is less than if that quantity is always changing?

I don't understand what you mean by that. Can you give an example?