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?

 

[blue_raver22]

I can provide a response to the concept of joint distribution of complicated variables. To approach a general case, we would need to use advanced mathematical techniques such as integration and probability theory. In the specific example given, we can use the concept of conditional probability to find the joint distribution of (X/Y+1). This involves calculating the probability of (X/Y+1) for different values of X and Y, and then integrating over the range of possible values for X and Y.

We can also use transformation techniques, such as the Jacobian transformation, to convert the joint distribution of X and Y to the joint distribution of (X/Y+1). This would involve using the densities of X and Y to calculate the density of (X/Y+1). However, this approach may be more complex and require advanced mathematical skills.

In addition, we can use simulation techniques to approximate the joint distribution of (X/Y+1). This involves generating a large number of random samples of X and Y, and then calculating the value of (X/Y+1) for each sample. By repeating this process multiple times, we can approximate the joint distribution of (X/Y+1).

Overall, finding the joint distribution of complicated variables requires a combination of mathematical techniques and may also require advanced statistical software. It is important to carefully consider the properties and assumptions of the variables and their distributions in order to accurately determine the joint distribution.