Transformations of variables support set

In summary, when defining support sets for transformed variables, it is important to consider the range of the original variables and ensure that the new variables also fall within that range.
  • #1
libragirl79
31
0
Hi,

I understand the transformations of variables concept, getting the Jacobian and so on, but I am having trouble with defining the support sets. For example, given that I have a joint pdf of x and y to be xy, and a new variable U=X/Y and V=XY, I get the X=sqrt(UV) and Y=sqrt(V/U), where x and y are from zero to one, (the joint being then f(u,v)=v/2u)
So I set it up: 0 < uv < 1 and 0 < v/u < 1, and combining the two, I get
0 < v < u < 1/v , but I don't think this is right...

Thanks for any help/input/suggestions!
 
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  • #2
The support set for the joint pdf of U and V should be 0 < U < 1, 0 < V < 1. This is because the original x and y are both from 0 to 1 and the new variables U and V must also follow this range in order to satisfy the original equation.
 

Related to Transformations of variables support set

1. What is a transformation of variables support set?

A transformation of variables support set refers to the range of values that a random variable can take after a mathematical transformation has been applied to it. This transformation can change the distribution of the variable and therefore, its support set.

2. Why is it important to consider transformations of variables support set?

It is important to consider transformations of variables support set because it can affect the validity and interpretation of statistical analyses. In particular, when conducting hypothesis testing, the assumption of normality is often made on the transformed data, which requires knowledge of the transformed support set.

3. How do you determine the support set of a transformed variable?

The support set of a transformed variable can be determined by first identifying the range of values that the original variable can take, and then applying the transformation function to each of these values. The resulting values will make up the support set of the transformed variable.

4. Can a transformation of variables change the support set from discrete to continuous or vice versa?

Yes, a transformation of variables can change the support set from discrete to continuous or vice versa. For example, if a variable is transformed using a logarithmic function, it can change the support set from discrete to continuous. Conversely, if a variable is transformed using a rounding function, it can change the support set from continuous to discrete.

5. Are there any guidelines for choosing an appropriate transformation of variables?

There are no strict guidelines for choosing an appropriate transformation of variables. However, some common transformations include logarithmic, exponential, and power transformations. It is important to consider the properties of the data and the research question when selecting a transformation.

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