- #1
Master1022
- 590
- 116
- Homework Statement
- If we have ## 0 < x_1 < \infty ## and ## 0 < x_2 < \infty ## and the transformations: ## y_1 = x_1 - x_2 ## and ## y_2 = x_1 + x_2 ##, find inequalities for each of ##y_1## and ##y_2##
- Relevant Equations
- Inequalities
Hi,
This is as part of a larger probability change of variables question, but it was this part that was giving me problems.
Question: If we have ## 0 < x_1 < \infty ## and ## 0 < x_2 < \infty ## and the transformations: ## y_1 = x_1 - x_2 ## and ## y_2 = x_1 + x_2 ##, find inequalities for each of ##y_1## and ##y_2##
Attempt:
Is there a general method to do these? The answer seems a bit arbitrary and was basically just stated and wasn't obvious to me how it was arrived at. Here is how I would attempt this:
0 < x_2 < \infty \rightarrow 0 < \frac{y_2 - y_1}{2} < \infty
and we can use the left inequality to get ## y_1 < y_2 ## and the right one to get ## y_2 < y_1 + \infty \rightarrow y_2 < \infty ##. Combining these give ## y_1 < y_2 < \infty ##.
Now for ## y_1##:
0 < x_1 < \infty \rightarrow 0 < \frac{y_1 + y_2}{2} < \infty
which we can break up into the left and right parts ## 0 < \frac{y_1 + y_2}{2} ## and ## \frac{y_1 + y_2}{2} < \infty ##. After subtracting ## y_2 ## from both sides, these yield ## -y_2 < y_1 ## and ## y_1 < \infty - y_2 \rightarrow y_1 < \infty ##. However, we can further restrict the upper bound of ## y_1 ## by using the inequality for ## y_2## to give: ## -y_2 < y_1 < y_2 ##.
How do I know whether this is sufficient for a solution (if this is even correct)? It just seems a bit arbitrary to me...
Any help would be greatly appreciated.
This is as part of a larger probability change of variables question, but it was this part that was giving me problems.
Question: If we have ## 0 < x_1 < \infty ## and ## 0 < x_2 < \infty ## and the transformations: ## y_1 = x_1 - x_2 ## and ## y_2 = x_1 + x_2 ##, find inequalities for each of ##y_1## and ##y_2##
Attempt:
Is there a general method to do these? The answer seems a bit arbitrary and was basically just stated and wasn't obvious to me how it was arrived at. Here is how I would attempt this:
0 < x_2 < \infty \rightarrow 0 < \frac{y_2 - y_1}{2} < \infty
and we can use the left inequality to get ## y_1 < y_2 ## and the right one to get ## y_2 < y_1 + \infty \rightarrow y_2 < \infty ##. Combining these give ## y_1 < y_2 < \infty ##.
Now for ## y_1##:
0 < x_1 < \infty \rightarrow 0 < \frac{y_1 + y_2}{2} < \infty
which we can break up into the left and right parts ## 0 < \frac{y_1 + y_2}{2} ## and ## \frac{y_1 + y_2}{2} < \infty ##. After subtracting ## y_2 ## from both sides, these yield ## -y_2 < y_1 ## and ## y_1 < \infty - y_2 \rightarrow y_1 < \infty ##. However, we can further restrict the upper bound of ## y_1 ## by using the inequality for ## y_2## to give: ## -y_2 < y_1 < y_2 ##.
How do I know whether this is sufficient for a solution (if this is even correct)? It just seems a bit arbitrary to me...
Any help would be greatly appreciated.
