Solving an inequality for a change of variables

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  • #1
Master1022
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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:
[tex] 0 < x_2 < \infty \rightarrow 0 < \frac{y_2 - y_1}{2} < \infty [/tex]
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##:
[tex] 0 < x_1 < \infty \rightarrow 0 < \frac{y_1 + y_2}{2} < \infty [/tex]
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.
 

Answers and Replies

  • #2
anuttarasammyak
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At first for each independently
[tex]?<y_1<?[/tex]
[tex]?<y_2<?[/tex]
Then compare ##y_1## and ##y_2##.
 
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  • #3
PeroK
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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:
[tex] 0 < x_2 < \infty \rightarrow 0 < \frac{y_2 - y_1}{2} < \infty [/tex]
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##:
[tex] 0 < x_1 < \infty \rightarrow 0 < \frac{y_1 + y_2}{2} < \infty [/tex]
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 perhaps the wrong approach - and may lead you round in circles. This is a case where you are better to see/guess the answer and then demonstrate that it is the case.

Can you see/guess what the answer must be? And then justify it?
 
  • #4
Master1022
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This is perhaps the wrong approach - and may lead you round in circles. This is a case where you are better to see/guess the answer and then demonstrate that it is the case.

Can you see/guess what the answer must be? And then justify it?
Okay thank you for the reply @PeroK . The answer reads:
"Firstly we know that:
[tex] 0 < y_1 + y_2 < \infty [/tex]
[tex] 0 < y_2 - y_1 < \infty [/tex]
This can be re-written as:
[tex] -y_2 < y_1 < y_2 [/tex]
[tex] 0 < y_2 < \infty [/tex] "

So not the most descriptive solution... Also, the wider problem was this (shown in image below), but it was only the "finding the support of y1 and y2" part I was struggling with - I am not sure if that helps to provide more context.

Screen Shot 2021-06-05 at 9.04.26 AM.png
 
  • #5
Master1022
611
116
At first for each independently
[tex]?<y_1<?[/tex]
[tex]?<y_2<?[/tex]
Then compare ##y_1## and ##y_2##.
Thanks for your reply @anuttarasammyak . I am not sure how I could obtain expressions independently in this case? Could you give an example of how I could obtain an inequality for ##y_1## different to what I had?
 
  • #6
anuttarasammyak
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Try ##y_2## first. Is it positive or negative ?
 
  • #7
Master1022
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Try ##y_2## first. Is it positive or negative ?
positive. The upper bound is ## \infty ##. Then perhaps by adding two inequalities, I can get ## 0 < 2 y_2 \rightarrow 0 < y_2 ##. Then if I wanted to try to something for ## y_1 ##, I am not sure what to do (other than what I previously did), because I am not completely sure how subtracting the inequalities work
 
  • #8
anuttarasammyak
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OK. Do you see minimum or maximum of ##y_1## ?
 
  • #9
PeroK
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@Master1022 so far from being pre-calculus mathematics, this is a much more advanced question on probability distributions!

Your approach still amounts to going round in circles. I've give you the answer to your initial question, as that is trivial compared to the actual question you are to attempt:$$0 < y_2 < \infty, \ \ -\infty < y_1 < y_2$$You should try to justify that answer if you can.
 
  • #10
Master1022
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@Master1022 so far from being pre-calculus mathematics, this is a much more advanced question on probability distributions!

Your approach still amounts to going round in circles. I've give you the answer to your initial question, as that is trivial compared to the actual question you are to attempt:$$0 < y_2 < \infty, \ \ -\infty < y_1 < y_2$$You should try to justify that answer if you can.
Okay thank you, I will aim to justify that answer. Apologies, yes I wasn't too sure where to put this question because I wasn't going to ask about the calculus parts. However, next time I have a question from this topic, I will put it into the calc. and beyond section
 
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  • #11
Master1022
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OK. Do you see minimum or maximum of ##y_1## ?
From the ## 0 < y_2 - y_1 ##, I can see a maximum for ## y_1 ## of ## y_2 ## (i.e. ## y_1 < y_2 ##)
 
  • #12
anuttarasammyak
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I said estimate them independently first. Then compare. Step by step.
 
  • #13
Master1022
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I said estimate them independently first. Then compare. Step by step.
The upper bound for ## y_1 ## is ## \infty ##.
 
  • #14
Master1022
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I just had a go at graphing the four inequalities (with ##y_1## and ##y_2##) and those help me see the limits more clearly. I think that helps to double check the seemingly arbitrary expressions.
 
  • #15
anuttarasammyak
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OK. And the lower one ?

[EDIT]
[tex]-\infty<y_1<\infty[/tex]
[tex]0<y_2<\infty[/tex]

[tex]y_1<y_2[/tex]
also
[tex]-y_1<y_2[/tex]
So in one
[tex]|y_1|<y_2[/tex]
You were already near to it in post #4.
 
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  • #16
FactChecker
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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.
Your work and answer look good to me. It is systematic and thorough.

Just to be sure, I ran a little simulation using random values for y1 and y2 and the satisfaction of the y constraints match the satisfaction of the x constraints.
 

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