Timeline for Monotonic Transformation Preserving Probabilities Intuition
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| Dec 11 at 20:00 | comment | added | whuber♦ | It might help to recognize this isn't about differentiability or continuous distributions or even about probability per se: it's set theory. The event described by "$X\gt a$" is the same as the event described by "$f(X)\gt f(a)$" when $f$ is strictly monotonic and increasing--and that's a direct, immediate consequence of the definition. There's nothing more to say! | |
| Dec 11 at 18:03 | comment | added | CormJack | @whuber okay in its most strait forward form. My understanding particularly from the attached image at the end is that when $f$ is monotonic $p(X > a) = p(f(X) > f(a))$ where $Y = f(X)$. I understand this to mean probabilities don’t change under monoatomic transformations. 1) I was struggling to understand this intuitively in the continuous case. I felt for example intervals of smaller numbers would be stretched less by say $x^2$ than larger intervals. I’m just missing a clear intuition for why probabilities are preserved. 2) Your understanding of $p(X > a) = p(f(X) > f(a))$ | |
| Dec 11 at 15:14 | comment | added | whuber♦ | It's not quite clear to me what you are asking, because the highlighted materials are a matter of definition and notation: the first highlight is the definition of "monotonically increasing" and the second is the fact that the probability of an event doesn't depend on how it is mathematically described. | |
| Dec 11 at 11:26 | comment | added | CormJack | @whuber are you able to help me understand your comment note in light of the image I’ve attached at the end of the post, and also the other comments which seem to support some notion of monophonic transformations preserving probabilities. Apologies if my terminology is off, I’d greatly appreciate your insight when you get a sec! | |
| Dec 6 at 2:42 | answer | added | Aksakal | timeline score: 0 | |
| Dec 6 at 1:53 | comment | added | CormJack | Thanks for your comments as always Whuber. I agree with both points and I have emended the end of the question. I think the example I have provided captures the idea I was trying to get and failing. it seems like there is some relative preservation? Again I’m not sure so your thoughts interpreting the attached, and the difference between this and preserving probabilities would be appreciated. Thanks! | |
| Dec 6 at 1:50 | history | edited | CormJack | CC BY-SA 4.0 |
added 350 characters in body
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| Dec 6 at 1:25 | comment | added | CormJack | Hi Ben, thank you very much for your answer! Will digest it soon. Apologies, I did cross post this. Sometimes the forums have been very slow so I was trying to maximise my chances, and honestly the range of different answers is really helpful For finding that thing that “clicks.” | |
| Dec 5 at 1:14 | comment | added | Ben | This question is cross-posted at SE.math (math.stackexchange.com/questions/5007245). | |
| Dec 5 at 1:13 | answer | added | Ben | timeline score: 3 | |
| Dec 4 at 21:45 | comment | added | whuber♦ | Your question (1) doesn't even employ a transformation, making it difficult to interpret. Maybe you are looking for something like stats.stackexchange.com/questions/14483? Regardless, monotonic transformations generally do not preserve probabilities or even probability densities. | |
| Dec 4 at 21:19 | history | asked | CormJack | CC BY-SA 4.0 |