Continuity -vs- Discreteness
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From: Huang on 12 Jan 2010 00:48 On Jan 11, 8:50 pm, Huang <huangxienc...(a)yahoo.com> wrote: > On Jan 10, 8:45 am, Huang <huangxienc...(a)yahoo.com> wrote: > > > > > > > On Jan 9, 10:20 am, Huang <huangxienc...(a)yahoo.com> wrote: > > > > On Jan 9, 9:29 am, Huang <huangxienc...(a)yahoo.com> wrote: > > > > > On Jan 8, 6:35 am, Huang <huangxienc...(a)yahoo.com> wrote: > > > > > > On Jan 7, 8:34 am, Huang <huangxienc...(a)yahoo.com> wrote: > > > > > > > On Jan 6, 9:16 pm, Huang <huangxienc...(a)yahoo.com> wrote: > > > > > > > > On Jan 6, 11:25 am, FredJeffries <fredjeffr...(a)gmail.com> wrote: > > > > > > > > > On Jan 4, 8:04 pm, Huang <huangxienc...(a)yahoo.com> wrote: > > > > > > > > > > We might even be able to say that the discrete universe is equivalent > > > > > > > > > to the continuous universe in the same sense that relative motions are > > > > > > > > > equivalent. > > > > > > > > >http://terrytao.wordpress.com/2007/05/23/soft-analysis-hard-analysis-... > > > > > > > > I believe that many of the results of calculus can be replicated using > > > > > > > discrete methods. In fact, I suspect that they all can. > > > > > > > > Lets suppose you could construct two bodies of formal mathematical > > > > > > > models. Calculus on the one hand, and a whole collection of discrete > > > > > > > models on the other hand which model the exact same things. Suppose > > > > > > > for a moment that these two bodies of knowledge are at our disposal. > > > > > > > We might like to show that there are some broad relationships between > > > > > > > these two collections of things. > > > > > > > > To my knowledge....this has not been done. I dont think that it has > > > > > > > even been explored to any great extent, if at all. > > > > > > > > So what the hell are you people waiting for ? Lazy sods ? I explain > > > > > > > all of these insights and you just sit on your fat asses ? Must I do > > > > > > > everything myself ? > > > > > > > To be more specific, I believe that it is possible to construct > > > > > > discrete models and continuous models which both return the same > > > > > > numerical results. It may even be possible to do this in a fairly > > > > > > clever way so that broad relationships can emerge by comparing these > > > > > > collections of things. > > > > > > > Once we have accomplished that, we should be able to say that these > > > > > > things are "equivalent" in the sense of Einstein, and that whether one > > > > > > wishes to model the universe as being discrete or continuous is purely > > > > > > a matter of choice. That the discrete universe and the continuous > > > > > > universe may be very different, but they are "equivalent" because the > > > > > > numbers crunch exactly the same. > > > > > > > I think that this is all possible using standard mathematics, no > > > > > > nonstandard math would be required for this. > > > > > The only reason to even mention these things is because anything that > > > > you can do under the assumption of existential indeterminacy - > > > > likewise must be possible without it. > > > > > The EQUIVALENCE of a continuous spacetime and discrete spacetime is > > > > one of the results obtailable by considering conjectural modelling. > > > > And we can use that result to explain WP duality. But this result > > > > (equivalence) should be obtainable under the assumption of existence > > > > i.e. under standard mathematics. > > > > > Clearly, we should be able to demonstrate such an equivalence in a > > > > very broad number of cases. This is NOT to say that continuity and > > > > discreteness are the same thing, indeed not. Rather, that a result > > > > obtained by using one approach should be deriveable using the other - > > > > to an arbitrary degree of accuracy. And we are using the word > > > > equivalnce in the sense of Einstein, i.e. the Equivalence Principle.. > > > > > Why and how mathematics could progress to the place it has achieved > > > > and not recognize this is quite mysterious to me. Perhaps it is simply > > > > because you cannot "prove" that continuity and discreteness are the > > > > same thing, indeed they simply are not. But Equivalence in the sense > > > > of Einstein is not so strict, all we care about is that the numbers > > > > crunch to the same result with arbitrary accuracy, and I believe this > > > > is possible using standard math. It is certainly the case in > > > > Conjectural Modelling.- Hide quoted text - > > > > > - Show quoted text - > > > > Which leads me to wonder about some criticism of GR which seems to be > > > absent. > > > > One of the early criticisms of calculus was that it dealt with the > > > "ghosts of departed quantities", specifically limits, and how they are > > > treated as if they are numbers. > > > > It seems that the same kind of problem is present when you consider > > > Einsteins Equivalence Principle, but in this case we are talking about > > > the "ghosts of a departed operator". Yes we do have the Lorentz > > > Transform, but what does that tell us anyway ? Does it say that things > > > are "equal" ??? No. It says that things are "equivalent". And > > > equivalence is not the same thing as equality under Einstein's usage > > > of the word equivalence. > > > > One should be able to demonstrate the "equivalence" of continuity and > > > discreteness in many cases, and I think that it is just slightly MORE > > > interesting than the equivalence of relative motions of moving > > > bodies......unless of course we prefer to walk around in a daze for > > > another 100 years under a self imposed state of deliberately chosen > > > confusion (wp duality).- Hide quoted text - > > > > - Show quoted text - > > > Here's my view on unification - based on the aforementioned > > considerations. > > > The universe should be accurately modellable with or without > > probability theory. > > The universe should be accurately modellable with or without > > randomness. > > The universe should be accurately modellable with or without > > existential indeterminacy. > > The universe should be accurately modellable as being either discrete > > or continuous. > > The universe should be accurately modellable as either deterministic > > or non-deterministic. > > > In my opinion, a model which successfully unifies physics should > > satisfy these requirements. > > > The problem is that these things seem quite impossible to incorporate > > into a single equation. You cannot have a single equation which is > > both probabilistic and at the same time non-probabilistic. The ONLY > > way to do that is by demonstrating and subsequently embracing > > EQUIVALENCE of various kinds of models, and considering all of these > > various approaches as different facets of single tool. > > > Whether there is paradox or not, that is the paradox. Unfortunately > > that is not going to change. To successfully MODEL your way around > > such a situation - you MUST use equivalence (in the sense of > > Einstein). Equivalence allows you to take many different kinds of > > tools which may seem immiscible and weld them together into a single > > tool....thats the only way to do it IMO.- Hide quoted text - > > > - Show quoted text - > > To get an idea of what such an equivalence might look like, consider > any standard physics formula such as the kinematic equations. In > standard mathematical form these are statements which are aguably > deterministic in some sense. > > Can we rewrite things using probability theory s.t. the kinematic > equations or the solutions therof are yielded as "expected > relationships" or "expected values" ? I think it could be written > pretty easily. > > Then, you would be forced to choose between probabilistic and non- > probabilistic approaches.....and there's not a damn reason why one > would choose one over the other when the results are identical to > aribitrary accuracy.- Hide quoted text - > > - Show quoted text - So, lets say you had some nice kinematic equations written out in standard form, and something written out probabilistically which yields the kinematic equations as some kind of "expected relationship". Is it currently possible to _transform_ between probabilistic and non- probabilistic expressions ? Can you write this as a formal transformation ? I doubt it. I would say it might be possible, but I dont think that it has ever been done. Ive never seen anything like that. This is where the recent work on chaos becomes very interesting perhaps....many people have tried to show that order and disorder can emerge in various studies of chaos, fractals and cellular automata. It's not quite the same thing as using something like a Lorentz Transform (if you will) to go back and forth between probabilistic and non-probabilistic expressions.......I cant imagine what such a trasform would even look like - if it can even exist. How can we ask such questions using mathematics ?
From: Huang on 12 Jan 2010 08:58 On Jan 11, 11:48 pm, Huang <huangxienc...(a)yahoo.com> wrote: > On Jan 11, 8:50 pm, Huang <huangxienc...(a)yahoo.com> wrote: > > > > > > > On Jan 10, 8:45 am, Huang <huangxienc...(a)yahoo.com> wrote: > > > > On Jan 9, 10:20 am, Huang <huangxienc...(a)yahoo.com> wrote: > > > > > On Jan 9, 9:29 am, Huang <huangxienc...(a)yahoo.com> wrote: > > > > > > On Jan 8, 6:35 am, Huang <huangxienc...(a)yahoo.com> wrote: > > > > > > > On Jan 7, 8:34 am, Huang <huangxienc...(a)yahoo.com> wrote: > > > > > > > > On Jan 6, 9:16 pm, Huang <huangxienc...(a)yahoo.com> wrote: > > > > > > > > > On Jan 6, 11:25 am, FredJeffries <fredjeffr...(a)gmail.com> wrote: > > > > > > > > > > On Jan 4, 8:04 pm, Huang <huangxienc...(a)yahoo.com> wrote: > > > > > > > > > > > We might even be able to say that the discrete universe is equivalent > > > > > > > > > > to the continuous universe in the same sense that relative motions are > > > > > > > > > > equivalent. > > > > > > > > > >http://terrytao.wordpress.com/2007/05/23/soft-analysis-hard-analysis-... > > > > > > > > > I believe that many of the results of calculus can be replicated using > > > > > > > > discrete methods. In fact, I suspect that they all can. > > > > > > > > > Lets suppose you could construct two bodies of formal mathematical > > > > > > > > models. Calculus on the one hand, and a whole collection of discrete > > > > > > > > models on the other hand which model the exact same things. Suppose > > > > > > > > for a moment that these two bodies of knowledge are at our disposal. > > > > > > > > We might like to show that there are some broad relationships between > > > > > > > > these two collections of things. > > > > > > > > > To my knowledge....this has not been done. I dont think that it has > > > > > > > > even been explored to any great extent, if at all. > > > > > > > > > So what the hell are you people waiting for ? Lazy sods ? I explain > > > > > > > > all of these insights and you just sit on your fat asses ? Must I do > > > > > > > > everything myself ? > > > > > > > > To be more specific, I believe that it is possible to construct > > > > > > > discrete models and continuous models which both return the same > > > > > > > numerical results. It may even be possible to do this in a fairly > > > > > > > clever way so that broad relationships can emerge by comparing these > > > > > > > collections of things. > > > > > > > > Once we have accomplished that, we should be able to say that these > > > > > > > things are "equivalent" in the sense of Einstein, and that whether one > > > > > > > wishes to model the universe as being discrete or continuous is purely > > > > > > > a matter of choice. That the discrete universe and the continuous > > > > > > > universe may be very different, but they are "equivalent" because the > > > > > > > numbers crunch exactly the same. > > > > > > > > I think that this is all possible using standard mathematics, no > > > > > > > nonstandard math would be required for this. > > > > > > The only reason to even mention these things is because anything that > > > > > you can do under the assumption of existential indeterminacy - > > > > > likewise must be possible without it. > > > > > > The EQUIVALENCE of a continuous spacetime and discrete spacetime is > > > > > one of the results obtailable by considering conjectural modelling. > > > > > And we can use that result to explain WP duality. But this result > > > > > (equivalence) should be obtainable under the assumption of existence > > > > > i.e. under standard mathematics. > > > > > > Clearly, we should be able to demonstrate such an equivalence in a > > > > > very broad number of cases. This is NOT to say that continuity and > > > > > discreteness are the same thing, indeed not. Rather, that a result > > > > > obtained by using one approach should be deriveable using the other - > > > > > to an arbitrary degree of accuracy. And we are using the word > > > > > equivalnce in the sense of Einstein, i.e. the Equivalence Principle. > > > > > > Why and how mathematics could progress to the place it has achieved > > > > > and not recognize this is quite mysterious to me. Perhaps it is simply > > > > > because you cannot "prove" that continuity and discreteness are the > > > > > same thing, indeed they simply are not. But Equivalence in the sense > > > > > of Einstein is not so strict, all we care about is that the numbers > > > > > crunch to the same result with arbitrary accuracy, and I believe this > > > > > is possible using standard math. It is certainly the case in > > > > > Conjectural Modelling.- Hide quoted text - > > > > > > - Show quoted text - > > > > > Which leads me to wonder about some criticism of GR which seems to be > > > > absent. > > > > > One of the early criticisms of calculus was that it dealt with the > > > > "ghosts of departed quantities", specifically limits, and how they are > > > > treated as if they are numbers. > > > > > It seems that the same kind of problem is present when you consider > > > > Einsteins Equivalence Principle, but in this case we are talking about > > > > the "ghosts of a departed operator". Yes we do have the Lorentz > > > > Transform, but what does that tell us anyway ? Does it say that things > > > > are "equal" ??? No. It says that things are "equivalent". And > > > > equivalence is not the same thing as equality under Einstein's usage > > > > of the word equivalence. > > > > > One should be able to demonstrate the "equivalence" of continuity and > > > > discreteness in many cases, and I think that it is just slightly MORE > > > > interesting than the equivalence of relative motions of moving > > > > bodies......unless of course we prefer to walk around in a daze for > > > > another 100 years under a self imposed state of deliberately chosen > > > > confusion (wp duality).- Hide quoted text - > > > > > - Show quoted text - > > > > Here's my view on unification - based on the aforementioned > > > considerations. > > > > The universe should be accurately modellable with or without > > > probability theory. > > > The universe should be accurately modellable with or without > > > randomness. > > > The universe should be accurately modellable with or without > > > existential indeterminacy. > > > The universe should be accurately modellable as being either discrete > > > or continuous. > > > The universe should be accurately modellable as either deterministic > > > or non-deterministic. > > > > In my opinion, a model which successfully unifies physics should > > > satisfy these requirements. > > > > The problem is that these things seem quite impossible to incorporate > > > into a single equation. You cannot have a single equation which is > > > both probabilistic and at the same time non-probabilistic. The ONLY > > > way to do that is by demonstrating and subsequently embracing > > > EQUIVALENCE of various kinds of models, and considering all of these > > > various approaches as different facets of single tool. > > > > Whether there is paradox or not, that is the paradox. Unfortunately > > > that is not going to change. To successfully MODEL your way around > > > such a situation - you MUST use equivalence (in the sense of > > > Einstein). Equivalence allows you to take many different kinds of > > > tools which may seem immiscible and weld them together into a single > > > tool....thats the only way to do it IMO.- Hide quoted text - > > > > - Show quoted text - > > > To get an idea of what such an equivalence might look like, consider > > any standard physics formula such as the kinematic equations. In > > standard mathematical form these are statements which are aguably > > deterministic in some sense. > > > Can we rewrite things using probability theory s.t. the kinematic > > equations or the solutions therof are yielded as "expected > > relationships" or "expected values" ? I think it could be written > > pretty easily. > > > Then, you would be forced to choose between probabilistic and non- > > probabilistic approaches.....and there's not a damn reason why one > > would choose one over the other when the results are identical to > > aribitrary accuracy.- Hide quoted text - > > > - Show quoted text - > > So, lets say you had some nice kinematic equations written out in > standard form, and something written out probabilistically which > yields the kinematic equations as some kind of "expected > relationship". > > Is it currently possible to _transform_ between probabilistic and non- > probabilistic expressions ? Can you write this as a formal > transformation ? I doubt it. I would say it might be possible, but I > dont think that it has ever been done. Ive never seen anything like > that. > > This is where the recent work on chaos becomes very interesting > perhaps....many people have tried to show that order and disorder can > emerge in various studies of chaos, fractals and cellular automata. > It's not quite the same thing as using something like a Lorentz > Transform (if you will) to go back and forth between probabilistic and > non-probabilistic expressions.......I cant imagine what such a > trasform would even look like - if it can even exist. > > How can we ask such questions using mathematics ?- Hide quoted text - > > - Show quoted text - To restate the question....suppose that we make a model like a cointoss. We actually make 2 different models. [1] The standard model from probability theory that uses random variables. [2] A non-probabilistic approach which uses standard physics, and the randomness is rendered as an emergent process related to sensitive dependence on initial conditions. Can we transform from [1] to [2] ? Clearly we can probably pronounce them to be equivalent (in the sense of Einstein), but I dont know if you can make a genuine transformation from [1] to [2].
From: Huang on 13 Jan 2010 00:20 On Jan 12, 7:58 am, Huang <huangxienc...(a)yahoo.com> wrote: > On Jan 11, 11:48 pm, Huang <huangxienc...(a)yahoo.com> wrote: > > > > > > > On Jan 11, 8:50 pm, Huang <huangxienc...(a)yahoo.com> wrote: > > > > On Jan 10, 8:45 am, Huang <huangxienc...(a)yahoo.com> wrote: > > > > > On Jan 9, 10:20 am, Huang <huangxienc...(a)yahoo.com> wrote: > > > > > > On Jan 9, 9:29 am, Huang <huangxienc...(a)yahoo.com> wrote: > > > > > > > On Jan 8, 6:35 am, Huang <huangxienc...(a)yahoo.com> wrote: > > > > > > > > On Jan 7, 8:34 am, Huang <huangxienc...(a)yahoo.com> wrote: > > > > > > > > > On Jan 6, 9:16 pm, Huang <huangxienc...(a)yahoo.com> wrote: > > > > > > > > > > On Jan 6, 11:25 am, FredJeffries <fredjeffr...(a)gmail.com> wrote: > > > > > > > > > > > On Jan 4, 8:04 pm, Huang <huangxienc...(a)yahoo.com> wrote: > > > > > > > > > > > > We might even be able to say that the discrete universe is equivalent > > > > > > > > > > > to the continuous universe in the same sense that relative motions are > > > > > > > > > > > equivalent. > > > > > > > > > > >http://terrytao.wordpress.com/2007/05/23/soft-analysis-hard-analysis-... > > > > > > > > > > I believe that many of the results of calculus can be replicated using > > > > > > > > > discrete methods. In fact, I suspect that they all can. > > > > > > > > > > Lets suppose you could construct two bodies of formal mathematical > > > > > > > > > models. Calculus on the one hand, and a whole collection of discrete > > > > > > > > > models on the other hand which model the exact same things. Suppose > > > > > > > > > for a moment that these two bodies of knowledge are at our disposal. > > > > > > > > > We might like to show that there are some broad relationships between > > > > > > > > > these two collections of things. > > > > > > > > > > To my knowledge....this has not been done. I dont think that it has > > > > > > > > > even been explored to any great extent, if at all. > > > > > > > > > > So what the hell are you people waiting for ? Lazy sods ? I explain > > > > > > > > > all of these insights and you just sit on your fat asses ? Must I do > > > > > > > > > everything myself ? > > > > > > > > > To be more specific, I believe that it is possible to construct > > > > > > > > discrete models and continuous models which both return the same > > > > > > > > numerical results. It may even be possible to do this in a fairly > > > > > > > > clever way so that broad relationships can emerge by comparing these > > > > > > > > collections of things. > > > > > > > > > Once we have accomplished that, we should be able to say that these > > > > > > > > things are "equivalent" in the sense of Einstein, and that whether one > > > > > > > > wishes to model the universe as being discrete or continuous is purely > > > > > > > > a matter of choice. That the discrete universe and the continuous > > > > > > > > universe may be very different, but they are "equivalent" because the > > > > > > > > numbers crunch exactly the same. > > > > > > > > > I think that this is all possible using standard mathematics, no > > > > > > > > nonstandard math would be required for this. > > > > > > > The only reason to even mention these things is because anything that > > > > > > you can do under the assumption of existential indeterminacy - > > > > > > likewise must be possible without it. > > > > > > > The EQUIVALENCE of a continuous spacetime and discrete spacetime is > > > > > > one of the results obtailable by considering conjectural modelling. > > > > > > And we can use that result to explain WP duality. But this result > > > > > > (equivalence) should be obtainable under the assumption of existence > > > > > > i.e. under standard mathematics. > > > > > > > Clearly, we should be able to demonstrate such an equivalence in a > > > > > > very broad number of cases. This is NOT to say that continuity and > > > > > > discreteness are the same thing, indeed not. Rather, that a result > > > > > > obtained by using one approach should be deriveable using the other - > > > > > > to an arbitrary degree of accuracy. And we are using the word > > > > > > equivalnce in the sense of Einstein, i.e. the Equivalence Principle. > > > > > > > Why and how mathematics could progress to the place it has achieved > > > > > > and not recognize this is quite mysterious to me. Perhaps it is simply > > > > > > because you cannot "prove" that continuity and discreteness are the > > > > > > same thing, indeed they simply are not. But Equivalence in the sense > > > > > > of Einstein is not so strict, all we care about is that the numbers > > > > > > crunch to the same result with arbitrary accuracy, and I believe this > > > > > > is possible using standard math. It is certainly the case in > > > > > > Conjectural Modelling.- Hide quoted text - > > > > > > > - Show quoted text - > > > > > > Which leads me to wonder about some criticism of GR which seems to be > > > > > absent. > > > > > > One of the early criticisms of calculus was that it dealt with the > > > > > "ghosts of departed quantities", specifically limits, and how they are > > > > > treated as if they are numbers. > > > > > > It seems that the same kind of problem is present when you consider > > > > > Einsteins Equivalence Principle, but in this case we are talking about > > > > > the "ghosts of a departed operator". Yes we do have the Lorentz > > > > > Transform, but what does that tell us anyway ? Does it say that things > > > > > are "equal" ??? No. It says that things are "equivalent". And > > > > > equivalence is not the same thing as equality under Einstein's usage > > > > > of the word equivalence. > > > > > > One should be able to demonstrate the "equivalence" of continuity and > > > > > discreteness in many cases, and I think that it is just slightly MORE > > > > > interesting than the equivalence of relative motions of moving > > > > > bodies......unless of course we prefer to walk around in a daze for > > > > > another 100 years under a self imposed state of deliberately chosen > > > > > confusion (wp duality).- Hide quoted text - > > > > > > - Show quoted text - > > > > > Here's my view on unification - based on the aforementioned > > > > considerations. > > > > > The universe should be accurately modellable with or without > > > > probability theory. > > > > The universe should be accurately modellable with or without > > > > randomness. > > > > The universe should be accurately modellable with or without > > > > existential indeterminacy. > > > > The universe should be accurately modellable as being either discrete > > > > or continuous. > > > > The universe should be accurately modellable as either deterministic > > > > or non-deterministic. > > > > > In my opinion, a model which successfully unifies physics should > > > > satisfy these requirements. > > > > > The problem is that these things seem quite impossible to incorporate > > > > into a single equation. You cannot have a single equation which is > > > > both probabilistic and at the same time non-probabilistic. The ONLY > > > > way to do that is by demonstrating and subsequently embracing > > > > EQUIVALENCE of various kinds of models, and considering all of these > > > > various approaches as different facets of single tool. > > > > > Whether there is paradox or not, that is the paradox. Unfortunately > > > > that is not going to change. To successfully MODEL your way around > > > > such a situation - you MUST use equivalence (in the sense of > > > > Einstein). Equivalence allows you to take many different kinds of > > > > tools which may seem immiscible and weld them together into a single > > > > tool....thats the only way to do it IMO.- Hide quoted text - > > > > > - Show quoted text - > > > > To get an idea of what such an equivalence might look like, consider > > > any standard physics formula such as the kinematic equations. In > > > standard mathematical form these are statements which are aguably > > > deterministic in some sense. > > > > Can we rewrite things using probability theory s.t. the kinematic > > > equations or the solutions therof are yielded as "expected > > > relationships" or "expected values" ? I think it could be written > > > pretty easily. > > > > Then, you would be forced to choose between probabilistic and non- > > > probabilistic approaches.....and there's not a damn reason why one > > > would choose one over the other when the results are identical to > > > aribitrary accuracy.- Hide quoted text - > > > > - Show quoted text - > > > So, lets say you had some nice kinematic equations written out in > > standard form, and something written out probabilistically which > > yields the kinematic equations as some kind of "expected > > relationship". > > > Is it currently possible to _transform_ between probabilistic and non- > > probabilistic expressions ? Can you write this as a formal > > transformation ? I doubt it. I would say it might be possible, but I > > dont think that it has ever been done. Ive never seen anything like > > that. > > > This is where the recent work on chaos becomes very interesting > > perhaps....many people have tried to show that order and disorder can > > emerge in various studies of chaos, fractals and cellular automata. > > It's not quite the same thing as using something like a Lorentz > > Transform (if you will) to go back and forth between probabilistic and > > non-probabilistic expressions.......I cant imagine what such a > > trasform would even look like - if it can even exist. > > > How can we ask such questions using mathematics ?- Hide quoted text - > > > - Show quoted text - > > To restate the question....suppose that we make a model like a > cointoss. We actually make 2 different models. > > [1] The standard model from probability theory that uses random > variables. > > [2] A non-probabilistic approach which uses standard physics, and the > randomness is rendered as an emergent process related to sensitive > dependence on initial conditions. > > Can we transform from [1] to [2] ? Clearly we can probably pronounce > them to be equivalent (in the sense of Einstein), but I dont know if > you can make a genuine transformation from [1] to [2].- Hide quoted text - > > - Show quoted text - Lets look at the coint toss example from standard probability theory. Customarily they present the standard explanation as a random variable { X | H, T } or somehting similar to that. But if you consider the total number of possible outcomes each as seperate elements of an outcome space, then you can write things a bit differently. For example, flip the coin 3 times and you have 8 possible outcomes. HHH, HHT, HTT, THH, THT, TTT, HTH, TTH These permutations may be regarded as elements of an outcome space in their own right, so that { X | HHH, HHT, HTT, THH, THT, TTT, HTH, TTH } These are really two very different things, but there may be a way to transform back and forth in a relatively acceptable way. It is very different to say that you have a random variable { X | H, T } for each of 3 individual trials, or, that you have a single trial with random the variable { X | HHH, HHT, HTT, THH, THT, TTT, HTH, TTH } These are two very different looking things with an identical result. There SHOULD be a way to transform from one to the other !!!
From: Huang on 13 Jan 2010 00:49 ...........Reposting because the thread is getting too long Lets look at the coint toss example from standard probability theory. Customarily they present the standard explanation as a random variable { X | H, T } or somehting similar to that. But if you consider the total number of possible outcomes each as seperate elements of an outcome space, then you can write things a bit differently. For example, flip the coin 3 times and you have 8 possible outcomes. HHH, HHT, HTT, THH, THT, TTT, HTH, TTH These permutations may be regarded as elements of an outcome space in their own right, so that { X | HHH, HHT, HTT, THH, THT, TTT, HTH, TTH } These are really two very different things, but there may be a way to transform back and forth in a relatively acceptable way. It is very different to say that you have a random variable { X | H, T } for each of 3 individual trials, or, that you have a single trial with random the variable { X | HHH, HHT, HTT, THH, THT, TTT, HTH, TTH } These are two very different looking things with an identical result. There SHOULD be a way to transform from one to the other !!!
From: Huang on 13 Jan 2010 09:55
On Jan 12, 11:49 pm, Huang <huangxienc...(a)yahoo.com> wrote: > ..........Reposting because the thread is getting too long > > Lets look at the coint toss example from standard probability theory. > Customarily they present the standard explanation as a random > variable > > { X | H, T } > > or somehting similar to that. > > But if you consider the total number of possible outcomes each as > seperate elements of an outcome space, then you can write things a > bit > differently. For example, flip the coin 3 times and you have 8 > possible outcomes. > > HHH, HHT, HTT, THH, THT, TTT, HTH, TTH > > These permutations may be regarded as elements of an outcome space in > their own right, so that > > { X | HHH, HHT, HTT, THH, THT, TTT, HTH, TTH } > > These are really two very different things, but there may be a way to > transform back and forth in a relatively acceptable way. > > It is very different to say that you have a random variable { X | H, > T } for each of 3 individual trials, > or, that you have a single trial with random the variable { X | HHH, > HHT, HTT, THH, THT, TTT, HTH, TTH } > > These are two very different looking things with an identical result. > There SHOULD be a way to transform from one to the other !!! Consider 3 trials of a random variable { X | H, T } . The result is no different than a single trial of the random variable { X | HHH, HHT, HTT, THH, THT, TTT, HTH, TTH } . How do we transform from one situation to the other ? Is it enough to say they are "equivalent" ?? |