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  1. #1 "Randomly" picking from an infinite number of choi 
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    Is it mathematically meaningful to talk about randomly picking something from an infinite set of possibilities? For example, is it possible to talk about randomly selecting any integer from the number line? Or randomly selecting any real number from all possible real numbers?

    It seems like if you try to calculate the odds of picking any given integer, the odds would be 1 over infinity, which would be zero - and how could you pick a number if every potential choice has a zero probability of being selected? On the other hand, you do have an infinite number of possible choices, which might somehow cancel that problem out. I don't really know enough about math to figure it out, so I'm asking here...


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  3. #2 Re: "Randomly" picking from an infinite number of 
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    Quote Originally Posted by Scifor Refugee
    Is it mathematically meaningful to talk about randomly picking something from an infinite set of possibilities? For example, is it possible to talk about randomly selecting any integer from the number line? Or randomly selecting any real number from all possible real numbers?

    It seems like if you try to calculate the odds of picking any given integer, the odds would be 1 over infinity, which would be zero - and how could you pick a number if every potential choice has a zero probability of being selected? On the other hand, you do have an infinite number of possible choices, which might somehow cancel that problem out. I don't really know enough about math to figure it out, so I'm asking here...
    While I have seen such a notion discussed, in elementary statistics text books no less, there is no probability distribution that would permit infinitely many mutually exclusive events of equal probability. The probability of picking something must be 1, and if each of an infinite number of mutually exclusive events has the same probbility then the sum is either infinite or 0 (if all have probability 0), a contradiction since they must sum to 1. Since all probabilities are non-negative there is no way to solve this problem by having things "cancel out".

    What one can do is impose a probability density function on the integers such that the probability varies from integer to integer and such that the sum of the probabilities is 1 (an infinite series). For instance you might assign each natural number n the probability n=1,2,3,... and all other integers probability 0.

    You can do something similar, but a bit more sophisticated with uncountably infinite sets. There you need the notion of a probability measure, or a density function. A common one in the case of one variable is the Gaussian density .


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    Thanks for that Dr Rocket. Can I ask a related question about infinity?

    A former teacher of maths once told me that the universe could not be infinitely old because it is impossible, mathemtically, to cover an infinite 'distance' and arrive at a specific point, ergo, it would not be possible to have a speicifc present if there had been an infinite past.

    Something about this argument sounded shady to me, but my maths was not good enough to say whether:

    a. his mathematics was faulty, or
    b. his analogy between time and number was faulty

    Any ideas?
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  5. #4  
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    Quote Originally Posted by sunshinewarrior
    Thanks for that Dr Rocket. Can I ask a related question about infinity?

    A former teacher of maths once told me that the universe could not be infinitely old because it is impossible, mathematically, to cover an infinite 'distance' and arrive at a specific point, ergo, it would not be possible to have a specific present if there had been an infinite past.

    Something about this argument sounded shady to me, but my maths was not good enough to say whether:

    a. his mathematics was faulty, or
    b. his analogy between time and number was faulty

    Any ideas?
    I'm not a mathematician! I have to admit that, sometimes, this argument sounds "shady" to me but at other times the idea the universe could not be infinitely old because it would not be possible to reach the present, given an infinite past, appears convincing.
    It does seem odd to me that altho' the Steady State Theory has largely been discredited this argument, as far as I am aware, did not appear to be used against it considering the Steady State theorists claimed the universe had always existed
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  6. #5  
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    Quote Originally Posted by sunshinewarrior
    Thanks for that Dr Rocket. Can I ask a related question about infinity?

    A former teacher of maths once told me that the universe could not be infinitely old because it is impossible, mathemtically, to cover an infinite 'distance' and arrive at a specific point, ergo, it would not be possible to have a speicifc present if there had been an infinite past.

    Something about this argument sounded shady to me, but my maths was not good enough to say whether:

    a. his mathematics was faulty, or
    b. his analogy between time and number was faulty

    Any ideas?
    It sounds to me like he may have been applying a mathematical model that is inappropriate.

    From the way that you state the position it sounds like he was saying something like "If the universe is a deterministic dynamical system then the behavior is determined by the doefficients of the differential equations that describe it plus the initial conditions. So, if the universe is infinitely old there are no initial conditions and hence later states cannot be determined."

    There is a problem with that statement. If you could measure the state at any point then you can apply the differential equations to determine the behavior at later points. If the equations are "nice" you may also be able to determine past behavior.

    If he is making some other assertion with respect to "infinity" then the problem is even more murky. Analytic functions, those describable by an infinite Taylor series, are completely determined by their value on any set with a limit point. So, for instance if you know the value of such a function on the unit interval, it is completely determined over the entire real line. Penrose at least believes that the universe is describable with analytic functions (of several variables).

    His argument to me sounds more philosophical than mathematical, and not particularly penetrating philosophy. In the 1950's Herman Bondi, a pretty decent mathematician, advocated a steady-state cosmology. With that comes an infinite time. Later knowledge shows that idea to not be a good one from the perspective of physics, but I think it does show that an infinite age for the universe is a possibility that would be accepted as mathematically valid.
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    Quote Originally Posted by DrRocket
    Quote Originally Posted by sunshinewarrior
    Thanks for that Dr Rocket. Can I ask a related question about infinity?

    A former teacher of maths once told me that the universe could not be infinitely old because it is impossible, mathemtically, to cover an infinite 'distance' and arrive at a specific point, ergo, it would not be possible to have a speicifc present if there had been an infinite past.

    Something about this argument sounded shady to me, but my maths was not good enough to say whether:

    a. his mathematics was faulty, or
    b. his analogy between time and number was faulty

    Any ideas?
    It sounds to me like he may have been applying a mathematical model that is inappropriate.

    From the way that you state the position it sounds like he was saying something like "If the universe is a deterministic dynamical system then the behavior is determined by the doefficients of the differential equations that describe it plus the initial conditions. So, if the universe is infinitely old there are no initial conditions and hence later states cannot be determined."

    There is a problem with that statement. If you could measure the state at any point then you can apply the differential equations to determine the behavior at later points. If the equations are "nice" you may also be able to determine past behavior.

    If he is making some other assertion with respect to "infinity" then the problem is even more murky. Analytic functions, those describable by an infinite Taylor series, are completely determined by their value on any set with a limit point. So, for instance if you know the value of such a function on the unit interval, it is completely determined over the entire real line. Penrose at least believes that the universe is describable with analytic functions (of several variables).

    His argument to me sounds more philosophical than mathematical, and not particularly penetrating philosophy. In the 1950's Herman Bondi, a pretty decent mathematician, advocated a steady-state cosmology. With that comes an infinite time. Later knowledge shows that idea to not be a good one from the perspective of physics, but I think it does show that an infinite age for the universe is a possibility that would be accepted as mathematically valid.
    I am of the belief that nothing is random. Because the only way to pick a number is to involve a nonrandom or humanly conceived method of choosing that some human is wise to. I contradict myself later but the contradiction is across two fields of living.

    In computers we are always hit with and abused with, our systems of randomly picking numbers are flawed. Because a human somewhere that wrote the randomized code. Can have a slight edge or at least understand how the random effect is taking place. Either by computer clock tics, or multiple passes over the data while using random timers. Day, month year, formulas or a million other variables.

    But the point is, it is random to the bulk of the populace. We have no idea of what will pop up. And we have no idea of how it pops up. To us it is an infinite possibility, when infinitely complex sets of numbers are chosen for us. Numbers over a billion places of accuracy are actually infinite to humans. They can imagine the accuracy but they cannot truly verify it or understand it, on a one to one basis.

    If you asked everyone in America, what number they would pick, one to infinity. Most would pick numbers that they know. Probably one through ten would be a large percentage. Some would pick a number with a trillion zeros.

    The fact that I am correct about that, by actual trial and error means that picking a number between one and infinity using humans which are considered the only truly random thing, life itself. We see that total randomness is not really possible.

    I could probably match a guess of a sample of one hundred individuals answers to their pick of an infinite number. By just choosing the number five. Certainly you can see a lack of randomness there.

    The time it would take a computer to sort or process an infinite amount of numbers would be to long to ever reach the random number. It would be infinitely long.

    Sincerely,


    William McCormick
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    I would have to conclude that picking an infinite number of random choices is in itself not random at all. You always know it's not going to be one that was chosen. It's possible to pick the same person over and over again with the same odds. To never pick that person ever again is not at all the same odds.
    Pleased to meet you. Hope you guess my name
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  9. #8  
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    Quote Originally Posted by (In)Sanity
    I would have to conclude that picking an infinite number of random choices is in itself not random at all. You always know it's not going to be one that was chosen. It's possible to pick the same person over and over again with the same odds. To never pick that person ever again is not at all the same odds.
    Huh ?
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    Infinity should not be used in the form x/infinity since it is not a number it is a concept.
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  11. #10  
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    Quote Originally Posted by sunshinewarrior
    Thanks for that Dr Rocket. Can I ask a related question about infinity?

    A former teacher of maths once told me that the universe could not be infinitely old because it is impossible, mathematically, to cover an infinite 'distance' and arrive at a specific point, ergo, it would not be possible to have a specific present if there had been an infinite past.

    Something about this argument sounded shady to me, but my maths was not good enough to say whether:

    a. his mathematics was faulty, or
    b. his analogy between time and number was faulty

    Any ideas?
    I can't answer your question but I don't think anyone else has either!
    Why do you believe this question about whether it is possible to have a universe, that has existed for an infinite time, is best answered by a mathematician rather than a physicist or philosopher?
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  12. #11 Re: "Randomly" picking from an infinite number of 
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    Quote Originally Posted by DrRocket
    While I have seen such a notion discussed, in elementary statistics text books no less, there is no probability distribution that would permit infinitely many mutually exclusive events of equal probability. The probability of picking something must be 1, and if each of an infinite number of mutually exclusive events has the same probbility then the sum is either infinite or 0 (if all have probability 0), a contradiction since they must sum to 1. Since all probabilities are non-negative there is no way to solve this problem by having things "cancel out".

    What one can do is impose a probability density function on the integers such that the probability varies from integer to integer and such that the sum of the probabilities is 1 (an infinite series). For instance you might assign each natural number n the probability n=1,2,3,... and all other integers probability 0.

    You can do something similar, but a bit more sophisticated with uncountably infinite sets. There you need the notion of a probability measure, or a density function. A common one in the case of one variable is the Gaussian density .
    Thanks for the information. Would you say this means that it's impossible to have infinitely many possibilities that are all equally likely? Or does it simply mean that math doesn't currently exist to describe such a situation?
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    Consider a procedure for picking such a number. A simple one would be to pick each bit randomly starting from the lowest order bit. So you have a 50/50 chance of the low order bit being 1, 50/50 of the next, etc. Think about that for a bit then see if the results make sense when you try to run the procedure for an infinite number of iterations.

    Basically, I think the math exists to say that the question doesn't really make sense rather than it's strictly impossible.
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  14. #13  
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    Quote Originally Posted by Halliday
    Quote Originally Posted by sunshinewarrior
    Thanks for that Dr Rocket. Can I ask a related question about infinity?

    A former teacher of maths once told me that the universe could not be infinitely old because it is impossible, mathematically, to cover an infinite 'distance' and arrive at a specific point, ergo, it would not be possible to have a specific present if there had been an infinite past.

    Something about this argument sounded shady to me, but my maths was not good enough to say whether:

    a. his mathematics was faulty, or
    b. his analogy between time and number was faulty

    Any ideas?
    I can't answer your question but I don't think anyone else has either!
    Why do you believe this question about whether it is possible to have a universe, that has existed for an infinite time, is best answered by a mathematician rather than a physicist or philosopher?
    1. Because his question was whether the universe could have existed or not for an infinite time was based on supposed mathematical principle.

    2. Because mathematicians have been involved in cosmological questions as much or more so than physicists and the same can be said for questions involving general relativity.

    3. Because philosophers don't answer questions, they just ask more of them.
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  15. #14 Re: "Randomly" picking from an infinite number of 
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    Quote Originally Posted by Scifor Refugee
    Quote Originally Posted by DrRocket
    While I have seen such a notion discussed, in elementary statistics text books no less, there is no probability distribution that would permit infinitely many mutually exclusive events of equal probability. The probability of picking something must be 1, and if each of an infinite number of mutually exclusive events has the same probbility then the sum is either infinite or 0 (if all have probability 0), a contradiction since they must sum to 1. Since all probabilities are non-negative there is no way to solve this problem by having things "cancel out".

    What one can do is impose a probability density function on the integers such that the probability varies from integer to integer and such that the sum of the probabilities is 1 (an infinite series). For instance you might assign each natural number n the probability n=1,2,3,... and all other integers probability 0.

    You can do something similar, but a bit more sophisticated with uncountably infinite sets. There you need the notion of a probability measure, or a density function. A common one in the case of one variable is the Gaussian density .
    Thanks for the information. Would you say this means that it's impossible to have infinitely many possibilities that are all equally likely? Or does it simply mean that math doesn't currently exist to describe such a situation?
    The only way to have infinitely many distinct events that are equally likely is for the likelihood to be zero. In which case there must be other events with non-zero probabilities that sum to 1.

    It is not that mathematics cannot deal with the question, it is that having infinitely many equally likely events with non-zero probability is impossible.

    Probability, following Kolmogorov, is a branch of what mathematicians call measure theory. A probability space is nothing more than a measure space with a postive measure in which the measure of the entire space is 1. If you have infinitely many disjoint events then their probabilities must add up to 1 or less (precisely 1 if their unionis the whole space). If they have the same probability then that probability must be 0, since an infinite sum with each summand fixed and positive is infinite.
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    Quote Originally Posted by Halliday
    Quote Originally Posted by sunshinewarrior
    Thanks for that Dr Rocket. Can I ask a related question about infinity?

    A former teacher of maths once told me that the universe could not be infinitely old because it is impossible, mathematically, to cover an infinite 'distance' and arrive at a specific point, ergo, it would not be possible to have a specific present if there had been an infinite past.

    Something about this argument sounded shady to me, but my maths was not good enough to say whether:

    a. his mathematics was faulty, or
    b. his analogy between time and number was faulty

    Any ideas?
    I can't answer your question but I don't think anyone else has either!
    Why do you believe this question about whether it is possible to have a universe, that has existed for an infinite time, is best answered by a mathematician rather than a physicist or philosopher?
    You seem sincerely motivated at getting to a conclusion or agreement. I would like to ask you. What is infinity to a human?

    I will share what I believe infinity is. I was taught in a strange school that looking back, the school system really did get deep into this stuff. Even on a grammar school level.

    To me if you are going to declare infinity as such a complexity or such a large volume, or such a small value.
    Then we have to declare what, "infinitely accurate" means to verifying our infinite or non-infinite object.

    I mean if we look at the sun and say, "at those temperatures and with those twisting solar flairs hopping up every so often there are 10^1000000000000000 atoms in the sun".
    Did we just end the infinite difficultly of actually trying to count the atoms at any given nano second, in the sun?
    While there seems to me to be an infinite number of variables to doing so?

    Varying surface temperature, density of the sun itself. What is actually in the core of the sun? Is the core symmetrical, or are the elements evenly distributed?

    To me the number of atoms in the sun is out of my scope to comprehend as an exacting scientific number. It would just be a guess. No one could say you are right or wrong.

    So in a way when you pick a number from 1 to infinity, depending on the number, no one would ever be able to tell if it exists, if you pick one so far out. By the time you told them the number they would be long dead. You would be long dead.

    So to me picking a number from one to infinity although it sounds real or like something to do. It seems to me to be an impossibility in actuality. I believe we as humans are limited to our picks. To numbers closer to our day to day life.


    What do you think?

    Sincerely,


    William McCormick,
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    Quote Originally Posted by (In)Sanity
    I would have to conclude that picking an infinite number of random choices is in itself not random at all. You always know it's not going to be one that was chosen. It's possible to pick the same person over and over again with the same odds. To never pick that person ever again is not at all the same odds.
    A friend of mine was a postmaster. He told me once that if you flip a coin the odds of it being heads or tails are 50/50 in theory. However you could flip a coin and it could land the same way hundreds or thousands of times the same way, in a row.

    In a post office all you do all day long is sort mail and packages. His information may be based upon the longest run of envelopes that all fell the same way in a long run, or process. When there was supposed to be a 50/50 chance each time a new envelope fell.

    He may have witnessed 1000/0 or better in real life. Is that because of the amount of experimenting they are doing? Or is it just a fluke? Or is that just part of the whole true 50/50 odds.

    One of my brothers was a partner in a gambling boat, and you hear a lot of stories about gambling Casinos. One I heard was that there are often runs of the same number coming in on the wheel. Or the same color coming in, and some of them were pretty off the odds.

    Is there such a thing as infinitely against the odds? Ha-ha.

    Sincerely,


    William McCormick
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  18. #17  
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    Out of all the math topics people in general really don't get, the two worst would likely be scale and randomness. Scale is a little off-topic for the moment, but here's a good link on understanding randomness. In the Introduction section, there's a little form for typing in a sequence of 100 coin flips (made up, not flipped) and see how random it was.

    This one's kind of hard to read, and I used to know a much better website with the same idea (which I can't find), but this works well enough and has a lot more info on randomness.
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  19. #18  
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    Quote Originally Posted by MagiMaster
    Out of all the math topics people in general really don't get, the two worst would likely be scale and randomness. Scale is a little off-topic for the moment, but here's a good link on understanding randomness. In the Introduction section, there's a little form for typing in a sequence of 100 coin flips (made up, not flipped) and see how random it was.

    This one's kind of hard to read, and I used to know a much better website with the same idea (which I can't find), but this works well enough and has a lot more info on randomness.
    There is a rather fundamental problem with this. There is no definition with any meaning for what it means for a physical process to be "random".

    Mathematics describes probability spaces and random processes, but those definitions are not verifiable in physical terms. The ONLY physical processes of which I am aware that are truly described by probability are quantum processes.

    ALL other physical processes are modeled probabilistically because of lack of complete knowledge of the process, usually ignorance of initial conditions. That applies to roulette wheels, dice, card shuffles, everything in the macroscopic world. So to ask if some process is "random" is not really meaningful. It is not, particularly when you are talking about small sample sizes.

    Yes, there are "tests for randomness", criteria for randomization in experimental technique, etc. They may be somewhat useful in a utilitarian sense, but they usually display a fundamental lack of understanding of what probability theory is and is not on the part of the person explaining them.

    Random number generators used in computer programs will commonly spit out the same sequence of "random: numbers time after time after time given the same seed.

    If you find a result that exhibits unexpected correlations, as in the case of the draft lottery used in the example at the web site to which you provided a link, that is good reason to look for physical phenomena that could have produced the noted effect. But it is not conclusive evidence that anything has happened. It is quite possible to produce runs of odd-looking outcomes from any probability distribution. In probability, anything that can happen will eventually happen. and in fact even events of probability zero can occur -- just not very often.

    If you put monkeys at typewriters, eventually one will produce the complete works of Shakespeare. But at a reasonable typing rate the expected time for this to occur exceeds the age of the universe, so don't hold your breath.

    By the same token I have seen odd results quoted as just random events of low probability that "happened" to occur. I stopped a component from being used in a very well known and expensive space launch because I refused to believe that a low-probability event was "random" -- and it turned out that the part exhibiting that low-probability (but apparently beneficial) characteristic was about to fail (It was replaced and the launch went off without a hitch).

    Properly applied statistics and probability can help you find problems before they become catastrophes. They can identify trends and cause one to investigate and find underlying causes. The applicationof that discipline is very beneficial in that respect. It applies quite well in models of games of chance over very long periods of time with large sample sizes. But statistics is often a poor substitute for physics.
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    Quote Originally Posted by DrRocket

    1. Because his question was whether the universe could have existed or not for an infinite time was based on supposed mathematical principle.

    2. Because mathematicians have been involved in cosmological questions as much or more so than physicists and the same can be said for questions involving general relativity.

    3. Because philosophers don't answer questions, they just ask more of them.
    on the last one!

    Ok, let me see if I can restate it in ways I thought of overnight to see if it makes it clearer whether there's a mathematical principle involved at all.

    "Could you sum your way up to infinity? And if so could you hit a specific number 'in' infinity which would allow you to perform calculations using that as an 'origin'?"

    I think this is what he's trying to say - that Aleph0, or whatever we represent 'regular' infinity by cannot be used as a number, therefore how can you say you have gone an infinite number of steps up the (natural) number scales and now arrived at a number you define as infinity 0, and all subsequent numbers are labelled, say AD (or AI) 2009 or some such.

    It may be that this is a serious, or ponderable mathmatical question.

    The next step is open to its own problems - even if this question is resolvable in terms of mathematics, it still doesn't mean it can then be applied to physics in a straightforward fashion.

    For the first question, only a mathematical answer will do.

    For the seond question too, I think a mathematician might be well placed to talk about its validity.

    Hence my posting in the maths section.
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  21. #20 Re: "Randomly" picking from an infinite number of 
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    Quote Originally Posted by DrRocket
    The only way to have infinitely many distinct events that are equally likely is for the likelihood to be zero. In which case there must be other events with non-zero probabilities that sum to 1.

    It is not that mathematics cannot deal with the question, it is that having infinitely many equally likely events with non-zero probability is impossible.

    Probability, following Kolmogorov, is a branch of what mathematicians call measure theory. A probability space is nothing more than a measure space with a postive measure in which the measure of the entire space is 1. If you have infinitely many disjoint events then their probabilities must add up to 1 or less (precisely 1 if their unionis the whole space). If they have the same probability then that probability must be 0, since an infinite sum with each summand fixed and positive is infinite.
    Suppose I tell you that I threw a coin in the air and it happened to land on a ruler. At what point on the ruler is the exact center of the coin? There are infinitely many points along the ruler that it could land on, all equally likely - and it definitely lands on one of them.
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  22. #21 Re: "Randomly" picking from an infinite number of 
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    Quote Originally Posted by Scifor Refugee
    Quote Originally Posted by DrRocket
    The only way to have infinitely many distinct events that are equally likely is for the likelihood to be zero. In which case there must be other events with non-zero probabilities that sum to 1.

    It is not that mathematics cannot deal with the question, it is that having infinitely many equally likely events with non-zero probability is impossible.

    Probability, following Kolmogorov, is a branch of what mathematicians call measure theory. A probability space is nothing more than a measure space with a postive measure in which the measure of the entire space is 1. If you have infinitely many disjoint events then their probabilities must add up to 1 or less (precisely 1 if their unionis the whole space). If they have the same probability then that probability must be 0, since an infinite sum with each summand fixed and positive is infinite.
    Suppose I tell you that I threw a coin in the air and it happened to land on a ruler. At what point on the ruler is the exact center of the coin? There are infinitely many points along the ruler that it could land on, all equally likely - and it definitely lands on one of them.
    Yep. The positions are all equally likely, and the likelihood is precisely 0. That is the nature of probability. It is NOT true that a probability 0 event cannot happen.

    Roughly speaking if an event can be expected to happen only finitely many times in an infinite number of trials it is of probability 0.

    More presicely speaking, if you have a continuous probability measure, a probability density function, on an interval (like a ruler) then the probability associated with any point (specific location in the case of the ruler) is 0.
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  23. #22 Re: "Randomly" picking from an infinite number of 
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    Quote Originally Posted by DrRocket

    Yep. The positions are all equally likely, and the likelihood is precisely 0. That is the nature of probability. It is NOT true that a probability 0 event cannot happen.

    Roughly speaking if an event can be expected to happen only finitely many times in an infinite number of trials it is of probability 0.

    More presicely speaking, if you have a continuous probability measure, a probability density function, on an interval (like a ruler) then the probability associated with any point (specific location in the case of the ruler) is 0.
    But if each of the points has a probability value of exactly zero, they surely wouldn't sum to one, no? You can integrate y=o from zero to infinity, and it's still always zero.
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  24. #23 Re: "Randomly" picking from an infinite number of 
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    Quote Originally Posted by Scifor Refugee
    Quote Originally Posted by DrRocket

    Yep. The positions are all equally likely, and the likelihood is precisely 0. That is the nature of probability. It is NOT true that a probability 0 event cannot happen.

    Roughly speaking if an event can be expected to happen only finitely many times in an infinite number of trials it is of probability 0.

    More presicely speaking, if you have a continuous probability measure, a probability density function, on an interval (like a ruler) then the probability associated with any point (specific location in the case of the ruler) is 0.
    But if each of the points has a probability value of exactly zero, they surely wouldn't sum to one, no? You can integrate y=o from zero to infinity, and it's still always zero.
    You have to use an integral rather than a sum. The integral of the density function over the interval is 1. But the integral over any single point is zero.

    For the case of a ruler, of say length 12 with the continuous version of "equally likely", you have the uniform distribution with is a constant 1/2 over the ruller. That integrates to 1. But the probability associated with any point is easily seen to be zero.

    To really do this right and to unify the notions of discrete probabilities and probability density functions you need the abstract theory of measure and integration. That takes a bit more background than what I have assumed to go this route.
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  25. #24 Re: "Randomly" picking from an infinite number of 
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    From my physics and chemistry classes I am familiar with the concept of applying a probability density function to a region of space in order to calculate the probability of something happening in a given area; that's how you calculate the shapes of electron orbitals around an atomic nucleus, as I'm guessing you probably already knew. I've never taken any serious statistics classes, but I can see how the same idea could be used to map a probability distribution over some other function or set of possible outcomes, or whatever. So, if I am understanding you correctly, the problem with my coin/ruler example is that my ruler is of finite length, while the entire number line is not?

    But the idea of something still being possible even though it has a zero probability of occurring seems deeply counter-intuitive to me; I thought that by definition something with a zero probability of occurring could not occur. Saying "It had a zero probability of happening, and it just happened" seems like a self-contradictory statement.
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  26. #25 Re: "Randomly" picking from an infinite number of 
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    Quote Originally Posted by DrRocket
    Quote Originally Posted by Scifor Refugee
    Quote Originally Posted by DrRocket

    Yep. The positions are all equally likely, and the likelihood is precisely 0. That is the nature of probability. It is NOT true that a probability 0 event cannot happen.

    Roughly speaking if an event can be expected to happen only finitely many times in an infinite number of trials it is of probability 0.

    More presicely speaking, if you have a continuous probability measure, a probability density function, on an interval (like a ruler) then the probability associated with any point (specific location in the case of the ruler) is 0.
    But if each of the points has a probability value of exactly zero, they surely wouldn't sum to one, no? You can integrate y=o from zero to infinity, and it's still always zero.
    You have to use an integral rather than a sum. The integral of the density function over the interval is 1. But the integral over any single point is zero.

    For the case of a ruler, of say length 12 with the continuous version of "equally likely", you have the uniform distribution with is a constant 1/2 over the ruller. That integrates to 1. But the probability associated with any point is easily seen to be zero.

    To really do this right and to unify the notions of discrete probabilities and probability density functions you need the abstract theory of measure and integration. That takes a bit more background than what I have assumed to go this route.
    Check my logic here. If a person did pick a number within an infinite set of numbers. As soon as he picked his number. Let us say it was a trillion to the trillionth power.

    That number would just be a number that a human can conceive. It would instantly become the highest "conceivable number" in an infinite set of numbers. But there is no highest conceivable number in an infinite set of numbers by definition. So there is some language barrier here. Because I am not lying.

    So in picking a number out of an infinite set of numbers, I already know that it will just be a number that we have the tools to notate, conceive or communicate.

    So no matter the number picked, it will just be within an infinitely small group of numbers that we can conceive. Compared to the infinite set that has no boundary or upper limit.

    Would you agree?

    Sincerely,


    William McCormick
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    What you're talking about there are the computable numbers, or possibly the definable numbers. Read the wiki articles for more information. Basically, if you've already written the number down (or written down how to write the number down), it's computable. If you've only managed to write down a description, it's definable. (Someone correct me if that's not quite right though.)
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  28. #27 Re: "Randomly" picking from an infinite number of 
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    Quote Originally Posted by Scifor Refugee
    From my physics and chemistry classes I am familiar with the concept of applying a probability density function to a region of space in order to calculate the probability of something happening in a given area; that's how you calculate the shapes of electron orbitals around an atomic nucleus, as I'm guessing you probably already knew. I've never taken any serious statistics classes, but I can see how the same idea could be used to map a probability distribution over some other function or set of possible outcomes, or whatever. So, if I am understanding you correctly, the problem with my coin/ruler example is that my ruler is of finite length, while the entire number line is not?

    But the idea of something still being possible even though it has a zero probability of occurring seems deeply counter-intuitive to me; I thought that by definition something with a zero probability of occurring could not occur. Saying "It had a zero probability of happening, and it just happened" seems like a self-contradictory statement.
    No, the problem with your ruler is not that it is of finite length. Finite length is not a problem at all. The fundamental problem is that it represents a continuum of posibilities, and not a discrete set.

    Bear with me here and I will try to explain probability theory in a limited space.

    Probability assigned to events numbers between 0 and 1. An event is really a set and rule that assigns numbers to sets is called a measure, in this case a probability measure. Not all sets qualify as events, but most do. If A and B are events then the event "A and B" is the intersection of the set A with the set B. And "A or B" is the union of A and B. For the case in which the unit interval, [0.1] and subsets are the probability space one way to assign probabilities to set is with a density function, a non-negative function f, and the probability is simply to integrate the density funtion over the set. This is most easily visualized for sets that are sub-intervals. So if the probability in question is the likelihood associated with say the selection of a point then then integral over an intervalis the likelihood that the point lies in that interval. That will in general be a positive number for a finite interval, but it would be zero for any particular point. But there would be some point chosen even though the probability of selecting any specific point is zero.

    Here is another way to think about events of probability zero occurring. This is only a heuristic argument, not a rigorous one, but it is commonly used in communications engineering classes. Think about an infinite experiment in which random trials are conducted and various outcomes can occur. The probability of a particular outcome, call it A, P(A) is where N(A) is the number of times that A occurs in n trials. Now if A only occurs a finite number of times then P(A)=0. In fact if then also P(A)=0. So even if A occurred infinitely many times in infinitely many trials it might still be or probability 0. This simply serves to illustrate that probability 0 events can actually occur and this creates no contradiction in probability theory.

    One problem is that the language of probability is rather badly abused in common useage. Probability is actually rather poorly understood even by many engineers and statisticians.
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    Quote Originally Posted by DrRocket
    Quote Originally Posted by Halliday
    Quote Originally Posted by sunshinewarrior
    Thanks for that Dr Rocket. Can I ask a related question about infinity?

    A former teacher of maths once told me that the universe could not be infinitely old because it is impossible, mathematically, to cover an infinite 'distance' and arrive at a specific point, ergo, it would not be possible to have a specific present if there had been an infinite past.

    Something about this argument sounded shady to me, but my maths was not good enough to say whether:

    a. his mathematics was faulty, or
    b. his analogy between time and number was faulty

    Any ideas?
    I can't answer your question but I don't think anyone else has either!
    Why do you believe this question about whether it is possible to have a universe, that has existed for an infinite time, is best answered by a mathematician rather than a physicist or philosopher?
    1. Because his question was whether the universe could have existed or not for an infinite time was based on supposed mathematical principle.

    2. Because mathematicians have been involved in cosmological questions as much or more so than physicists and the same can be said for questions involving general relativity.

    3. Because philosophers don't answer questions, they just ask more of them.
    Tell me if I am wrong? But you are making a crack about philosophers there?

    Now I can understand the frustration if you have to deal with a poor philosopher. However the poor philosophers are usually avoiding a question. Socrates is my choice of a philosopher. He was also an accomplished scientist. Very accomplished in nature and the Universe.

    But he was also honest, and realized that modern man in his day, had little time, little vocabulary, beyond what they needed to go to work 14 hours a day. So he devoted his life to getting to the truth. It seemed the fastest way to bring forth true advancement for mankind.

    Rather then racing off with the latest scientific discovery about global warming, and gaining favor with the king or emperor or law maker. He decided to teach truth, internal truth. To set basics that could be used to cross check, higher and newer finds.

    He was very successful, some of us today still try to practice what he taught. Guitarist is saying "This is off topic". But I do not think so.

    We have all of us, to my knowledge admitted the basic definitions can be contradicting, misunderstood and misused. Basically that they are wrong, if only by popular conception. New or old.

    The words and their meanings are the building blocks for anything, anyone of us, are going to say. Good bad or ugly.

    If the meanings or definitions are admittedly flawed there can be no one in charge of science and mathematics. By all scientific method proofs that I know of. Because anything we say will be flawed. That is why many that actually work with things hands on, do not talk about it. It opens up a can of worms.

    You can have building inspectors, of the highest caliper, engineers and architects of the highest training, workers of the utmost character and quality. But if the building materials come flawed, off spec or mislabeled. All of those finest builders in the world, will be very embarrassed. All of them.

    Sometimes we get to debating this math stuff, and none of us have the time to get to the gritty basics. So we argue, and to be honest I don't think any of us look good, we can't, we have no basics to plant our argument on.

    So a good philosopher in these times should be asking questions. Because he cannot honestly give answers, without all the input necessary to respond to the questions.

    And to be honest neither can anyone else. I understand that if it is your field of expertise, you do not want to admit it has collapsed at its foundation.

    It makes the pay check you are getting, immoral to some degree.

    If you admit to it, you are stating that, you are also a bit lax, or ineffective in getting it fixed. I am not being facetious at all. I am real about this. All the fields I work in suffer from the same collapse.

    I am sure the new president is finding out that he just took over a collapse.

    My point, do not be hard on real philosophers. Some of them are darn good mathematicians. Like Socrates.

    If you asked a stupid question of Socrates. He would ask an even stupider question that you would race to answer with ease. Except your answer would highlight the facetious nature of your first question. He was so good they killed him.

    Sincerely,


    William McCormick
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  30. #29 A simple example of picking from an infintie set 
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    A common example of a random process is the random walk. At each step of the random walk a new random direction is picked. After selecting a random direction the next step is to select a random distance to walk. A new point in the random walk is reached and the process continues.

    A random direction is a random choice from an infinite set. There are an infinite number of directions to choose from in the interval 0 to 360 degrees. If the distance walked is restricted from 0 some arbitrary maximum distance say d, then there are an infinite number of choices to choose.

    The main difference between this example and the other examples is that the size of the numbers in the set of possible is bounded.

    A typical criterium for choosing a random number is to choose it uniformly. That means every direction has an equal chance of being selected. One way to think of this with an infinite set is that all intervals of the same size have the same chance of being selected.

    How about this for a direction selection technique using a coin.

    Initially the set of directions is [0, 360)
    Split the interval in half mid=(start+end)/2
    [start, mid) and [mid, end)
    Flip a coin.
    Pick the 1st interval if its heads, the second half if its tails
    Go back and split the interval

    This process picks a direction uniformly. It can also be used to pick a distance along a line of fixed length.
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  31. #30 Re: A simple example of picking from an infintie set 
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    Quote Originally Posted by hokie
    A common example of a random process is the random walk. At each step of the random walk a new random direction is picked. After selecting a random direction the next step is to select a random distance to walk. A new point in the random walk is reached and the process continues.

    A random direction is a random choice from an infinite set. There are an infinite number of directions to choose from in the interval 0 to 360 degrees. If the distance walked is restricted from 0 some arbitrary maximum distance say d, then there are an infinite number of choices to choose.

    The main difference between this example and the other examples is that the size of the numbers in the set of possible is bounded.

    A typical criterium for choosing a random number is to choose it uniformly. That means every direction has an equal chance of being selected. One way to think of this with an infinite set is that all intervals of the same size have the same chance of being selected.

    How about this for a direction selection technique using a coin.

    Initially the set of directions is [0, 360)
    Split the interval in half mid=(start+end)/2
    [start, mid) and [mid, end)
    Flip a coin.
    Pick the 1st interval if its heads, the second half if its tails
    Go back and split the interval

    This process picks a direction uniformly. It can also be used to pick a distance along a line of fixed length.
    You can also formulate random walks in the language of topological groups. For any given probability measure on a group, you can think of that measure as defining the probability of the "next step" which is taken by selecting a group element according to the probability distribution then addition that element to the group element that represents the current position, and you can do this even for non-abelian groups by deciding to add on the right or left with the element selected at random. After the nth step, starting from the identity element, the probability measure for the position is the n-fold (left or right in the non-abelian case) convolution product of the measure with itself.

    Dimension plays a role in the nature of the randon walks, and for most probability mesures on ordinary Euclidean space the random walk is recurrent in dimensions 1 and 2 and non-recurrent in dimensions 3 and above. In lay terns, a drunk on the street corner will continue to bump his head against the lampost indefinitely, but and inebriated astronaut will wander off to infinity.
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    Quote Originally Posted by MagiMaster
    What you're talking about there are the computable numbers, or possibly the definable numbers. Read the wiki articles for more information. Basically, if you've already written the number down (or written down how to write the number down), it's computable. If you've only managed to write down a description, it's definable. (Someone correct me if that's not quite right though.)

    Did you notice that, the fellow gave the standard coin toss results. He did not get into the more obscure, but everyday life, one hundred in a row, coin tosses. These things do happen. I am sure some Spanish guy is buried with a knife in him,when he was that lucky. ha-ha.

    I was just watching a special about rogue waves. For years scientists said that in the lab tank no wave could achieve a higher peak or crest, then the length of the tank permits.

    Sailors for hundreds of years have been saying as a rebuttal, "I was just caught in a rogue wave". They were called liars.

    Then it came down to steel ships that did not sink, being photographed with the damage that only a rogue wave could do.

    Still scientists protested the rogue wave as a drunken sailors tail. The truth was he may have only started drinking when no one believed what he had lived through.

    This had been going on for years and years. The scientists once again were caught with no scientific evidence but a lot of destructive criticism.

    I was personally caught in a rogue wave, that was 90 feet high. I was in a small boat on a very nice sunny day, in late summer. Less then ten miles offshore.

    I got so sick from the instant horror and disorientation. That I got literally sick. My legs were under so much stress from the boat raising up 90 feet in two seconds that, I got sick. I almost damaged my legs, with the force I needed to compensate for the wave. I was pulling tendons. The metal captains chairs bent under the stress.

    It looked like a plateau of water, coming at you at about fifty miles an hour. A wall of water straight up is what greeted the boats stern. If we were not under power at about ten to twenty knots we may have taken on water. The wave hit us at about thirty to forty knots. The speed of the boat, slowed the rise of the boat up the wall of water. But the boat turned and caught the waters flow, that was moving up the wave.

    It is funny but the angle of the wave kicks the back of the boat over to one side or the other. Then the flow of the water on the surface of wave catches the keel of the boat, and turns it.

    In order to ride them, without getting washed out. We had to power down the waves. And power up the waves. Sometimes when we hit a large wave, we would drop into a 90 foot valley, and hit speeds in a cabin boat of over 50 miles an hour. The boat was chattering on the surface like we were going over rocks. The speedometer was pinned. Something you usually wish for. Ha-ha.

    My point is now with radar they know these happen everyday, and if you are unlucky enough to be caught in one, you know all to well how insane it is.

    These waves were considered by the top scientists in the world as infinitely impossible to occur.

    For years I had recounting this tale to captains in the area. That they should not take gambling boats out ten miles. They changed the law in my area. Probably due to good captains that had heard of it before. And were concerned.

    My conclusion is the question or request is flawed. You cannot actually pick a number between zero and infinity. The question should have been pick a number between zero and a trillion or something like that.

    Because if you go much higher, you are going to be creating or defining a number, not picking one.

    In other words infinity is not a value. It is an unknown. You cannot pick a number between zero and an unknown value.

    Edited or maybe the sailors kept spelling them rouge waves. Sorry about that. Ha-ha.





    Sincerely,


    William McCormick
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    Mod note:

    William. I remind you once again that this is a mathematics sub-forum. Just in case it is not perfectly obvious, that means the only welcome posts here will have at least some bearing on the subject.

    Your life experiences, while being admittedly quite interesting and varied, by the sound of it, are simply not relevant here.

    -G-
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    Quote Originally Posted by Guitarist
    Mod note:

    William. I remind you once again that this is a mathematics sub-forum. Just in case it is not perfectly obvious, that means the only welcome posts here will have at least some bearing on the subject.

    Your life experiences, while being admittedly quite interesting and varied, by the sound of it, are simply not relevant here.

    -G-
    I am sorry.

    I was trying to highlight that what we have been told, by the experts. That seems so real with the weight of their degrees and diplomas. And the years and years of just accepting it. Sharing it with others as truth. And gambling our lives on it.
    Is often not even true or the reality. I was trying to get an analogy to infinite. I have been told infinite is a lot of things. I have been told things were designed with an infinite level of safety. I am just trying to share my lesson on infinity with the younger guys on the forum.

    I practiced drawing sideways eights as a kid in school. Infinity had my teacher dreaming of things having to do with infinity, she would forget what she was doing.

    But I am just realizing stuff about infinity, I was just trying to put across with analogies to other fields, an attempt to show how the word infinite might have gotten screwed up.

    In other words the reality is, we have never seen anything man made designed with an infinite margin of safety. It may be infinitely better then the one that just collapsed in a tropical breeze, but still not infinitely safe.

    Really all I am saying is that I believe we have turned yet another word into a slang word, with different meanings in different circles.

    My finale thought on infinity is. It is the unknown in some direction.

    I am done. Sorry.



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    William McCormick
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    It has always puzzled me that one can put a "uniform distribution" on an uncountable set such as the interval [0,1], but not on a countably infinite set such as the integers.

    Is there any reason one could not define a variation of measure theory for use in countably infinite settings? Countable additivity would have to be replaced with finite additivity, but everything else could be the same.

    Or would this lead to a contradiction somehow?
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  36. #35  
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    Quote Originally Posted by plektix
    It has always puzzled me that one can put a "uniform distribution" on an uncountable set such as the interval [0,1], but not on a countably infinite set such as the integers.

    Is there any reason one could not define a variation of measure theory for use in countably infinite settings? Countable additivity would have to be replaced with finite additivity, but everything else could be the same.

    Or would this lead to a contradiction somehow?
    The problem with a uniform distribution on a countably infinite set is that a regular measure is determined by its value on singleton sets. To be uniform the measure would have to assign the same number to each point, and cannot therefore assign the number 1 to the complete set of integers.

    Replacement of countable additivity with finite additivity adds all sorts of complications to the theory of measure and integration. I think that the subject of finitely additive measures has been studied, though I have not studied it myself. You cna find some discussion in Part I of Linear Operators by Dunford and Schwartz.

    Measure theory itself works just fine on finite, countably infinite and uncountably infinite spaces. In fact, since the work of Kolmogorov probability theory is just a specialized branch of measure theory. To really study probability properly is to study the theory of positive finite measures in great detail. The measures studied are countably additive.

    So far as I am aware teh restriction to countably additive measures is done in order to preserve the machinery of the usual theory of measure and integration, particularly the crucial limiting theorems. I do not know of any other compelling reason for it. If you are so inclined you could develop the theory of probability starting from finitely additive measures and see how far you can get. I think you will have trouble with some of the pillars such as the central limit theorem.
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  37. #36  
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    Quote Originally Posted by DrRocket
    In fact, since the work of Kolmogorov probability theory is just a specialized branch of measure theory.
    As any financial mathematics textbook will testify too - risk neutral measures anyone?
    As is often the case with technical subjects we are presented with an unfortunate choice: an explanation that is accurate but incomprehensible, or comprehensible but wrong.
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    megabrain;
    Infinity should not be used in the form x/infinity since it is not a number it is a concept.
    That seems a reasonable answer.

    Using basic probability theory, the population of discrete members has to be defined, with all members having none zero probability.
    An unbounded (sub for infinite) set does not qualify.

    yes, no, maybe?
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    Quote Originally Posted by phyti
    megabrain;
    Infinity should not be used in the form x/infinity since it is not a number it is a concept.
    That seems a reasonable answer.

    Using basic probability theory, the population of discrete members has to be defined, with all members having none zero probability.
    An unbounded (sub for infinite) set does not qualify.

    yes, no, maybe?
    No.

    First there are uses for infinity, and in measure theory it is quite common for positive measures to take values in the extended real numbers, the real numbers plus . In the extended real numbers division by is not a defined operation -- the extended reals are not a field.

    It is quite possible to put a perfectly valid probability measure on an unbounded set. The usual Gaussian probaility density on the real line being a very important example. Unbounded sets and infinite sets are not the same thing, although it is trivial that an unbounded set must be infinite.

    You can also put a discrete probability measure on an infinite, even an uncountably infinite set. It just has some restrictions. For one it cannot be a finite measure. Second if finite it cannot be non-zero on an uncountable set. But you can have a finite measure, hence by normalization a probability measure, on an infinite set. The measure that assigns to each positive integer n the value being an examaple. Note that the positive integers are both infinite and unbounded.

    What you cannot do is have a regular uniform measure on an infinite discrete set, or an uniform measure on an unbounded interval. In short if points are events you cannot an infinite number of them with equal probability, unless that probability is zero. And if intervals are events with probability proportional to length then you cannot have intervals of infinite length with non-zero probability.
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    Quote Originally Posted by river_rat
    Quote Originally Posted by DrRocket
    In fact, since the work of Kolmogorov probability theory is just a specialized branch of measure theory.
    As any financial mathematics textbook will testify too - risk neutral measures anyone?
    I'll just stick to Loeve and leave the "financial mathematics" books to others.
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    My handbook has been Shreve and Karatzas.

    Something slightly different, what do you think of Fremlin's measure theory monstrosity?
    As is often the case with technical subjects we are presented with an unfortunate choice: an explanation that is accurate but incomprehensible, or comprehensible but wrong.
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  42. #41 Re: "Randomly" picking from an infinite number of 
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    Quote Originally Posted by Scifor Refugee
    But the idea of something still being possible even though it has a zero probability of occurring seems deeply counter-intuitive to me; I thought that by definition something with a zero probability of occurring could not occur. Saying "It had a zero probability of happening, and it just happened" seems like a self-contradictory statement.
    I think it's apples and oranges here.

    Case 1 is an object composed of discrete elements, which can be subdivided into finite lengths.
    Relation of part/whole is: a segment/ruler = a ratio <1

    Case 2 is an object considered as a continuum, with elements of 0 probability, because there is no resolution of its elements.
    Relation of part/whole is: a point/ruler = 0

    If it must land on the ruler, the probability for the ruler is 1 in both cases.
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    Quote Originally Posted by DrRocket
    If you are so inclined you could develop the theory of probability starting from finitely additive measures and see how far you can get. I think you will have trouble with some of the pillars such as the central limit theorem.
    My interest in uniform distributions on the integers comes partly from the following question, which I heard was asked in a job interview:

    "What is the probability that two randomly chosen integers are relatively prime?"

    The question can't be answered using standard measure-theoretic probability, yet there is an answer that makes sense. I'll let you all puzzle over it before I post an answer.

    DrRocket-you are probably right that some (many?) theorems from standard probability theory would not be importable into finitely additive probability theory. But this wouldn't make the finite version less valid, only less useful. I'll think more about it when I get the chance.
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  44. #43 "Randomly" picking from an infinite number of choi 
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    Hi,

    This is really very intrested topic.And it is the probabitlity topic in maths.I am very appeciated this topic.


    kael87

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  45. #44 drawing random numbers 
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    Drawing a random number IS a mathematically precise concept. If is a probability space (that is, a measure space with total measure = 1), then one thinks of the various measureable subsets as possible events. The probability of an event taking place is the measure of the set.

    In this context, a random variable is simply a measureable function

    .

    The right intuition to have is that the various points in represent different possible outcomes of an experiment. The value of X at any one of these points represents the value of some observable associated to the given outcome.

    For example, suppose a particle is equally likely to be anywhere on the interval [a,b]. If dx is Lebesgue measure, then dx/(b-a) is a probability measure on [a,b]. Let X:[a,b]-->R be the function X(t) = t. Then X is the random variable that represents the random position of the particle on the interval.
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  46. #45  
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    Quote Originally Posted by river_rat
    My handbook has been Shreve and Karatzas.

    Something slightly different, what do you think of Fremlin's measure theory monstrosity?
    I have never heard of Fremlin's measure theory monstrosity. What is it ?
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  47. #46  
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    [quote="plektix"]
    Quote Originally Posted by DrRocket


    DrRocket-you are probably right that some (many?) theorems from standard probability theory would not be importable into finitely additive probability theory. But this wouldn't make the finite version less valid, only less useful. I'll think more about it when I get the chance.
    I did not say that using finitely additive measure was invalid. I do not say it was not useful.

    What I said is that if you drop the requirement for countable additivity you lose a tremendous amount of the classical machinery of measure and integration. You wind up with a rather different subject and a different bag of tools. You also need to be careful in reading the literature as the standard set of assumptions involves countably additive measures.
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