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#1 2007-07-26 16:51:36

John E. Franklin
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Registered: 2005-08-29
Posts: 3,588

discrete or continuous Uni.Ran.Var.?

This question was posed and I don't
understand it.  It was put in Euler Ave in
someone else's message.  I was wondering
how you can tell if they are discrete or continuous
variables from the wording??

"let x is in [1,3] and y is in [1,4] whare x and y are uniform random variables. what is the probability that y>x?"

If discrete then 11  21  22  31 32 33 have xy where x>=y.  Six cases.
If discrete also then 12 13 14 23 24 34 have xy where y > x.  Six cases.

If continuous, I have no clue at this time.

2/3's was mentioned by mathsy in passing, but not sure if this is continuous or something else I don't get.


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#2 2007-07-26 17:54:51

Ricky
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Re: discrete or continuous Uni.Ran.Var.?

[1,3] indicates an interval on the real numbers.  In short, continuous.  How to answer the question?  Start by noting that 1/4th of the time, y is going to be greater than 3.  Try to continue from there.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#3 2007-07-26 20:13:07

George,Y
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Re: discrete or continuous Uni.Ran.Var.?

Rectangle of X and Y distribution:
(1,1) (3,1) (3,4) (1,4)

(X,Y) is equally distributed in this rectangle with f(x,y)=1/6=1/2*1/3, on the assumption that X and Y are independent.

draw a line y=x, any (x,y) above this obeys y>x, whereas any (x,y) below this obeys y<x
y=x cut the rectangle at (1,1) and (3,3)
The area of the rectangular above the line is 4
Hence the probability that Y>X is
1/6*4=2/3

Last edited by George,Y (2007-07-26 20:30:28)


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#4 2007-07-26 20:29:23

George,Y
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Re: discrete or continuous Uni.Ran.Var.?

BTW

you know what? according to the assumption of "continuous probability" X<Y has the probability 1/3 and X=Y has the probability 0! Moreover, any point within the distribution rectangle has the probability 0! -Any X and Y combination such as X=2 and Y=3 has the probability 0, meaning NOT POSSIBLE AT ALL!!

That's my another reason against continuous assumption. Perhaps I can try a probability magazine and tell them why I object continuous reals.

Last edited by George,Y (2007-07-26 20:35:30)


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#5 2007-07-27 01:11:24

Ricky
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Re: discrete or continuous Uni.Ran.Var.?

George, do you not realize the same problem exists when you choose a random integer?  Your problem is not with continuousness, rather only infinite sets.  And you must realize that your problem also extends to the rational numbers.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#6 2007-07-27 04:18:56

John E. Franklin
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Re: discrete or continuous Uni.Ran.Var.?

I drew the diagram you described.
Now I recall a similar problem mathsy
did in another thread once.  Thanks a lot!


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#7 2007-07-27 13:10:14

George,Y
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Re: discrete or continuous Uni.Ran.Var.?

Ricky wrote:

George, do you not realize the same problem exists when you choose a random integer?  Your problem is not with continuousness, rather only infinite sets.  And you must realize that your problem also extends to the rational numbers.

Yes, if the integers has the infinite amount, that's true. Anything with infinity, no matter infinite large or infinitesimal, is equivalently illogical.


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#8 2007-07-27 13:14:09

George,Y
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Re: discrete or continuous Uni.Ran.Var.?

continuousness is based on using infinitesimal implicitly.


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#9 2007-07-27 13:15:21

George,Y
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Re: discrete or continuous Uni.Ran.Var.?

postgallery.php?pid=76017&filename=UniRanVar.PNG

Exactly!

Cute image~


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#10 2007-07-27 13:37:03

Ricky
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Re: discrete or continuous Uni.Ran.Var.?

Yes, if the integers has the infinite amount, that's true. Anything with infinity, no matter infinite large or infinitesimal, is equivalently illogical.

Then you believe there exists a largest integer?  Or is all of math just illogical?


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#11 2007-07-28 02:01:49

George,Y
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Re: discrete or continuous Uni.Ran.Var.?

"Then you believe there exists a largest integer"
--I am afraid it's the only logical option out. Besides, experiments have shown many failures of existence of infinte amount. Just one example: the gravity force, used to be thought as proportional to the inverse of the distance squared. And it follows that if the two object has no distance to each other, infinte gravity force. However, the nature refuses this assumption, gravity force cannot reach the ideal amount obeying formula even when the distance is small enough. And it's not alone-many physics formula share this micro-breaking-down feature. Just coincidence? I doubt so.

Any way, in Post 4 I have projected the paradox in probability-Can you make a way out of it?

Last edited by George,Y (2007-07-28 02:05:41)


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#12 2007-07-28 03:30:44

mathsyperson
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Re: discrete or continuous Uni.Ran.Var.?

Of course there isn't a largest integer. For any integer you can think of (N), there exists an integer that is larger (N+1).

As for the gravity example, you're right in that theoretically the force would be infinite if two objects were in exactly the same place, but nuclear forces repel objects at very small distances with far greater force than gravity attracts them, meaning that two objects can never be in the same place and that situation would never happen.

Continuous probability being 0 everywhere is the same as saying that

.

Your problem is just a result of you not believing in the same theory of limits as everyone else.


Why did the vector cross the road?
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#13 2007-07-28 08:02:07

Ricky
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Re: discrete or continuous Uni.Ran.Var.?

Your problem is just a result of you not believing in the same theory of limits as everyone else.

It's not only that.  He doesn't believe the existence of reals or the density of rationals.  In fact, the only number system he believes in is a weird, finite, form of integers.  He also completely ignores the success that standard mathematics has had.  For example, Maxwell's Equations which are based upon the completeness of the reals.

Another thing he doesn't seem to comprehend is that even if the world were truly discrete, then it has to be by such a small amount that the error bars are almost non-existent.  For example, calculate the area under the curve x^2 from 0 to 1 using the fact that "space" is discrete and can only exist in units of 10^(-100).  Now do the same calculation using the fact that space is continuous using integral calculus.  What is the difference in area?  Which calculation was quicker?


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#14 2007-08-01 15:44:25

George,Y
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Re: discrete or continuous Uni.Ran.Var.?

Ricky wrote:

Another thing he doesn't seem to comprehend is that even if the world were truly discrete, then it has to be by such a small amount that the error bars are almost non-existent.  For example, calculate the area under the curve x^2 from 0 to 1 using the fact that "space" is discrete and can only exist in units of 10^(-100).  Now do the same calculation using the fact that space is continuous using integral calculus.  What is the difference in area?  Which calculation was quicker?

You completely got me, Ricky. That's my point-the infinite & continuous system may have simulated the finite world so well that people even presume that the further is true while the latter is not. That's a huge problem hindering scientists to discover nature. I mean, how hard it could be for a detective if s/he always assumes the wrong suspect before collecting enough evidence to defy his old thinking? We may provide multi-assumptions to scientists so that they know there Is another math-tool, another assumption.

To mathyperson:

Columb's Law- can it be explained by a repelling force?

Your explaination of probability still uses limit, meaning within a small time interval, the probability exists and is quite proportional to the interval length. However, I do want to ask a question: Does 8:00 exactly, not 7:59 to 8:01, not 7:59'59" to 8:00'01" or anything else, ever exist?

BTW: I have a friend studying electrical engineering in the best technology university in my country.

I asked: Are your courses so simple as to entail only formulas discovered a hundred year ago?

Replied him: Not necessarily. From acadamic year 3 we have courses in micro-electric, where quantum effects take place of simple formulas. We have to examine dynamics & details of materials perspectively instead of applying universal formula.
-The Maxwell equation is not so universal.


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#15 2007-08-01 15:52:42

George,Y
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Re: discrete or continuous Uni.Ran.Var.?

Anyway, an editor shares pretty the same point with you, Ricky. smile


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#16 2007-08-01 16:23:14

MathsIsFun
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Re: discrete or continuous Uni.Ran.Var.?

I enjoy your contributions, George, they make me think harder about continuous vs discrete.


"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

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#17 2007-08-01 19:06:26

Ricky
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Posts: 3,791

Re: discrete or continuous Uni.Ran.Var.?

You completely got me, Ricky. That's my point-the infinite & continuous system may have simulated the finite world so well that people even presume that the further is true while the latter is not. That's a huge problem hindering scientists to discover nature. I mean, how hard it could be for a detective if s/he always assumes the wrong suspect before collecting enough evidence to defy his old thinking? We may provide multi-assumptions to scientists so that they know there Is another math-tool, another assumption.

You think applications of math become better upon assuming there is a discrete amount to distance/time/anything.  Please show me how this is true.  I've already provided one counter-example in post 13.

BTW: I have a friend studying electrical engineering in the best technology university in my country.

I asked: Are your courses so simple as to entail only formulas discovered a hundred year ago?

Replied him: Not necessarily. From acadamic year 3 we have courses in micro-electric, where quantum effects take place of simple formulas. We have to examine dynamics & details of materials perspectively instead of applying universal formula.
-The Maxwell equation is not so universal.

I did not say Maxwell equations are universal.  But they are still used today for a great deal of things.  Of course they aren't used for the quantum world because they were discovered before we even knew a quantum world existed.  But quantum properties don't apply to the macroscopic world.  The probabilities become so small they effectively get canceled out.  One might say they approach 0, but of course, you don't think such an approach is possible.

Anyway, an editor shares pretty the same point with you, Ricky.

I don't understand what you're trying to say here.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#18 2007-08-01 23:42:54

mathsyperson
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Re: discrete or continuous Uni.Ran.Var.?

George,Y wrote:

Does 8:00 exactly, not 7:59 to 8:01, not 7:59'59" to 8:00'01" or anything else, ever exist?

What's the alternative if it doesn't?
Start at 7:59. Two minutes later it's 8:01. So how could the time have changed like that if it wasn't 8:00 at some point?


Why did the vector cross the road?
It wanted to be normal.

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#19 2007-08-04 12:46:04

John E. Franklin
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Registered: 2005-08-29
Posts: 3,588

Re: discrete or continuous Uni.Ran.Var.?

I think he is saying the present is really thin, in fact zero thickness in time units.


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#20 2007-08-05 16:15:55

George,Y
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Re: discrete or continuous Uni.Ran.Var.?

Ricky, your counter-example is Maxwell Equations if I got it right. You have agreeed that microly it doesn't count. That's enough. The only difference we can discover about an infinitesimal system and a sufficient small system lies in whether microly the former applies. The success of an infinitesimal system in macro world isn't enough because as you have admitted, the error can be so small that the infinitesimal system might just "steal" the success of the sufficient small system.

"So how could the time have changed like that if it wasn't 8:00 at some point?"
-mathyperson you are objecting the abrupt change of time. Well, it definately looks odd. But water, wood and  gold these kind of things just abruptly add one smallest piece of themselves. Besides, electricity does the same too. Looking at cell multiplication, we might have another model of adding one piece of the self.

Anyway, I meant if 8:00 does exist -as you might agree, but according to the probability theory, anything Cannot occur at this moment because of the absolute 0 probability on it.


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#21 2007-08-05 17:32:04

Ricky
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Re: discrete or continuous Uni.Ran.Var.?

Ricky, your counter-example is Maxwell Equations if I got it right.

Another thing he doesn't seem to comprehend is that even if the world were truly discrete, then it has to be by such a small amount that the error bars are almost non-existent.  For example, calculate the area under the curve x^2 from 0 to 1 using the fact that "space" is discrete and can only exist in units of 10^(-100).  Now do the same calculation using the fact that space is continuous using integral calculus.  What is the difference in area?  Which calculation was quicker?

George, what your saying certainly is possibly true.  Then again, what isn't?  You don't have evidence for it, only your belief that there are contradictions.  Something not making sense to you is not a contradiction.  And math has worked.  We see it working every day.  I've said this before, and I'll say it again.  You got something better, something that can be as successful as math has, let's have it.  Heck, even something that works in only one situation where traditional math doesn't, give it up.  Otherwise, your argument is not grounded in logic nor in application.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#22 2007-08-06 02:28:48

George,Y
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Re: discrete or continuous Uni.Ran.Var.?

I have just posted one good counter-example:

If 8:00 does exist,

why any event has no probability to occur at this moment theoretically yet happens in real life?

For example, when do you usually get up? 7:40-8:30 most of the times? Okay, let's build up a probability distribution concentrated in 7:40-8:30. We don't even need it to be uniformly distributed. Now tell me what's the probability that you got at 8:00 sharp? Or what's the probability that you got up at the Exact time that you got up this morning?

0? yes! 0 means not any. You have no probability to get up that moment- you Cannot get up yet you Did get up. Tell me why this cannot testify maths wrong?


"your argument is not grounded in logic "
No, I obey logic faithfully, whereas you always state we can allow arbitary settings like axioms that are contradicting themselves in logic.


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#23 2007-08-06 08:56:07

Ricky
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Re: discrete or continuous Uni.Ran.Var.?

I have just posted one good counter-example:

If 8:00 does exist,

why any event has no probability to occur at this moment theoretically yet happens in real life?

For example, when do you usually get up? 7:40-8:30 most of the times? Okay, let's build up a probability distribution concentrated in 7:40-8:30. We don't even need it to be uniformly distributed. Now tell me what's the probability that you got at 8:00 sharp? Or what's the probability that you got up at the Exact time that you got up this morning?

0? yes! 0 means not any. You have no probability to get up that moment- you Cannot get up yet you Did get up. Tell me why this cannot testify maths wrong?

You are attempting to apply methods of discrete probability to a continuous problem.  Are you really all that surprised that you came to a contradiction?

"your argument is not grounded in logic "
No, I obey logic faithfully, whereas you always state we can allow arbitary settings like axioms that are contradicting themselves in logic.

Poor wording.  What I meant was that you claim there are contradictions when you have failed to show a single one.  For example, the above does not reach a contradiction because you are applying a branch of mathematics (discrete probability theory) to a distinctly different branch (continuous probability theory).  Yet you continually shout that there are contradictions, no matter how many times you've failed to find one.

Just because something doesn't make sense to you does not mean there is a contradiction.  Just because something is not intuitive also does not mean there is a contradiction.  Infinity is not intuitive.  Density and countability are certainly not intuitive.  To show a contradiction, you must reach a conclusion that something known to be false is true, or something known to be true is false.  You have yet to do so in any post here.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#24 2007-08-06 21:41:26

George,Y
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Re: discrete or continuous Uni.Ran.Var.?

Okay, how do you solve that problem, Ricky? Tell me how do you build a model to get up at morning.

Or, if you build a continuous model, you can only tell the probability to get up before 8:00 and after 8:00? Just fine. Then what's 1 minus these two probabilities? 0? yes-it's just the probability of getting up at 8:00 sharp. Still no where to hide, Ricky.

Just as I have said long before, you always use bans to avoid facing the contradiction. Every contradiction can be banned from exploring, smart!

Last edited by George,Y (2007-08-06 21:44:46)


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#25 2007-08-06 21:45:25

George,Y
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Re: discrete or continuous Uni.Ran.Var.?

MathsIsFun wrote:

I enjoy your contributions, George, they make me think harder about continuous vs discrete.

Thanks, glad you are one of my first readers.:)


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