Math Is Fun Forum

Ricky

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Registered: 2005-12-04

Posts: 3,791

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

You can not interpret probability of a continuous variable in the same way you do a discrete.  A 0 probability in a continuous system does not imply that something can't happen.  That is a property of a discrete system.

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"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..."

George,Y

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Registered: 2006-03-12

Posts: 1,379

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

So what 0 probability exactly mean in a continuous system, Ricky?

0 means some probability?

Then how much?

infinitesimal?

Or a small one?

How small?

No matter how small it is, it has some, then the total probability on infinite points is Infinity. Because
c(infinity)=infinity.

So may be the best interpretation of 0 is infinitesimal, right? good answer.

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X'(y-Xβ)=0

Ricky

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Registered: 2005-12-04

Posts: 3,791

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

0 does not mean anything.  There is absolutely no significance in it other than the limit presented earlier in this thread.  All values in a continuous system have a probability of 0.

Edit: The above isn't entirely true.  The probability of each point must be 0, because if it was any other real number than 0, the sum of the of probabilities would be infinite.

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"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..."

George,Y

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Registered: 2006-03-12

Posts: 1,379

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

The probability of each point must be 0, because if it was any other real number than 0, the sum of the of probabilities would be infinite.

Yes, hence the probability is 0, meaning not any probability. If any, the sum of all the probabilities would be infinite.

How do you conceive not any probability? -I interpret it as no chance at all, impossible. No matter how many trials you have, you cannot make it happen once, otherwise the probability of the event cannot be 0.

Then why again and again these "impossible" things just happen in real life? -You get up every morning, Ricky.

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X'(y-Xβ)=0

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

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

How do you conceive not any probability? -I interpret it as no chance at all, impossible. No matter how many trials you have, you cannot make it happen once, otherwise the probability of the event cannot be 0.

Yes, it is tempting to do so because it applies to discrete probabilities.  But that is not the case here and you must refrain from doing it.  This is a different system, one with different properties.  Just because it seems intuitive does not mean it is right.  Math is highly not intuitive.

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"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..."

George,Y

Member

Registered: 2006-03-12

Posts: 1,379

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

Ricky, you should distinguish between something intuitive and something logical.

You may not accept an intuitive perception, but you have to accept a logic one.

The 0 probability here either means possible or impossible, some chance or no chance at all. Not a third choice, only yes or no. That's how logic works.

If you just vaguously avoid talking about either case and try to ban the discussion, you are not reasoning but trying to make arbitary rules.

It will be a great pity for maths to be arbitary and illogical.

BTW:
1)A has no B.
2)A has some B.

Logically, if one in 1) and 2) is False, the other must be True.

This applies to any A and B(universal), such as  A=" the event occuring on some moment" , B= "probability"

I rather obey the universal rule of logic.

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X'(y-Xβ)=0

Ricky

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Registered: 2005-12-04

Posts: 3,791

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

The 0 probability here either means possible or impossible, some chance or no chance at all. Not a third choice, only yes or no. That's how logic works.

And until you give up this perception up, you will never understand continuous probability.  You say it is logical.  Show me the logic that ends with the conclusion, "Therefore, 0 means impossible."

Many times in mathematics, we have these great systems but they don't apply to everything.  We can not apply concepts of discrete probability to problems of continuous.  We find that we can't solve any problems of probabilities of the real numbers if we try to use discrete methods, everything will be 1/infinity.  No matter what you take this number to mean, it will result in contradiction.  Assume a random variable x is picked from [0, 10].  What is the chance that x will be less than or equal to 1?  I hope we all agree the answer is 10%.

But let us try to apply discrete methods as you would have us do George.  Let P(a) be the probability that the real number a will be choosen.  Let P(a) = w, where w is some element of any group G.  I hope we agree that any number system must follow the general properties of a Group.  Note that for any a, P(a) = w, that is, all probabilities are constant, just like in a discrete system.  Then:

Is the probability that any a in [0, 1] will be choosen.  However, [0, 1] and (1, 10] must have the same cardinality, and as such:

That is, you are just as equally likely to choose a number from [0, 1] as you are to do so in (1, 10].  Contradiction.

How do we fix this problem George?  How do we solve my proposed problem?  I welcome any solution you have to propose, so long as it is grounded in logic, as you so rightfully require.

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"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..."

George,Y

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Registered: 2006-03-12

Posts: 1,379

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

"How do we fix this problem George?  How do we solve my proposed problem? "

Simple, dump the whole trick of Reals.

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X'(y-Xβ)=0

Ricky

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Registered: 2005-12-04

Posts: 3,791

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

Ok, say goodbye to all forms of calculus, real analysis, complex analysis, topology, ZFC set theory, many parts of field theory, game theory, and the like.  You just eliminated half of maths, much of which have important results in solving real life problems that we no longer know how to solve.

Edit: George Y... welcome back to the 16th Century.

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"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..."

George,Y

Member

Registered: 2006-03-12

Posts: 1,379

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

Ok, say goodbye to all forms of calculus, real analysis, complex analysis, topology, ZFC set theory, many parts of field theory, game theory, and the like.  You just eliminated half of maths, much of which have important results in solving real life problems that we no longer know how to solve.

Edit: George Y... welcome back to the 16th Century.

Don't exaggerate, Ricky. A finite system admitting the last error isn't that uncapable.

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X'(y-Xβ)=0

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

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

Ok George lets dump the reals.  And I suppose you wish to dump the rationals too, because they would have the same problem as they are dense as well.  What system do we replace it with?  What decimal representations are valid and which are not?  How many digits of precision are we allowed to have?

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"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..."

George,Y

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Registered: 2006-03-12

Posts: 1,379

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

So far physics scientists have discovered the tiniest thing( including space, photon) in the world is about 10^-33 metres scale, and the largest about 10^33 metres scale. I guess 10,000 digits is quite enough. 10^-5,000~10^5,000. And actually,  even if it is not enough to describe the nature, this precision is far more than enough for us to apply because we are not able to construct things that detailed.

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X'(y-Xβ)=0

Ricky

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Registered: 2005-12-04

Posts: 3,791

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

So we are allowed to use the 33 digits for accuracy.  To calculate the the circumference of the earth's orbit and be accurate within 1 cm, how many digits of pi must we use?  I believe the answer is 64, but I have to double check that with a professor who used this in a presentation on pi.

According to you, George, we can't do that.

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"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..."

mathsyperson

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Registered: 2005-06-22

Posts: 4,900

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

George is allowing us 10000 digits, so that wouldn't be a problem.

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

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

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

Ah, sorry, I missed that.  Then simply calculate the surface area of a sphere with the radius of the known universe with the same problem.

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"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..."

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

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

Oh, and I forgot to mention how arbitrary 5,000 was picked.  No physical or real world reason, just some really big number.  Is that really what we are going to base math on?

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"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..."