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The Man Who Invented Modern Probability

nautil.us

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Re: The Man Who Invented Modern Probability

#21

Post by nawitus »

mag487 wrote: post #16
nawitus wrote: post #11

Earlier quoted context omitted.

I didn't know that. Any link to a proof?

There's only a countable number of formulas of ZFC. Therefore, there's only a countable number ways of writing out a formula of ZFC that uniquely specifies a real number. Therefore, only countably many real numbers exist that have such definitions. And all countable sets of reals have zero Lebesgue measure.

Additional question: what's the Kolmogorov complexity of a irrational numberwhich is not described with a ZFC formula? And how is it described?

Re: The Man Who Invented Modern Probability

#22

Post by SatvikBeri »

dxbydt wrote: post #9
tikhonj wrote: post #4

> Kolmogorov armed a group of researchers with electromechanical calculators and charged them with the task of calculating the rhythmical structures of Russian poetry And this is what "big data" and NLP were like before computers :). One can only imagine what he could have done with a modern computer. (Which is particular hard to imagine because he's responsible for advances which ultimately led to these same modern…

>And this is what "big data" and NLP were like before computers :) aside, I recently interviewed a candidate for a "Big Data" position. He got quite chatty about Kolmogorov & Measure Theory, so I quickly cooked up an interesting problem to relax his nerves. I opened the interview with "If the side of a cube comes from a uniform distribution between 1 and 5 inches, what's the expected volume of the cube ?" I thought h…

If the side length isn't constrained to integers, the expected volume is 39 inches cubed.

In a bit more detail: take the integral of x^3 (the volume function) from 1 to 5 and you get 5^4/4 - 1^4/4 = 624/4 = 156. Then we divide by the length of the line segment (4) to get 39.

If the side lengths have to be integers, then it's just (1 + 8 + 27 + 64 + 125)/5 = 45 inches.

Edit: corrected division mistake, thanks to dxbydt

Edit 2: and corrected a second mistake thanks to tzs.

Re: The Man Who Invented Modern Probability

#24

Post by consz »

nawitus wrote: post #20
nwhitehead wrote: post #14

Earlier quoted context omitted.

Consider the set P of programs that take no input and generate an infinite stream of digits. Each program in P has a finite length and is written out of a finite set of symbols, so there must only be a finite number of programs of any given length. That makes P countable. Let Q be the set of numbers described by programs in P. Each program from P describes exactly one number, so Q must also be countable. The set of i…

Your proof seems to only prove that a finite number of programs (described by you) which can produce a finite number of irrational numbers, while there are infinite number of irrational numbers. But we're surely not talking about a finite number of programs. I'm not even sure what the Kolmogorov complexity of a "complex irrational number" means. If you need the sequence of digits and you cannot use an algorithm to pr…

The set of irrational numbers which can be defined algorithmically are programable, eg. they are defined by a program of finite length.

Re: The Man Who Invented Modern Probability

#25

Post by dxbydt »

SatvikBeri wrote: post #22
dxbydt wrote: post #9

Earlier quoted context omitted.

>And this is what "big data" and NLP were like before computers :) aside, I recently interviewed a candidate for a "Big Data" position. He got quite chatty about Kolmogorov & Measure Theory, so I quickly cooked up an interesting problem to relax his nerves. I opened the interview with "If the side of a cube comes from a uniform distribution between 1 and 5 inches, what's the expected volume of the cube ?" I thought h…

If the side length isn't constrained to integers, the expected volume is 39 inches cubed. In a bit more detail: take the integral of x^3 (the volume function) from 1 to 5 and you get 5^4/4 - 1^4/4 = 624/4 = 156. Then we divide by the length of the line segment (4) to get 39. If the side lengths have to be integers, then it's just (1 + 8 + 27 + 64 + 125)/5 = 45 inches. Edit: corrected division mistake, thanks to dxbyd…

not 59, 39.

Re: The Man Who Invented Modern Probability

#26

Post by tzs »

relaia wrote: post #13
dxbydt wrote: post #9

Earlier quoted context omitted.

>And this is what "big data" and NLP were like before computers :) aside, I recently interviewed a candidate for a "Big Data" position. He got quite chatty about Kolmogorov & Measure Theory, so I quickly cooked up an interesting problem to relax his nerves. I opened the interview with "If the side of a cube comes from a uniform distribution between 1 and 5 inches, what's the expected volume of the cube ?" I thought h…

27?

Nope. To see where the subtlety lies in this problem, simplify it. Change it to a discrete problem: the side can be 1, 3, or 5, each with probability 1/3. The expected value of the side length is still 3.

1/3 of the cubes have volume 1, 1/3 have volume 27, and 1/3 have volume 125. The expected value of the volume is 51. Think about why it is 51, not 27, and you'll see why it is not 27 for the continuous case.

In general, if you have a random variable X, and some function of that random variable, f(X), then it is NOT true that expected value of f(X) = f(expected value of X).

Re: The Man Who Invented Modern Probability

#27

Post by drdeca »

SatvikBeri wrote: post #22
dxbydt wrote: post #9

Earlier quoted context omitted.

>And this is what "big data" and NLP were like before computers :) aside, I recently interviewed a candidate for a "Big Data" position. He got quite chatty about Kolmogorov & Measure Theory, so I quickly cooked up an interesting problem to relax his nerves. I opened the interview with "If the side of a cube comes from a uniform distribution between 1 and 5 inches, what's the expected volume of the cube ?" I thought h…

If the side length isn't constrained to integers, the expected volume is 39 inches cubed. In a bit more detail: take the integral of x^3 (the volume function) from 1 to 5 and you get 5^4/4 - 1^4/4 = 624/4 = 156. Then we divide by the length of the line segment (4) to get 39. If the side lengths have to be integers, then it's just (1 + 8 + 27 + 64 + 125)/5 = 45 inches. Edit: corrected division mistake, thanks to dxbyd…

156/4=39

Re: The Man Who Invented Modern Probability

#28

Post by SatvikBeri »

dxbydt wrote: post #25
SatvikBeri wrote: post #22

Earlier quoted context omitted.

If the side length isn't constrained to integers, the expected volume is 39 inches cubed. In a bit more detail: take the integral of x^3 (the volume function) from 1 to 5 and you get 5^4/4 - 1^4/4 = 624/4 = 156. Then we divide by the length of the line segment (4) to get 39. If the side lengths have to be integers, then it's just (1 + 8 + 27 + 64 + 125)/5 = 45 inches. Edit: corrected division mistake, thanks to dxbyd…

not 59, 39.

Heh, failed at the simplest step. Thanks for the correction.

Re: The Man Who Invented Modern Probability

#29

Post by mag487 »

nawitus wrote: post #21
mag487 wrote: post #16

Earlier quoted context omitted.

There's only a countable number of formulas of ZFC. Therefore, there's only a countable number ways of writing out a formula of ZFC that uniquely specifies a real number. Therefore, only countably many real numbers exist that have such definitions. And all countable sets of reals have zero Lebesgue measure.

Additional question: what's the Kolmogorov complexity of a irrational numberwhich is not described with a ZFC formula? And how is it described?

I haven't really studied algorithmic information theory, but I'd assume that Kolmogorov complexity isn't defined for uncomputable/undefinable numbers. Or maybe it's defined as "infinite," but either way, my guess is that such numbers are simply ignored. (Interestingly, the function which takes a computable number and outputs its Kolmogorov complexity is itself uncomputable!)

Re: The Man Who Invented Modern Probability

#30

Post by tzs »

SatvikBeri wrote: post #22
dxbydt wrote: post #9

Earlier quoted context omitted.

>And this is what "big data" and NLP were like before computers :) aside, I recently interviewed a candidate for a "Big Data" position. He got quite chatty about Kolmogorov & Measure Theory, so I quickly cooked up an interesting problem to relax his nerves. I opened the interview with "If the side of a cube comes from a uniform distribution between 1 and 5 inches, what's the expected volume of the cube ?" I thought h…

If the side length isn't constrained to integers, the expected volume is 39 inches cubed. In a bit more detail: take the integral of x^3 (the volume function) from 1 to 5 and you get 5^4/4 - 1^4/4 = 624/4 = 156. Then we divide by the length of the line segment (4) to get 39. If the side lengths have to be integers, then it's just (1 + 8 + 27 + 64 + 125)/5 = 45 inches. Edit: corrected division mistake, thanks to dxbyd…

> Edit: corrected division mistake, thanks to dxbydt

Well, one of the mistakes:

> If the side lengths have to be integers, then it's just (1 + 8 + 27 + 125)/4 = 40.25 inches

If I'm not mistaken, 4 is an integer, and so should not have cruelly snubbed when you sent out invitations to your summation of the cubes of the integers from 1 to 5. Sure, 4 is not a cool integer like the others in 1 to 5...he's got a reputation as a square, but that doesn't mean he doesn't have feelings.

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