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The math that explains why bell curves are everywhere

quantamagazine.org

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Re: The math that explains why bell curves are everywhere

#81

Post by CoastalCoder »

srean wrote: post #80

The way I understand this is that adding of random variables is a smoothening operation on their densities (more generally the distributions, but let me speak of densities only). A little more formally, additions over random variables are convolutions of their densities. Repeated additions are repeated convolutions. A single convolution can be understood as a matrix multiplication by a specific symmetric matrix. Repe…

> Anyone familiar with linear algebra will know that repeated matrix multiplication by non degenerate matrices reveals it's eigenvectors.

TIL that I'm not "familiar" with linear algebra ;)

But seriously, thanks for sharing that knowledge.

Re: The math that explains why bell curves are everywhere

#82

Post by GuB-42 »

hodgehog11 wrote: post #79
bicepjai wrote: post #40

This is one of my favorite philosophical questions to ponder. I always ask it in interviews as a warmup to get their thoughts. I’ve noticed that interviewees often curl up, thinking it’s a technical question, so I’ve been modifying the question one after the other to make it less scary. The interviews are for data scientist roles.

It's amazing that you find so many that are uncomfortable with this question. I literally teach a first-year data science course and I ask the students this very question. I spend half a lecture on it and put it in their assessment. This is one of the most fundamental things to understand in statistics. If you don't have at least some degree of comfort with this, you have no business working with data in a profession…

You can be comfortable about the concept, but not comfortable about the interview.

The way I understand it, OP asked this as a way to open the conversation, while candidates interpreted it as a math problem to solve, unintentionally getting their mind into "exam" mode.

Re: The math that explains why bell curves are everywhere

#83

Post by srean »

CoastalCoder wrote: post #81
srean wrote: post #80

The way I understand this is that adding of random variables is a smoothening operation on their densities (more generally the distributions, but let me speak of densities only). A little more formally, additions over random variables are convolutions of their densities. Repeated additions are repeated convolutions. A single convolution can be understood as a matrix multiplication by a specific symmetric matrix. Repe…

> Anyone familiar with linear algebra will know that repeated matrix multiplication by non degenerate matrices reveals it's eigenvectors. TIL that I'm not "familiar" with linear algebra ;) But seriously, thanks for sharing that knowledge.

If you are not speaking in jest (I strongly suspect you are), knowledge of linear algebra is one of the biggest bang for buck one can get as an investment in mathematical knowledge.

So humble and basic a field. So wide it's consequences and scope.

Re: The math that explains why bell curves are everywhere

#84

Post by andyjohnson0 »

DroneBetter wrote: post #2

I hate Quanta a lot a vast amount of fluff for less than a college statistics professor would (hopefully) be able to impart with a chalkboard in 10 minutes, when Quanta has the ability to prepare animated diagrams like 3Blue1Brown but chooses not to use it they could go down myriad paths, like how it provides that random walks on square lattices are asymptotically isotropic, or give any other simple easy-to-understan…

I probably don't have your mathematical sophistication - but I like and appreciate Quanta precisely because it helps people like me to understand a little bit about challenging things. This enriches my tiny life, and I hope it also makes the world a fractionally better place for us all.

Perhaps you're just not in their intended audience?

Re: The math that explains why bell curves are everywhere

#85

Post by Sharlin »

CoastalCoder wrote: post #81
srean wrote: post #80

The way I understand this is that adding of random variables is a smoothening operation on their densities (more generally the distributions, but let me speak of densities only). A little more formally, additions over random variables are convolutions of their densities. Repeated additions are repeated convolutions. A single convolution can be understood as a matrix multiplication by a specific symmetric matrix. Repe…

> Anyone familiar with linear algebra will know that repeated matrix multiplication by non degenerate matrices reveals it's eigenvectors. TIL that I'm not "familiar" with linear algebra ;) But seriously, thanks for sharing that knowledge.

Yeah, I don't think this was revealed on my undergrad linalg course, and neither during all my years of using linalg in computer graphics =D

Re: The math that explains why bell curves are everywhere

#86

Post by wodenokoto »

> the “steadfast order of the universe” that eventually overcame any and all deviations from the bell.

I can’t believe the author wrote that without explaining why it’s called the bell curve.

I find the article spends a lot of time talking about repeating games without really getting to the meat of it.

If you throw a dice a million times the result is still following a uniform distribution.

It isn’t until you start summing random events that the normal distribution occurs.

Re: The math that explains why bell curves are everywhere

#87

Post by CoastalCoder »

srean wrote: post #83
CoastalCoder wrote: post #81

Earlier quoted context omitted.

> Anyone familiar with linear algebra will know that repeated matrix multiplication by non degenerate matrices reveals it's eigenvectors. TIL that I'm not "familiar" with linear algebra ;) But seriously, thanks for sharing that knowledge.

If you are not speaking in jest (I strongly suspect you are), knowledge of linear algebra is one of the biggest bang for buck one can get as an investment in mathematical knowledge. So humble and basic a field. So wide it's consequences and scope.

My first sentence was in jest. I've used LA for various things, but haven't had many dealings with eigenvectors. So that information was genuinely new to me.

My expression of gratitude was sincere.

Re: The math that explains why bell curves are everywhere

#88

Post by Sharlin »

srean wrote: post #83
CoastalCoder wrote: post #81

Earlier quoted context omitted.

> Anyone familiar with linear algebra will know that repeated matrix multiplication by non degenerate matrices reveals it's eigenvectors. TIL that I'm not "familiar" with linear algebra ;) But seriously, thanks for sharing that knowledge.

If you are not speaking in jest (I strongly suspect you are), knowledge of linear algebra is one of the biggest bang for buck one can get as an investment in mathematical knowledge. So humble and basic a field. So wide it's consequences and scope.

Their point was that "familiarity" apparently means different things for different people :P Someone using linalg in computer graphics applications may say they're familiar with it even though they've never heard the term "eigenvector". I'm not actually sure about what you mean – how does repeated multiplication reveal eigenvectors?

Re: The math that explains why bell curves are everywhere

#89

Post by srean »

Sharlin wrote: post #88
srean wrote: post #83

Earlier quoted context omitted.

If you are not speaking in jest (I strongly suspect you are), knowledge of linear algebra is one of the biggest bang for buck one can get as an investment in mathematical knowledge. So humble and basic a field. So wide it's consequences and scope.

Their point was that "familiarity" apparently means different things for different people :P Someone using linalg in computer graphics applications may say they're familiar with it even though they've never heard the term "eigenvector". I'm not actually sure about what you mean – how does repeated multiplication reveal eigenvectors?

Consider a diagonalizable matrix A. For example, a real symmetric matrix. Start with any vector b and keep multiplying it with A.

    A A A ... A b
The vector that the result will converge to is a scaled version of one of the eigenvectors of the matrix A.

But which one ? The one with the largest eigenvalue among all eigenvectors not orthogonal to b.

https://en.wikipedia.org/wiki/Power_iteration

Re: The math that explains why bell curves are everywhere

#90

Post by srean »

CoastalCoder wrote: post #87
srean wrote: post #83

Earlier quoted context omitted.

If you are not speaking in jest (I strongly suspect you are), knowledge of linear algebra is one of the biggest bang for buck one can get as an investment in mathematical knowledge. So humble and basic a field. So wide it's consequences and scope.

My first sentence was in jest. I've used LA for various things, but haven't had many dealings with eigenvectors. So that information was genuinely new to me. My expression of gratitude was sincere.

Understood and thanks for the opportunity of sharing together in the joy of something so amusing.
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