Comments on: Four characterizations of the normal distribution http://www.johndcook.com/blog/2008/03/13/four-characterizations-of-the-normal-distribution/ The blog of John D. Cook Sat, 11 Feb 2012 22:42:11 -0500 http://wordpress.org/?v=2.8.4 hourly 1 By: Agustín http://www.johndcook.com/blog/2008/03/13/four-characterizations-of-the-normal-distribution/comment-page-1/#comment-110893 Agustín Fri, 28 Oct 2011 20:47:43 +0000 http://www.johndcook.com/blog/2008/03/13/four-characterizations-of-the-normal-distribution/#comment-110893 New comments I want to consult you about the internal tanh distribution Y=tanh[(x/p)^n] After performing a lot of calculus to evaluate the medium I produce "an approximate solution" µ= p* Gamma[0,71158 + 0,68926/n]/Gamma[0,71158] If I want to calculate the kth gross moment I apply µ= (p^k)* Gamma[0,71158 +k* 0,68926/n]/Gamma[0,71158] Are those expressions a strong solutions in the n>2 zone? One of my son told me that these expressions are so exact to be approximations New comments
I want to consult you about the internal tanh distribution
Y=tanh[(x/p)^n]
After performing a lot of calculus to evaluate the medium I produce “an approximate solution”

µ= p* Gamma[0,71158 + 0,68926/n]/Gamma[0,71158]
If I want to calculate the kth gross moment I apply
µ= (p^k)* Gamma[0,71158 +k* 0,68926/n]/Gamma[0,71158]
Are those expressions a strong solutions in the n>2 zone?
One of my son told me that these expressions are so exact to be approximations

]]> By: Agustín http://www.johndcook.com/blog/2008/03/13/four-characterizations-of-the-normal-distribution/comment-page-1/#comment-110892 Agustín Fri, 28 Oct 2011 20:45:02 +0000 http://www.johndcook.com/blog/2008/03/13/four-characterizations-of-the-normal-distribution/#comment-110892 I want to consult you about the internal tanh distribution Y=tanh[(x/p)^n] After performing a lot of calculus to evaluate the medium I produce "an approximate solution" µ= p* Gamma[0,71158 + 0,68926/n]/Gamma[0,71158] If I wnat to calculate the kth gross moment I apply µ= (p^k)* Gamma[0,71158 +k* 0,68926/n]/Gamma[0,71158] Are those expressions a strong solutions in the n>2 zone? One of my son told me that these expressions are so exact to be approximations I want to consult you about the internal tanh distribution
Y=tanh[(x/p)^n]
After performing a lot of calculus to evaluate the medium I produce “an approximate solution”

µ= p* Gamma[0,71158 + 0,68926/n]/Gamma[0,71158]
If I wnat to calculate the kth gross moment I apply
µ= (p^k)* Gamma[0,71158 +k* 0,68926/n]/Gamma[0,71158]
Are those expressions a strong solutions in the n>2 zone?
One of my son told me that these expressions are so exact to be approximations

]]> By: Agustín F. CORREA http://www.johndcook.com/blog/2008/03/13/four-characterizations-of-the-normal-distribution/comment-page-1/#comment-110396 Agustín F. CORREA Wed, 26 Oct 2011 16:03:09 +0000 http://www.johndcook.com/blog/2008/03/13/four-characterizations-of-the-normal-distribution/#comment-110396 I am using the tanh internal distribution .I think it is a camouflaged gamma dsitrbution Tanh [(x/p)^n]. It produces the p parameter (76,1%) in a satisfactory way The Mode is calculaterd in a fast iteration routine. Medium and gross moments are calculated by a simple gamma function ( of my own) Sometimes it is difficult to give a suitable explanation when n< 1 because it does not produce a mode, it seems I am in front of a Poisson random dsitribution .I am not sure of it I am using the tanh internal distribution .I think it is a camouflaged gamma dsitrbution
Tanh [(x/p)^n]. It produces the p parameter (76,1%) in a satisfactory way
The Mode is calculaterd in a fast iteration routine.
Medium and gross moments are calculated by a simple gamma function ( of my own)
Sometimes it is difficult to give a suitable explanation when n< 1 because it does not produce a mode, it seems I am in front of a Poisson random dsitribution .I am not sure of it

]]> By: John http://www.johndcook.com/blog/2008/03/13/four-characterizations-of-the-normal-distribution/comment-page-1/#comment-110344 John Wed, 26 Oct 2011 12:13:02 +0000 http://www.johndcook.com/blog/2008/03/13/four-characterizations-of-the-normal-distribution/#comment-110344 Mark: You are correct. There must be some additional hypothesis, and now I don't know what it was. Mark: You are correct. There must be some additional hypothesis, and now I don’t know what it was.

]]> By: Mark http://www.johndcook.com/blog/2008/03/13/four-characterizations-of-the-normal-distribution/comment-page-1/#comment-110276 Mark Wed, 26 Oct 2011 06:10:26 +0000 http://www.johndcook.com/blog/2008/03/13/four-characterizations-of-the-normal-distribution/#comment-110276 Hi John, I should probably look at the references you mention first but I'm going to fire away my query first anyway! My query is about the claim in (3) that the MLE for the mean of a random variable and the sample mean only co-incide for the normal. There are other distributions where this is true too (Poisson, exponential, Bernoulli,...)? I would also be interested in the reference discussing the original use of "normal" (in the sense of orthogonal or perpendicular) in this context. Hi John,

I should probably look at the references you mention first but I’m going to fire away my query first anyway!

My query is about the claim in (3) that the MLE for the mean of a random variable and the sample mean only co-incide for the normal. There are other distributions where this is true too (Poisson, exponential, Bernoulli,…)?

I would also be interested in the reference discussing the original use of “normal” (in the sense of orthogonal or perpendicular) in this context.

]]> By: Matthew Handy http://www.johndcook.com/blog/2008/03/13/four-characterizations-of-the-normal-distribution/comment-page-1/#comment-67280 Matthew Handy Tue, 22 Feb 2011 17:48:07 +0000 http://www.johndcook.com/blog/2008/03/13/four-characterizations-of-the-normal-distribution/#comment-67280 Can you say more about the two things that were perpendicular that led to the distribution being called Normal? (In the UK, we routinely use the term "normal" to describe things that are perpendicular.) Can you say more about the two things that were perpendicular that led to the distribution being called Normal? (In the UK, we routinely use the term “normal” to describe things that are perpendicular.)

]]> By: Matt http://www.johndcook.com/blog/2008/03/13/four-characterizations-of-the-normal-distribution/comment-page-1/#comment-67270 Matt Tue, 22 Feb 2011 17:12:20 +0000 http://www.johndcook.com/blog/2008/03/13/four-characterizations-of-the-normal-distribution/#comment-67270 The normal distribution maximising entropy is in Shannon's seminal 1948 paper <a href="http://cm.bell-labs.com/cm/ms/what/shannonday/paper.html" rel="nofollow">"A Mathematical Theory of Communication"</a> (Section 20). Shannon founded all this information theory stuff, so I'd be surprised if it appears anywhere before that. The normal distribution maximising entropy is in Shannon’s seminal 1948 paper “A Mathematical Theory of Communication” (Section 20). Shannon founded all this information theory stuff, so I’d be surprised if it appears anywhere before that.

]]> By: John http://www.johndcook.com/blog/2008/03/13/four-characterizations-of-the-normal-distribution/comment-page-1/#comment-31663 John Wed, 27 Jan 2010 18:41:46 +0000 http://www.johndcook.com/blog/2008/03/13/four-characterizations-of-the-normal-distribution/#comment-31663 Agustín: The fact that the normal distribution can be negative is not such a handicap in application. Obviously it could be for some applications, but often negative values are so rare that this isn't a problem. For example, women's heights are normally distributed with mean 64 and standard deviation 3. That distribution could have negative values, but they would be 22 standard deviations away from the mean, something with probability less than 10^-100. Sometimes people use a truncated normal, a normal distribution restricted to an interval, to avoid negative values. Sometimes people use a <a href="http://www.johndcook.com/blog/2009/09/29/achievement-is-log-normal/" rel="nofollow">log normal</a> distribution, which is equivalent to assuming the log of the data is normally distributed. Or you could use a gamma distribution with a large shape parameter; it's approximately normal but positive valued. Agustín: The fact that the normal distribution can be negative is not such a handicap in application. Obviously it could be for some applications, but often negative values are so rare that this isn’t a problem. For example, women’s heights are normally distributed with mean 64 and standard deviation 3. That distribution could have negative values, but they would be 22 standard deviations away from the mean, something with probability less than 10^-100.

Sometimes people use a truncated normal, a normal distribution restricted to an interval, to avoid negative values. Sometimes people use a log normal distribution, which is equivalent to assuming the log of the data is normally distributed. Or you could use a gamma distribution with a large shape parameter; it’s approximately normal but positive valued.

]]> By: Correa, Agustín http://www.johndcook.com/blog/2008/03/13/four-characterizations-of-the-normal-distribution/comment-page-1/#comment-31662 Correa, Agustín Wed, 27 Jan 2010 18:25:45 +0000 http://www.johndcook.com/blog/2008/03/13/four-characterizations-of-the-normal-distribution/#comment-31662 It is very simple; sometimes negative values are prohibited . So , how the Gaussian distribution may work in these kind of situations?? It is very simple; sometimes negative values are prohibited . So , how the Gaussian distribution may work in these kind of situations??

]]> By: John http://www.johndcook.com/blog/2008/03/13/four-characterizations-of-the-normal-distribution/comment-page-1/#comment-31661 John Wed, 27 Jan 2010 18:12:00 +0000 http://www.johndcook.com/blog/2008/03/13/four-characterizations-of-the-normal-distribution/#comment-31661 Thanks, Jason. Many things are normal <b>in the middle</b> but not many things are normal in the tails. So it all depends on how you're going to use the normal distribution. For example, I wrote a pair of posts, one explaining why human heights <a href="http://www.johndcook.com/blog/2008/07/20/why-heights-are-normally-distributed/" rel="nofollow">are</a> normally distributed and another explaining why they <a href="http://www.johndcook.com/blog/2008/07/20/why-heights-are-not-normally-distributed/" rel="nofollow">are not</a>. It all depends on what question you're asking. Thanks, Jason.

Many things are normal in the middle but not many things are normal in the tails. So it all depends on how you’re going to use the normal distribution. For example, I wrote a pair of posts, one explaining why human heights are normally distributed and another explaining why they are not. It all depends on what question you’re asking.

]]> By: Jason B http://www.johndcook.com/blog/2008/03/13/four-characterizations-of-the-normal-distribution/comment-page-1/#comment-31660 Jason B Wed, 27 Jan 2010 18:05:09 +0000 http://www.johndcook.com/blog/2008/03/13/four-characterizations-of-the-normal-distribution/#comment-31660 I once sat through a lecture by simulationist Averill Law (author of a seminal modeling and simulation text) where he advised the students that nothing in a system he has simulated followed the normal distribution. His assertion was that nothing anywhere followed the normal distribution. I suppose this means he never tried to model errors. Great post. I once sat through a lecture by simulationist Averill Law (author of a seminal modeling and simulation text) where he advised the students that nothing in a system he has simulated followed the normal distribution. His assertion was that nothing anywhere followed the normal distribution.

I suppose this means he never tried to model errors.

Great post.

]]> By: Tweets that mention Four characterizations of the normal distribution — The Endeavour -- Topsy.com http://www.johndcook.com/blog/2008/03/13/four-characterizations-of-the-normal-distribution/comment-page-1/#comment-31620 Tweets that mention Four characterizations of the normal distribution — The Endeavour -- Topsy.com Wed, 27 Jan 2010 04:32:33 +0000 http://www.johndcook.com/blog/2008/03/13/four-characterizations-of-the-normal-distribution/#comment-31620 [...] This post was mentioned on Twitter by Probability Fact, Mert Dikmen. Mert Dikmen said: RT @ProbFact: There are numerous ways to motivate the normal distribution. http://bit.ly/5kGlVm [...] [...] This post was mentioned on Twitter by Probability Fact, Mert Dikmen. Mert Dikmen said: RT @ProbFact: There are numerous ways to motivate the normal distribution. http://bit.ly/5kGlVm [...]

]]> By: Correa, Agustín http://www.johndcook.com/blog/2008/03/13/four-characterizations-of-the-normal-distribution/comment-page-1/#comment-30243 Correa, Agustín Tue, 05 Jan 2010 16:04:09 +0000 http://www.johndcook.com/blog/2008/03/13/four-characterizations-of-the-normal-distribution/#comment-30243 The Gaussian does not fit well distributions as granulometric's. I'd rather prefer the Tanh distribution or the log normal distribution Median , mode, and medium used not to be equal. The Gaussian does not fit well distributions as granulometric’s.
I’d rather prefer the Tanh distribution or the log normal distribution
Median , mode, and medium used not to be equal.

]]> By: Correa, Agustín http://www.johndcook.com/blog/2008/03/13/four-characterizations-of-the-normal-distribution/comment-page-1/#comment-30242 Correa, Agustín Tue, 05 Jan 2010 15:59:09 +0000 http://www.johndcook.com/blog/2008/03/13/four-characterizations-of-the-normal-distribution/#comment-30242 Why then we sometimes prefer the Tanh distribution or the log normal distribution?? regards Why then we sometimes prefer the Tanh distribution or the log normal distribution??
regards

]]> By: John http://www.johndcook.com/blog/2008/03/13/four-characterizations-of-the-normal-distribution/comment-page-1/#comment-38 John Thu, 13 Mar 2008 11:10:13 +0000 http://www.johndcook.com/blog/2008/03/13/four-characterizations-of-the-normal-distribution/#comment-38 Sure. (1) is in any book on probability. (2) and (3) can be found in "Probability Theory: The Logic of Science" by E. T. Jaynes, chapter 7. (4) can be found in Rao's book linked to in my post, page 162. Sure. (1) is in any book on probability. (2) and (3) can be found in “Probability Theory: The Logic of Science” by E. T. Jaynes, chapter 7. (4) can be found in Rao’s book linked to in my post, page 162.

]]> By: John B http://www.johndcook.com/blog/2008/03/13/four-characterizations-of-the-normal-distribution/comment-page-1/#comment-37 John B Thu, 13 Mar 2008 10:55:11 +0000 http://www.johndcook.com/blog/2008/03/13/four-characterizations-of-the-normal-distribution/#comment-37 <em>There are two strategies for estimating the mean of a random variable from a sample: the arithmetic mean of the samples, and the maximum likelihood value. Only for the normal distribution do these coincide.</em> Can you link to references so that we can read more about these characterizations? For example, I'd never heard the quoted characterization of the normal distribution, and I'd like to read more. There are two strategies for estimating the mean of a random variable from a sample: the arithmetic mean of the samples, and the maximum likelihood value. Only for the normal distribution do these coincide.

Can you link to references so that we can read more about these characterizations? For example, I’d never heard the quoted characterization of the normal distribution, and I’d like to read more.

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