Comments on: Sums of uniform random values http://www.johndcook.com/blog/2009/02/12/sums-of-uniform-random-values/ The blog of John D. Cook Sat, 11 Feb 2012 01:10:06 -0500 http://wordpress.org/?v=2.8.4 hourly 1 By: Mark http://www.johndcook.com/blog/2009/02/12/sums-of-uniform-random-values/comment-page-1/#comment-100475 Mark Fri, 26 Aug 2011 19:39:42 +0000 http://www.johndcook.com/blog/?p=1491#comment-100475 Personally, considering how very "non-normal" the uniform distribution is, I am always amazed how close to a normal we get summing just two uniforms (CLT requires n->infinity and n=2 is a long, long way from that)! I believe the sum (or possibly average) of n uniforms is sometimes called Bartlett's distribution but I have no reference for the origin of this name (likely because it follows Stigler's law). Personally, considering how very “non-normal” the uniform distribution is, I am always amazed how close to a normal we get summing just two uniforms (CLT requires n->infinity and n=2 is a long, long way from that)!

I believe the sum (or possibly average) of n uniforms is sometimes called Bartlett’s distribution but I have no reference for the origin of this name (likely because it follows Stigler’s law).

]]> By: Developer http://www.johndcook.com/blog/2009/02/12/sums-of-uniform-random-values/comment-page-1/#comment-73365 Developer Mon, 28 Mar 2011 08:18:35 +0000 http://www.johndcook.com/blog/?p=1491#comment-73365 Hi John, Could you provide a simple code e.g., Python code, to implement the given formula? I will understand much easier the concept then. I had also difficulty in very first part while the rest was understandable! "Suppose you have uniform random samples from the interval [0, 1]." >>> s = scipy.rand(k,n) "If you add a large number (k) of such samples together, the sum ..."(????) >>> s1 = s.flatten() #all samples to one big sample or >>> s1 = scipy.sum(s) #k sums however none of the results above have normal distribution! Thanks. Developer Hi John,
Could you provide a simple code e.g., Python code, to implement the given formula? I will understand much easier the concept then.
I had also difficulty in very first part while the rest was understandable!
“Suppose you have uniform random samples from the interval [0, 1].”
>>> s = scipy.rand(k,n)
“If you add a large number (k) of such samples together, the sum …”(????)
>>> s1 = s.flatten() #all samples to one big sample
or
>>> s1 = scipy.sum(s) #k sums
however none of the results above have normal distribution!
Thanks.
Developer

]]> By: John http://www.johndcook.com/blog/2009/02/12/sums-of-uniform-random-values/comment-page-1/#comment-28866 John Mon, 14 Dec 2009 21:53:31 +0000 http://www.johndcook.com/blog/?p=1491#comment-28866 Tanvir, we may be talking about PDF versus CDF. In my blog post, I gave the PDF. Feller (page 285 of the third edition, problem 20) gives the CDF. Tanvir, we may be talking about PDF versus CDF. In my blog post, I gave the PDF. Feller (page 285 of the third edition, problem 20) gives the CDF.

]]> By: Tanvir Zaman http://www.johndcook.com/blog/2009/02/12/sums-of-uniform-random-values/comment-page-1/#comment-28805 Tanvir Zaman Mon, 14 Dec 2009 04:29:42 +0000 http://www.johndcook.com/blog/?p=1491#comment-28805 In Feller's book the expression has n! instead of (n-1)! and (x-k)^n instead of (x-k)^(n-1) Would you please explain why is this difference? In Feller’s book the expression has n! instead of (n-1)! and (x-k)^n instead of (x-k)^(n-1)
Would you please explain why is this difference?

]]> By: electronicresourceguy http://www.johndcook.com/blog/2009/02/12/sums-of-uniform-random-values/comment-page-1/#comment-17013 electronicresourceguy Wed, 06 May 2009 03:33:10 +0000 http://www.johndcook.com/blog/?p=1491#comment-17013 What about the margins John? The normal distribution still has value outside of -6 to 6, the sum of 12 uniform random variables does not. Considering that anything beyond a 6 standard deviation span of the normal is generally accepted to be insignificant, it probably does not matter. What about the margins John?

The normal distribution still has value outside of -6 to 6, the sum of 12 uniform random variables does not.
Considering that anything beyond a 6 standard deviation span of the normal is generally accepted to be insignificant, it probably does not matter.

]]> By: gappy http://www.johndcook.com/blog/2009/02/12/sums-of-uniform-random-values/comment-page-1/#comment-13267 gappy Fri, 13 Feb 2009 04:00:48 +0000 http://www.johndcook.com/blog/?p=1491#comment-13267 John, just a couple of notes: Feller's classic for free is available on scribd.com I am not sure how it is possible that this and other good books are there, but they have been there for quite a while. Second, it's interesting how all modern probability books (Durrett, Pollard, Kallenberg, even Billingsley) do not have such beautifully worked-out numerical examples. John,
just a couple of notes: Feller’s classic for free is available on scribd.com I am not sure how it is possible that this and other good books are there, but they have been there for quite a while.

Second, it’s interesting how all modern probability books (Durrett, Pollard, Kallenberg, even Billingsley) do not have such beautifully worked-out numerical examples.

]]> By: John http://www.johndcook.com/blog/2009/02/12/sums-of-uniform-random-values/comment-page-1/#comment-13257 John Thu, 12 Feb 2009 22:31:27 +0000 http://www.johndcook.com/blog/?p=1491#comment-13257 It's interesting to compare numerical error and programming error. Generating 12 uniform values is probably a few times more expensive than something like Marsaglia's ziggurat algorithm. On the other hand, the uniform average method is so simple, it's hard to imagine programming it incorrectly. A bug could more easily hide in code for Marsaglia's algorithm. It’s interesting to compare numerical error and programming error. Generating 12 uniform values is probably a few times more expensive than something like Marsaglia’s ziggurat algorithm. On the other hand, the uniform average method is so simple, it’s hard to imagine programming it incorrectly. A bug could more easily hide in code for Marsaglia’s algorithm.

]]> By: John Venier http://www.johndcook.com/blog/2009/02/12/sums-of-uniform-random-values/comment-page-1/#comment-13252 John Venier Thu, 12 Feb 2009 21:38:12 +0000 http://www.johndcook.com/blog/?p=1491#comment-13252 Thanks for doing the math, John. I was curious how good it was. Sounds plenty good. I think we (and I intentionally include myself) sometimes worry too much about the quality of PRNGs and approximations. Fortunately these days it is not usually not much harder to make the best choice than it is to make a bad one; the real trick is knowing when your choice is bad and that there is a better alternative. But if it looks like a Herculean task to get a slightly better PRNG we may do well to recall what was used (very successfully) in the Manhattan project. But I digress. In fact, I wrote a long comment and redacted almost all of it due to a bad case of digression to the mean. The key consideration is the symmetry of the underlying distribution. A few practical experiments are enlightening, and may well change your attitude about a lot of statistics. There are many similar places in statistics where the practical seems at odds with the theory and especially the classical presentation of statistics. I think this is a fascinating topic in its own right and have examined it quite a bit, but it is out of place here. Thanks for doing the math, John. I was curious how good it was. Sounds plenty good. I think we (and I intentionally include myself) sometimes worry too much about the quality of PRNGs and approximations. Fortunately these days it is not usually not much harder to make the best choice than it is to make a bad one; the real trick is knowing when your choice is bad and that there is a better alternative. But if it looks like a Herculean task to get a slightly better PRNG we may do well to recall what was used (very successfully) in the Manhattan project. But I digress.

In fact, I wrote a long comment and redacted almost all of it due to a bad case of digression to the mean. The key consideration is the symmetry of the underlying distribution. A few practical experiments are enlightening, and may well change your attitude about a lot of statistics. There are many similar places in statistics where the practical seems at odds with the theory and especially the classical presentation of statistics. I think this is a fascinating topic in its own right and have examined it quite a bit, but it is out of place here.

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