Talk:Uniform distribution (continuous)

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Contents 1 Changed definition 2 PDF 3 Math error in the graph 4 Standard Uniform

[edit] Changed definition

I changed the definition for the continuous case so that F(F(U(x))=U(x) where U(x) is the uniform distribution and F is the continuous Fourier transform. If this change survives, I will alter the graph of the uniform distribution to reflect the change, something along the lines of the graph in the rectangular function. I'm trying to eventually bring the boxcar function, the rectangular function, and the uniform distribution into coherence. Paul Reiser 19:27, 19 Feb 2005 (UTC)

And now I've changed it to make it clear that although that view may be more-or-less harmless, it must not be considered obligatory. Michael Hardy 01:32, 20 Feb 2005 (UTC)

Hi Michael - Would you have an objection to making the definition for the uniform distribuiton to be 1/2 at the transition points and making it clear that its not obligatory? The advantages that I see are:

• It will be consistent with the Heaviside step and rectangle function articles where the transitions are set to 1/2. We don't want to alter these, I think, because then the statement that the Fourier transform of the Sinc function is the rectangle function has to be modified by adding "except at the transition points". This would hold true also for any function built up with the Heaviside function.
• Assuming the Heaviside stays the same, we won't need to modify the definition of the uniform distribution in terms of the Heaviside step function by adding "except at the transition points". Paul Reiser 04:22, 20 Feb 2005 (UTC)

I've added an explanation to go along with the mid-point definition. This also has the advantage that we can keep the definition of the Heaviside step function in terms of the sign function which is unambiguously defined at the transition point. Now we can use "equals" when defining the uniform distribution in terms of either the Heaviside or the sign functions. 69.143.60.69 01:21, 23 Feb 2005 (UTC)

PS - Above user is me, Paul Reiser (not logged in due to disk crash)

I have no objection to mentioning that that convention is convenient in certain contexts, but I don't want to see it made conspicuous in the tables and such, because a different convention is appropriate in a different context, and also it may erroneously appear to be very important to adopt a certain convention at the boundaries. Michael Hardy 01:58, 7 Apr 2005 (UTC)

Whatever definition we choose will be conspicuous. Do you have an idea about how to stress more strongly that, in the probability distribution context, the values at the transition points are not important? PAR 03:17, 7 Apr 2005 (UTC)

Michael - I saw your changes, and made a few more. We have to enter the same kind of rewording in the Heaviside and rectangle function articles as well. PAR 03:37, 7 Apr 2005 (UTC)

[edit] PDF

So why does the PDF defined like a heaviside step function? The probability of a point is zero, so defining it at exactly a and b is irrelevant since p(a) = p(b) = 0. Cburnett 05:37, Apr 6, 2005 (UTC)

I forgot to move the discussion page from the old uniform distribution to here. I inserted it above, and I hope it explains that. PAR 21:56, 6 Apr 2005 (UTC)

[edit] Math error in the graph

I am not able to edit the picture. It has a − b where clearly b − a is needed. Also the stubby little hyphen it uses for a minus sign, with no spacing, is nearly illegible. And the half-way points on the vertical line are obnoxious. At best they're unnecessary, and when applied to maximum likelihood problems, they are very misleading. Michael Hardy 02:28, 25 Apr 2005 (UTC)

Hi Michael - I fixed the a-b error in the graph, thanks for pointing that out. As for the half-way points, they are not absolutely obnoxious, they are only relatively obnoxious to someone used to dealing with maximum likelihood problems. To someone used to dealing with Fourier analysis and closure under L2 integration, any other representation is relatively obnoxious. We've gone over this before (see above), but I'm willing to change the picture to whatever consensus we can arrive at. PAR 03:01, 25 Apr 2005 (UTC)

Paul, I think I understand where you are coming from, but I have to say that in my limited experience everyone in statistics defines the uniform distribution over an open interal (a;b), with zero mass allocated to the points a and b. I'm willing to bet that this is the most common definition one will find in statistics textbooks, and it has the advantage of being straightforward and not requiring much explanation. This is what I would suggest we use for the infobox, both the plots and the formulas. It's of course Ok to discuss alternative definitions in the body of the article, but let's keep the infoboxes as simple as possible – they are terse enough to begin with, and anything that might strike readers as unusual is better discussed in the article itself. I would suggest you keep the current plots but move them out of the infobox and integrate them into the discussion of the alternative definition. --MarkSweep 06:58, 25 Apr 2005 (UTC)

As I pointed out in the previous section, the probability of any point is zero so defining them, in terms of probability, is pointless (haha, pun intended). I also think it should be an open or closed (again, it doesn't matter) interval but not half-valued-endpoints like heaviside step. Cburnett 19:51, Apr 25, 2005 (UTC)

Ok, I will change it soon, to be 1/(b-a) at the transition points, but this will have ripple effects. We want the following articles to be consistent:

• The Heaviside step function article - what is the definition at the transition point? - if other than 1/2, then its no longer definable in terms of sgn function without disclaimer
• The Rectangular function article - what is the definition at the transition point? - if other than 1/2, then its no longer definable in terms of sgn function without disclaimer
• The Sinc function article (?)

Please let me know your thoughts on these as well. Although I work with them, I have never tracked down the "correct" definitions. Also, any help fixing these articles would be appreciated. PAR 21:12, 25 Apr 2005 (UTC)

I think all those other ones can stay as they are. The uniform distribution article is specifically about probability and P(a) = P(b) = 0. I guess I don't see the necessity of keeping functions synchronized with a probability distribution. Though I think noting that the Heaviside half-value convention can be used with the uniform distribution if it's necessary/helpful precisely because P(a) = P(b) = 0. Cburnett 22:56, Apr 25, 2005 (UTC)

I fixed the plot so that P(a)=P(b_=1/(a-b). I was looking at "what links here" for the uniform article, and the beta distribution gave as a limiting case the uniform distribution which had this behavior. I can change it either way without a lot of bother, but we should settle on zero or 1/(a-b) for the transition points and make all other articles consistent. What do we want for the transition points? PAR 06:05, 26 Apr 2005 (UTC)

I don't see why you need to mark the end points at all on the pdf graph. Having the mid points would be ugly, and in fact the decision is arbitrary, as the article notes. What is wrong with solid horizontal line segments joined by dashed vertical line segments? --Henrygb 23:21, 18 May 2005 (UTC)

Because then somebody would complain that the value at the transition point was vague, and wouldn't it be best to pick one and go with it. Sorry for the flip answer, but its probably true, and I tend to agree. Please read this whole page, and you'll see we have been discussing this at length. PAR 00:58, 19 May 2005 (UTC)

[edit] Standard Uniform

I've commented out the "standard uniform" section because I can't make sense of it. I assume that the writer is trying to say something about random variates uniformly distributed, but the difference of two variates uniformly distributed between 0 and 1 is not a uniform distribution between 0 and 1. Maybe it should be uniform between -1 and 1? PAR 21:31, 26 Apr 2005 (UTC)

I uncommented the definition since that's correct for sure. As for the property, you're right. I was heading in the right direction but missed it. If and U₂ = 1 − U₁ then . I think this holds if you replace "1" with "b". Right? Cburnett 21:51, Apr 26, 2005 (UTC)

Yes - I'll put that in. I'll use (0,1) since its in the standard section. PAR 23:14, 26 Apr 2005 (UTC)