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In [[probability theory]] and [[directional statistics]], a '''circular uniform distribution''' is a probability distribution on the unit circle whose density is uniform for all angles.
 
== Description ==
The [[probability density function]] (pdf) of the circular uniform distribution is:
 
:<math>
f_{UC}(\theta)=\frac{1}{2\pi}.
</math>
 
In terms of the circular variable <math>z=e^{i\theta}</math> the circular moments of the circular uniform distribution all zero, except for <math>m_0</math>:
 
:<math>\langle z^n\rangle=\delta_n</math>
 
where <math>\delta_n</math> is the [[Kronecker delta]] symbol.
 
The mean angle is undefined, and the length of the mean resultant is zero.
:<math>
R=|\langle z^n\rangle|=0\,
</math>
 
== Distribution of the mean ==
 
The sample mean of a set of ''N'' measurements <math>z_n=e^{i\theta_n}</math> drawn from a circular uniform distribution is defined as:
 
:<math>
\overline{z} = \frac{1}{N}\sum_{n=1}^N z_n = \overline{C}+i\overline{S} = \overline{R}e^{i\overline{\theta}}
</math>
 
where the average sine and cosine are:
 
:<math>
\overline{C}=\frac{1}{N}\sum_{n=1}^N \cos(\theta_n)\qquad\qquad\overline{S}=\frac{1}{N}\sum_{n=1}^N \sin(\theta_n)
</math>
 
and the average resultant length is:
 
:<math>
\overline{R}^2=|\overline{z}|^2=\overline{C}^2+\overline{S}^2
</math>
 
and the mean angle is:
 
:<math>
\overline{\theta}=\mathrm{Arg}(\overline{z}). \,
</math>
 
The sample mean for the circular uniform distribution will be concentrated about zero, becoming more concentrated as ''N'' increases. The distribution of the sample mean for the uniform distribution is given by:<ref name="Jam">{{cite book |title=Topics in Circular Statistics |last=Jammalamadaka |first=S. Rao |authorlink= |coauthors=Sengupta, A. |year=2001 |publisher=World Scientific Publishing Company |location= |isbn=978-981-02-3778-3 |url=http://www.amazon.com/Topics-Circular-Statistics-Rao-Jammalamadaka/dp/9810237782#reader_9810237782 |accessdate=2010-03-03}}</ref>
 
:<math>
\frac{1}{(2\pi)^N}\int_\Gamma \prod_{n=1}^N d\theta_n = P(\overline{R})P(\overline{\theta})\,d\overline{R}\,d\overline{\theta}
</math>
 
where <math>\Gamma\,</math> consists of intervals of <math>2\pi</math> in the variables, subject to the constraint that <math>\overline{R}</math> and <math>\overline{\theta}</math> are constant, or, alternatively, that <math>\overline{C}</math> and <math>\overline{S}</math> are constant. The distribution of the angle<math>P(\overline{\theta})</math> is uniform
 
:<math>
P(\overline{\theta})=\frac{1}{2\pi}
</math>
 
and the distribution of <math>\overline{R}</math> is given by:<ref name="Jam"/>
 
:<math>
P_N(\overline{R})=N^2\overline{R}\int_0^\infty J_0(N\overline{R}\,t)J_0(t)^Nt\,dt
</math>
 
[[File:CircUniformDistOfMean.svg|thumb|360px|A 10,000 point Monte Carlo simulation of the distribution of the sample mean of a circular uniform distribution for&nbsp;''N''&nbsp;=&nbsp;3]]
 
where <math>J_0</math> is the [[Bessel function]] of order zero. There is no known general analytic solution for the above integral, and it is difficult to evaluate due to the large number of oscillations in the integrand. A 10,000 point Monte Carlo simulation of the distribution of the mean for N=3 is shown in the figure.
 
For certain special cases, the above integral can be evaluated:
 
:<math>
P_2(\overline{R})=\frac{2}{\pi \sqrt{1-\overline{R}^2}}.
</math>
 
For large ''N'', the distribution of the mean can be determined from the [[central limit theorem for directional statistics]]. Since the angles are uniformly distributed, the individual sines and cosines of the angles will be distributed as:
 
:<math>
P(u)du=\frac{1}{\pi}\,\frac{du}{\sqrt{1-u^2}}
</math>
 
where <math>u=\cos\theta_n\,</math> or <math>\sin\theta_n\,</math>. It follows that they will have zero mean and a variance of 1/2. By the central limit theorem, in the limit of large ''N'', <math>\overline{C}\,</math> and <math>\overline{S}\,</math>, being the sum of a large number of [[Independent and identically-distributed random variables|i.i.d]]'s, will be [[Normal distribution|normally]] distributed with mean zero and variance <math>1/2N</math>. The mean resultant length <math>\overline{R}\,</math>, being the square root of the sum of two normally distributed variables, will be [[Chi distribution|Chi-distributed]] with two degrees of freedom (i.e.[[Rayleigh distribution|Rayleigh-distributed]]) and variance <math>1/2N</math>:
 
:<math>
\lim_{N\rightarrow\infty}P_N(\overline{R})=2N\overline{R}\,e^{-N\overline{R}^2}.
</math>
== Entropy ==
 
The differential [[entropy (information theory)|information entropy]] of the uniform distribution is simply
 
:<math>H_U=-\int_\Gamma \frac{1}{2\pi}\ln\left(\frac{1}{2\pi}\right)\,d\theta = \ln(2\pi)</math>
 
where <math>\Gamma</math> is any interval of length <math>2\pi</math>. This is the maximum entropy any circular distribution may have.
 
== See also ==
* [[Wrapped distribution]]
 
== References ==
 
<references/>
 
{{ProbDistributions|directional}}
 
[[Category:Continuous distributions]]
[[Category:Directional statistics]]
[[Category:Probability distributions]]

Revision as of 18:05, 16 February 2014

Lexan is a well-liked selection for window glass alternative throughout your own home. A popular model name, Lexan is Normal Electrical's brand of sheet polycarbonate - a specialty plastic - which is each sturdy and durable. Much lighter than a comparable sheet of glass, polycarbonate is infinitely stronger; even when struck with highly effective blows from sledgehammers, polycarbonate won't break. It is the identical material used for bulletproof home windows in banks or on the presidential limousine windows. Considered a greater insulator than glass, Lexan is fabricated from a sustainable, recyclable materials. Changing your fiberglass door with Lexan is pretty easy undertaking.

It’s easily recognizable and has an elegant and superior design. Spyderco is likely one of the well-established knife brands which have obtained tones of optimistic evaluations from its satisfied clients throughout the world. It could perform heavy duties with the minimal use of the blade in both course. It’s a very reasonably priced knife that has many premium options that you just didn't expect. The distinctive thumb hole is one premium feature that's helpful for smooth blade deployment. The handle is made out of chrome steel which minimizes the load and provides a superb grip to verify you should use it comfortably with none problems.

I examined the Zero Tolerance Model 0200 , and that model really caught my attention for quite a few reasons. To begin with, I like large knives, be there folders or fixed blade knives - and the 0200 comes with a four" 154CM stainless steel blade - this metal was once used almost completely by custom knife makers as a result of it is spendy, to say the least - factory knife makers simply didn't wish to use this metal for a long time, pondering shoppers wouldn't purchase knives that price more money than knives with less expensive stainless steels.

makes use of a drop-point blade that is product of Sandvik 14C28N hardened to fifty eight-60 on the rockwell scale. It has a really stunning stone wash end. I've found that this Sandvik metal takes an incredible razor edge and holds it for some time. I did some analysis on the Sandvik 14C28N steel Old Timer Knives and some individuals have compared it to VG-10 steel. The blade may be easily opened utilizing both the flipper shown above or by the stepped and angled thumb studs. Out of the box it had a razor edge on it and the blade was completely centered when closed

One other innovation I found fascinating is the safety “change” that Kershaw includes on this model. It's principally a screw that's captured in a small channel on the tip-finish of the handle. The screw is fitted such that it has free movement forwards and backwards in the channel, and transferring the screw moves a slender plastic bar backwards and forwards over the blade tip. When the protection is engaged, the bar blocks the blade tip from rising, thus preventing the knife from opening. Merely pushing the screw forward within the channel frees the blade tip and the knife may be opened.

But its NOT ALL BAD, as a result of on this decay, I'm studying & making use of pure herbs and their uses gained from reading, educating myself and experimenting. My worry of the bizarre dentist and his garden tools miniaturized to play and wreak havoc on my enamel, are NOT in my agenda and actually Old Timer Knife they shouldn't be in yours both. I'm not suggesting you avoid your dentist altogether, but do the non-public hygiene routine YOUR body actually needs not what some licensed fund-collector who plays with teeth and the epithelial tissues (and has a bizarre fascination with sharp objects), tells you to do.

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