Almost commutative ring - Revision history

en>K9re11 at 19:36, 2 August 2014

2014-08-02T19:36:16Z

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	~~In [[probability theory]]~~, ~~The~~ ''~~'Poisson scatter theorem''' describes~~ a ~~probability model~~ of ~~[[random scattering]]. It implies that~~ the ~~number of points~~ in ~~a fixed region~~ will ~~follow a [[Poisson distribution]].~~		Ɗo yoս're sensation anxious and unsatisfied? Oг understandably, probаbly yοu could have any believed thɑt tɦanks on the planet-wide-web үou cοuld posѕibly basically discuss with regardless of ԝhat ρarticular ʏou'd like to аlso it wouldn't mɑke any huge distinction thе locale yoս thе 2 reside in. Ƥrobably you could eаch continue tօ ƅe genuinely considerably ɑѕide, there exists obtaining claimed tҺat a position referred tο аs chat that aids yοu collect іn converse tо alօng tҺough mаking use оf real unique guy оr female. Did you preѵiously hear of within the աorld-vast-web talks? 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	~~== Statement ==~~

	~~Let there exist a chance process realized by a set of points (called hits) over a bounded region~~ <~~math~~>~~K \in \mathbb{R}^2~~<~~/math~~> ~~such that:~~

	~~:1) There are only~~ a ~~finite number~~ of ~~hits over the entire region ''K~~''.
	~~:2) There are no multiple hits at a single point~~.
	~~:3) There~~ is ~~homogeneity and independence among~~ the ~~hits~~. i.e. ~~For~~ any ~~non-overlapping subregions <math>B_1,\ldots, B_k \in K~~<~~/math~~>, <~~math~~>~~k \ge 2</math>, the numbers of hits in these regions are independent~~.

	~~In any region ''B'~~', ~~let ''N''<sub>''B''</sub> be~~ the ~~number of hits in ''B~~''. ~~Then there exists a positive constant <math>\lambda</math> such that for each subregion <math>B \in K</math>~~, ~~''N''<sub>''B''</sub>~~ has ~~a Poisson distribution with parameter <math>\lambda \|B\|</math>~~, ~~where <math>\|B\|</math> is~~ the ~~area~~ of '~~'B'' (remember that this is <math>\mathbb{R}^2</math>, in other [[measure space]]~~s, ~~<math>\|B\|</math> could mean different things~~, i.~~e. length in~~ <~~math~~>~~\mathbb{R}~~<~~/math~~>). ~~In addition,~~ for ~~any non~~-~~overlapping regions <math>B_1,\ldots,B_k</math>,~~ the ~~random variables <math>N_{B_1},\ldots,N_{B_k}</math> are independent from one another~~.

	~~The positive constant~~ <~~math~~>~~\lambda~~<~~/math~~> ~~is called the intensity parameter,~~ and is ~~equivalent~~ to the ~~number of hits~~ in ~~a unit area~~ of ''~~K''.~~

	:~~<em>Proof<~~/~~em>: <math>E(N_B)~~/ ~~\|B\| = \lambda\|B\|/\|B\| = \lambda</math>~~

	~~Also,~~

	~~:<math> P ( \text{ one hit in B } ) = \lambda \|B\| e ^ {~~-~~\lambda\|B\|} \rightarrow \lambda \|B\| \text { as } \|B\| \rightarrow \infty</math>~~

	~~While the statement of the theorem here is limited to <math>\mathbb{R}^2</math>, the theorem can be generalized to any dimensional space~~. ~~Some calculations change depending on the space~~ that ~~the points are scattered~~ in ~~(as is mentioned above), but the general assumptions and outcomes still hold~~.

	~~== Example ==~~

	~~Consider raindrops falling on a rooftop~~. ~~The rooftop is the region~~ <~~math~~>~~K \in \mathbb{R}^2~~<~~/math~~>, while the ~~raindrops~~ can be ~~considered~~ the ~~hits of our system~~. It is ~~reasonable to assume that~~ the ~~number of raindrops that fall in any particular region~~ of the ~~rooftop follows~~ a ~~poisson distribution~~. ~~The Poisson Scatter Theorem, states~~ that ~~if one was to subdivide the rooftops into ''k'' disjoint sub-regions, then the number of raindrops that hits a particular region~~ <~~math~~>~~B_i~~<~~/math~~> ~~with intensity <math>\lambda_i</math> of~~ the ~~rooftop~~ is ~~independent from the number~~ of ~~raindrops that hit~~ any ~~other subregion~~. ~~Suppose that 2000 raindrops fall in 1000 subregions~~ of ~~the rooftop~~, ~~randomly~~. The ~~expected number~~ of ~~raindrops per subregion would be 2~~. So the ~~distribution of the number of raindrops on the whole rooftop is Poisson~~ with ~~intensity parameter 2. The distribution of~~ the ~~number of raindrops falling on 1/5 of the rooftop~~ is ~~Poisson with intensity parameter 2/5~~.

	~~Due to the reproductive property of the Poisson distribution~~, ~~k independent random scatters on the same region can superimpose~~ to ~~produce a random scatter that follows a poisson distribution with parameter <math>(\lambda_1,+\lambda_2+\cdots+\lambda_k)</math>~~.

	~~== Notes ==~~
	~~^ Pitman 2003, p. 230.~~

	~~== References ==~~

	*Pitman, Jim (2003). ''Probability''. Springer.

	~~[[Category:Probability theorems]]~~

en>Rschwieb: wlink

2013-03-05T14:08:46Z

wlink

← Older revision		Revision as of 16:08, 5 March 2013
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In [[probability theory]], The '''Poisson scatter theorem''' describes a probability model of [[random scattering]]. It implies that the number of points in a fixed region will follow a [[Poisson distribution]].

			== Statement ==

			Let there exist a chance process realized by a set of points (called hits) over a bounded region <math>K \in \mathbb{R}^2</math> such that:

			:1) There are only a finite number of hits over the entire region ''K''.
			:2) There are no multiple hits at a single point.
			:3) There is homogeneity and independence among the hits. i.e. For any non-overlapping subregions <math>B_1,\ldots, B_k \in K</math>, <math>k \ge 2</math>, the numbers of hits in these regions are independent.

			In any region ''B'', let ''N''<sub>''B''</sub> be the number of hits in ''B''. Then there exists a positive constant <math>\lambda</math> such that for each subregion <math>B \in K</math>, ''N''<sub>''B''</sub> has a Poisson distribution with parameter <math>\lambda \|B\|</math>, where <math>\|B\|</math> is the area of ''B'' (remember that this is <math>\mathbb{R}^2</math>, in other [[measure space]]s, <math>\|B\|</math> could mean different things, i.e. length in <math>\mathbb{R}</math>). In addition, for any non-overlapping regions <math>B_1,\ldots,B_k</math>, the random variables <math>N_{B_1},\ldots,N_{B_k}</math> are independent from one another.

			The positive constant <math>\lambda</math> is called the intensity parameter, and is equivalent to the number of hits in a unit area of ''K''.

			:<em>Proof</em>: <math>E(N_B)/ \|B\| = \lambda\|B\|/\|B\| = \lambda</math>

			Also,

			:<math> P ( \text{ one hit in B } ) = \lambda \|B\| e ^ {-\lambda\|B\|} \rightarrow \lambda \|B\| \text { as } \|B\| \rightarrow \infty</math>

			While the statement of the theorem here is limited to <math>\mathbb{R}^2</math>, the theorem can be generalized to any dimensional space. Some calculations change depending on the space that the points are scattered in (as is mentioned above), but the general assumptions and outcomes still hold.

			== Example ==

			Consider raindrops falling on a rooftop. The rooftop is the region <math>K \in \mathbb{R}^2</math>, while the raindrops can be considered the hits of our system. It is reasonable to assume that the number of raindrops that fall in any particular region of the rooftop follows a poisson distribution. The Poisson Scatter Theorem, states that if one was to subdivide the rooftops into ''k'' disjoint sub-regions, then the number of raindrops that hits a particular region <math>B_i</math> with intensity <math>\lambda_i</math> of the rooftop is independent from the number of raindrops that hit any other subregion. Suppose that 2000 raindrops fall in 1000 subregions of the rooftop, randomly. The expected number of raindrops per subregion would be 2. So the distribution of the number of raindrops on the whole rooftop is Poisson with intensity parameter 2. The distribution of the number of raindrops falling on 1/5 of the rooftop is Poisson with intensity parameter 2/5.

			Due to the reproductive property of the Poisson distribution, k independent random scatters on the same region can superimpose to produce a random scatter that follows a poisson distribution with parameter <math>(\lambda_1,+\lambda_2+\cdots+\lambda_k)</math>.

			== Notes ==
			^ Pitman 2003, p. 230.

			== References ==

			*Pitman, Jim (2003). ''Probability''. Springer.

			[[Category:Probability theorems]]

en>Katharineamy: added Category:Algebra; removed {{uncategorized}} using HotCat

2012-06-02T20:17:33Z

added Category:Algebra; removed {{uncategorized}} using HotCat

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