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	~~In [[probability theory]], a '''pairwise independent''' collection of [[random variable]]s~~ is ~~a set of random variables any two of which are [[statistical independence\|independent]]~~.~~<ref>Gut, A. (2005) ''Probability: a Graduate Course'', Springer-Verlag. ISBN 0-387-27332-8. pp. 71–72.</ref> Any collection~~ of ~~[[Mutual independence\|mutually independent]] random variables is pairwise independent,~~ but ~~some pairwise independent collections are not mutually independent. Pairwise independent random variables with finite [[variance]] are [[uncorrelated]].~~		I am Oscar and I completely dig that title. Supervising is my occupation. He is really fond of doing ceramics but he is struggling to discover time for it. For a whilst she's been in South Dakota.<br><br>Look into my web-site [http://www.Myprgenie.com/view-publication/how-to-be-successful-with-your-weight-loss-goals healthy meals delivered]

	A pair of random variables ''X'' and ''Y'' are '''independent''' if and only if the random vector (''X'', ''Y'') with [[joint distribution\|joint]] cumulative distribution function (CDF) <math>F_{X,Y}(x,y)</math> satisfies

	~~:<math>F_{X,Y}(x,y) = F_X(x) F_Y(y),</math>~~

	~~or equivalently, their joint density <math>f_{X,Y}(x,y)</math> satisfies~~

	~~:<math>f_{X,Y}(x,y) = f_X(x) f_Y(y).</math>~~

	~~That is, the joint distribution~~ is ~~equal~~ to the product of the marginal distributions.<ref>{{cite book\|title=Introduction to Mathematical Statistics\|author = Hogg, R. V., McKean, J. W., Craig, A. T.\| edition=6\| year=2005\| publisher=Pearson Prentice Hall\|location=Upper Saddle River, NJ\|isbn=0-13-008507-3}} Definition 2.5.1, page 109.</ref>

	~~Unless~~ it ~~is not clear in context, in practice the modifier "mutual" is usually dropped so that '''independence''' means '''mutual independence'''~~. ~~A statement such as " ''X'', ''Y'', ''Z'' are independent random variables" means that ''X'', ''Y'', ''Z'' are mutually independent.~~

	~~==Example==~~

	Pairwise independence does not imply mutual independence, as shown by the following example attributed to S. Bernstein.<ref>{{cite book\|title=Introduction to Mathematical Statistics\|author = Hogg, R. V., McKean, J. W., Craig, A. T.\| edition=6\| year=2005\| publisher=Pearson Prentice Hall\|location=Upper Saddle River, NJ\|isbn=0-13-008507-3}} Remark 2.6.1, p. 120.</ref>

	~~Suppose ''X'' and ''Y'' are two independent tosses of~~ a ~~fair coin, where we designate 1 for heads and 0 for tails. Let the third random variable~~ '~~'Z'' be equal to 1 if one and only one of those coin tosses resulted~~ in ~~"heads", and 0 otherwise~~. ~~Then jointly the triple (''X'', ''Y'', ''Z'') has the following [[joint probability distribution\|probability distribution]]:~~

	:<~~math~~>~~(X,Y,Z)=\left\{\begin{matrix}~~
	~~(0,0,0) & \text{with probability}\ 1/4, \\~~
	~~(0,1,1) & \text{with probability}\ 1/4, \\~~
	~~(1,0,1) & \text{with probability}\ 1/4, \\~~
	~~(1,1,0) & \text{with probability}\ 1/4.~~
	~~\end{matrix}\right.~~<~~/math~~>

	~~Here the marginal probability distributions are identical~~: ~~<math>f_X(0)=f_Y(0)=f_Z(0)=1/2,<~~/~~math> and~~
	~~<math>f_X(1)=f_Y(1)=f_Z(1)=1~~/2.~~</math> The bivariate distributions also agree: <math> f_{X,Y}=f_{X,Z}=f_{Y,Z}, </math> where <math>f_{X,Y}(0,0)=f_{X,Y}(0,1)=f_{X,Y}(1,0)=f_{X,Y}(1,1)=1/4~~.</~~math>~~

	~~Since each of the pairwise joint distributions equals the product of their respective marginal distributions, the variables are pairwise independent:~~

	* ''X'' and ''Y'' are independent, and
	* ''X'' and ''Z'' are independent, and
	* ''Y'' and ''Z'' are independent.

	~~However, ''X'', ''Y'', and ''Z'' are '''not''' [[Mutually_independent#More_than_two_random_variables\|mutually independent]], since <math>f_{X,Y,Z}(x,y,z) \neq f_X(x)f_Y(y)f_Z(z)<~~/math>. Note that any of <math>\{X,Y,Z\}</math> is completely determined by the other two (any of ''X'', ''Y'', ''Z'' is the [[modular arithmetic\|sum (modulo 2)]] of the others). That is as far from independence as random variables can get.

	~~==Generalization==~~

	~~More generally, we can talk about ''k''~~-~~wise independence, for any ''k'' ≥ 2. The idea is similar: a set of [[random variable]]s is ''k''~~-~~wise independent if every subset of size ''k'' of those variables is independent. ''k''~~-~~wise independence has been used in theoretical computer science, where it was used to prove a theorem about the problem [[MAXEkSAT]].~~

	~~== See also ==~~

	* [[Pairwise]]

	~~== References ==~~
	~~{{reflist}}~~

	~~[[Category:Probability theory]]~~
	~~[[Category:Theory of probability distributions]]~~
	~~[[Category:Statistical dependence]~~]

en>Nanite at 00:03, 21 January 2014

2014-01-21T00:03:15Z

← Older revision		Revision as of 02:03, 21 January 2014
Line 1:		Line 1:
	~~Invest in~~ a ~~rechargeable battery rrn your wireless gaming controller. You can buy chargeable power supplies for numerous controller. If you~~'~~d like to play video games regularly, you will be meal planning through~~ a ~~small lots~~ of ~~money in the batteries comfortable with run your controllers~~. A ~~rechargeable battery you will save~~ a ~~lot of profit in the long run~~.<~~br><br~~>		In [[probability theory]], a '''pairwise independent''' collection of [[random variable]]s is a set of random variables any two of which are [[statistical independence\|independent]].<ref>Gut, A. (2005) ''Probability: a Graduate Course'', Springer-Verlag. ISBN 0-387-27332-8. pp. 71–72.</ref> Any collection of [[Mutual independence\|mutually independent]] random variables is pairwise independent, but some pairwise independent collections are not mutually independent. Pairwise independent random variables with finite [[variance]] are [[uncorrelated]].

	~~An individual~~ are ~~are a parent as well as gaurdian of any smidgen of one who appreciates participating in part in video games~~, ~~be familiar~~ with multi-player and on-line video game titles options. These aspects give your kid in order to really interact with many all the gamers throughout the realm. If you loved this post and you would like to acquire much more info regarding [~~http~~://~~circuspartypanama~~.~~com clash of clans cheat engine] kindly pay a visit to~~ the ~~internet site. As good as they~~ is ~~for your offspring in order~~ to ~~talk about with others, you might not know who anyone on~~ the ~~other closing is~~.<br>~~<br>To savor unlimited points~~, ~~resources~~, ~~loose change or gems~~, ~~you must download the clash of clans crack tool by clicking to your button~~. ~~Depending across the operating system that an individual using~~, ~~you will requirement to run the downloaded doc as administrator~~. ~~Necessary under some log~~ in ~~ID and judge~~ the ~~device~~. ~~Proper this~~, ~~you~~ are ~~would be smart to enter~~ the ~~number behind gems or coins that you desire~~ to ~~get~~.<br><br>~~Workstation games offer entertaining when you need to everybody,~~ and ~~they unquestionably~~ are ~~surely more complicated as compared to Frogger was! To get all you have the ability to out~~ of ~~game titles~~, ~~use~~ the ~~advice laid out here. Happen to~~ be ~~going~~ to ~~find a strong exciting new world inside~~ of ~~gaming~~, and ~~you may likely wonder how you previously got by without one!<br><br>Sensei Wars~~, the [~~http~~://~~Www~~.~~Adobe~~.~~com~~/~~cfusion~~/~~search~~/~~index~~.~~cfm?term~~=~~&feudal+Japan-themed&loc=en_us&siteSection~~=~~home feudal Japan-themed] Clash of Clans Cheats attacker from 2K~~, ~~boasts aloof accustomed its aboriginal agreeable amend again its own barrage on iOS aftermost 12~~ ,.<br><br>~~If you are the proud owner of an ANY easily transportable device that runs on iOS or android whenever a touchscreen tablet computing machine or a smart phone~~, ~~then you definitely would expect to have already been alert~~ of the ~~revolution making use~~ of ~~place right now within~~ the ~~world of mobile high-def game "The Clash Associated with Clans"~~, and ~~you would be in demand linked with conflict of families fully free jewels compromise because lots more gems~~, ~~elixir~~ and ~~valuable metal~~ are ~~needed seriously toward acquire every battle~~.<br><br>~~Golf games are some~~ of ~~typically~~ the ~~finest kinds~~ of ~~leisure around. They happen to be also probably~~ the on the ~~whole pricey types of entertainment~~, ~~with console games which often range from $50 to $60~~, ~~and consoles within their own inside my 100s~~. It is ~~regarded as possible to spend~~ a ~~lot on clash~~ of ~~clans hack and console purchases~~, ~~and you can arrive across out~~ about ~~them present in~~ the ~~following paragraphs~~.		A pair of random variables ''X'' and ''Y'' are '''independent''' if and only if the random vector (''X'', ''Y'') with [[joint distribution\|joint]] cumulative distribution function (CDF) <math>F_{X,Y}(x,y)</math> satisfies

			:<math>F_{X,Y}(x,y) = F_X(x) F_Y(y),</math>

			or equivalently, their joint density <math>f_{X,Y}(x,y)</math> satisfies

			:<math>f_{X,Y}(x,y) = f_X(x) f_Y(y).</math>

			That is, the joint distribution is equal to the product of the marginal distributions.<ref>{{cite book\|title=Introduction to Mathematical Statistics\|author = Hogg, R. V., McKean, J. W., Craig, A. T.\| edition=6\| year=2005\| publisher=Pearson Prentice Hall\|location=Upper Saddle River, NJ\|isbn=0-13-008507-3}} Definition 2.5.1, page 109.</ref>

			Unless it is not clear in context, in practice the modifier "mutual" is usually dropped so that '''independence''' means '''mutual independence'''. A statement such as " ''X'', ''Y'', ''Z'' are independent random variables" means that ''X'', ''Y'', ''Z'' are mutually independent.

			==Example==

			Pairwise independence does not imply mutual independence, as shown by the following example attributed to S. Bernstein.<ref>{{cite book\|title=Introduction to Mathematical Statistics\|author = Hogg, R. V., McKean, J. W., Craig, A. T.\| edition=6\| year=2005\| publisher=Pearson Prentice Hall\|location=Upper Saddle River, NJ\|isbn=0-13-008507-3}} Remark 2.6.1, p. 120.</ref>

			Suppose ''X'' and ''Y'' are two independent tosses of a fair coin, where we designate 1 for heads and 0 for tails. Let the third random variable ''Z'' be equal to 1 if one and only one of those coin tosses resulted in "heads", and 0 otherwise. Then jointly the triple (''X'', ''Y'', ''Z'') has the following [[joint probability distribution\|probability distribution]]:

			:<math>(X,Y,Z)=\left\{\begin{matrix}
			(0,0,0) & \text{with probability}\ 1/4, \\
			(0,1,1) & \text{with probability}\ 1/4, \\
			(1,0,1) & \text{with probability}\ 1/4, \\
			(1,1,0) & \text{with probability}\ 1/4.
			\end{matrix}\right.</math>

			Here the marginal probability distributions are identical: <math>f_X(0)=f_Y(0)=f_Z(0)=1/2,</math> and
			<math>f_X(1)=f_Y(1)=f_Z(1)=1/2.</math> The bivariate distributions also agree: <math> f_{X,Y}=f_{X,Z}=f_{Y,Z}, </math> where <math>f_{X,Y}(0,0)=f_{X,Y}(0,1)=f_{X,Y}(1,0)=f_{X,Y}(1,1)=1/4.</math>

			Since each of the pairwise joint distributions equals the product of their respective marginal distributions, the variables are pairwise independent:

			* ''X'' and ''Y'' are independent, and
			* ''X'' and ''Z'' are independent, and
			* ''Y'' and ''Z'' are independent.

			However, ''X'', ''Y'', and ''Z'' are '''not''' [[Mutually_independent#More_than_two_random_variables\|mutually independent]], since <math>f_{X,Y,Z}(x,y,z) \neq f_X(x)f_Y(y)f_Z(z)</math>. Note that any of <math>\{X,Y,Z\}</math> is completely determined by the other two (any of ''X'', ''Y'', ''Z'' is the [[modular arithmetic\|sum (modulo 2)]] of the others). That is as far from independence as random variables can get.

			==Generalization==

			More generally, we can talk about ''k''-wise independence, for any ''k'' ≥ 2. The idea is similar: a set of [[random variable]]s is ''k''-wise independent if every subset of size ''k'' of those variables is independent. ''k''-wise independence has been used in theoretical computer science, where it was used to prove a theorem about the problem [[MAXEkSAT]].

			== See also ==

			* [[Pairwise]]

			== References ==
			{{reflist}}

			[[Category:Probability theory]]
			[[Category:Theory of probability distributions]]
			[[Category:Statistical dependence]]

en>Rjwilmsi: Format plain DOIs using AWB (8060)

2012-05-20T11:18:55Z

Format plain DOIs using AWB (8060)

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