en>Mgkrupa: /* References */ Added {{Functional Analysis}} footer

2014-01-10T03:20:30Z

References: Added {{Functional Analysis}} footer

← Older revision		Revision as of 05:20, 10 January 2014
Line 1:		Line 1:
	~~Travel can be a great educational experience for every member~~ of the ~~family~~.<br><br>		The '''order in probability''' notation is used in [[probability theory]] and [[statistical theory]] in direct parallel to the [[big-O notation]] that is standard in [[mathematics]]. Where the [[big-O notation]] deals with the convergence of sequences or sets of ordinary numbers, the order in probability notation deals with [[Convergence of random variables\|convergence of sets of random variables]], where convergence is in the sense of [[convergence in probability]].<ref>Dodge, Y. (2003) ''The Oxford Dictionary of Statistical Terms'', OUP. ISBN 0-19-920613-9</ref>

	~~As long as you are careful~~, the ~~[http~~:~~//search.Huffingtonpost.com/search?q~~=~~developing&s_it=header_form_v1 developing] world [http:~~//~~tinyurl.com~~/~~nqbkz6z ugg outlet] can be~~ a ~~good place~~ to ~~visit~~, ~~and it can show your kids how life outside of their comfort zone is like, this is also a great time to teach them the value [http:~~//~~tinyurl.com~~/~~nqbkz6z ugg boots] of money.<br~~>~~You should get the National Park season pass if you go to parks often, these are great family outings, most only cost $50 and~~ can be ~~used all year round. For more about parks and places to visit check out online website.~~<br><br>~~Don~~'~~t forget to pack your child~~'~~s car seats when you go on a road trip. Chances are [http:~~//~~tinyurl.com~~/~~nqbkz6z ugg boots usa] that you will not get a high quality car seat with a rental car~~, ~~[http~~:/~~/tinyurl.com/nqbkz6z http://tinyurl.com/nqbkz6z] the seats a rental car agency will provide will most likely be of low quality and very uncomfortable~~ for ~~your child~~.<br>~~<br>Keep in mind that many people stay in your hotel~~. ~~Be courteous to others; keep your noise level down and close doors quietly~~. ~~Even at mid-day others may be napping~~, ~~someone may be napping or recovering from a bad case of jet lag~~. ~~You don~~'~~t have to take it to an extreme~~, ~~but make sure a reasonable volume level is maintained~~.<br><br>~~Sometimes you can choose the area or the actual seat that you want on the plane~~, ~~but there are times when you have no choice. Once you have paid for your tickets~~, ~~check~~ the ~~booking site often so that you can take advantage~~ of ~~this feature as soon as possible.~~<br><br>~~Research your destination online to find reviews of hotels, resorts and attractions before deciding on your travel plans.~~<br><br>~~When you are planning~~ a ~~vacation~~, ~~you should think about your destinations [http~~:/~~/tinyurl~~.~~com~~/~~nqbkz6z ugg boots outlet~~] and ~~recreational activities; then~~, ~~and only pack what you need~~ to ~~fulfill~~ that ~~itinerary~~. ~~Efficiency ensures a great trip that~~ is ~~both fun and enjoyable~~.<br>~~Make sure your dog is easily accustomed to new places if you plan on traveling with him. They should be leashed as well as capable of travel where people are present.~~<br><br>~~You can roll your [http~~:/~~/Search~~.~~Usa~~.~~gov~~/~~search?query=clothes clothes] or use~~ a ~~travel vacuum bag instead~~ of ~~folding them. This little tip saves you lots of space in your luggage. This should really cut down on the baggage amounts on your trip.~~<br>~~This article has supplied you with good information for travel planning. These strategies were compiled~~ to ~~give new travelers the basics~~, ~~as well as other great techniques that are easy to do when going on a trip~~.		For a set of random variables ''X<sub>n</sub>'' and a corresponding set of constants ''a<sub>n</sub>'' (both indexed by ''n'', which need not be discrete), the notation

			:<math>X_n=o_p(a_n) \,</math>

			means that the set of values ''X<sub>n</sub>''/''a<sub>n</sub>'' converges to zero in probability as ''n'' approaches an appropriate limit.
			Equivalently, ''X''<sub>''n''</sub> = o<sub>''p''</sub>(''a''<sub>''n''</sub>) can be written as ''a''<sub>''n''</sub> o<sub>''p''</sub>(1) where ''X''<sub>''n''</sub> = o<sub>''p''</sub>(1) is defined as,

			:<math>\lim_{n \to \infty} P(\|X_n\| \geq \varepsilon) = 0,</math>

			for every positive ε.<ref>Yvonne M. Bishop, Stephen E. Fienberg, Paul W. Holland. (1975,2007) ''Discrete multivariate analysis'', Springer. ISBN 0-387-72805-8, ISBN 978-0-387-72805-6</ref>

			The notation,
			:<math>X_n=O_p(a_n), \,</math>
			means that the set of values ''X<sub>n</sub>''/''a<sub>n</sub>'' is stochastically bounded. That is, for any ε > 0, there exists a finite M > 0 such that,
			:<math>P(\|X_n/a_n\| > M) < \varepsilon,\ \forall n.</math>

			<!--
			Let ''H''(''n'') be a [[function (mathematics)]] of ''n'', and ''f''(''X'') be a [[real-valued function]] of a [[random variable]] ''X'', where ''X'' has a [[sample size]] of ''n'' observations. Then [[limit]] as ''n'' goes to [[infinity]] of the [[probability]] that ''f''(''X'')/''H''(''n'') is below an [[arbitrary]] [[positive]] [[real number]] is [[zero]]. That is,

			:<math>\forall \varepsilon >0, \lim_{n\rightarrow\infty}\mathbb{P}\left(\left\|f(X)/H(n)\right\|\geq\varepsilon\right)=0.</math>

			-->

			==Example==

			If <math>(X_n)</math> is a stochastic sequence such that each element has finite variance, then
			:<math>X_n - E(X_n) = O_p(\sqrt{\operatorname{var}(X_n)}) \,</math>
			(see Theorem 14.4-1 in Bishop et al.)

			If, moreover, <math>a_n^{-2}\operatorname{var}(X_n) = \operatorname{var}(a_n^{-1}X_n)</math> is a null sequence for a sequence <math>(a_n)</math> of real numbers, then <math>a_n^{-1}(X_n - E(X_n))</math> converges to zero in probability by [[Chebyshev's inequality]], so
			:<math>X_n - E(X_n) = o_p(a_n)</math>.
			==References==
			<references/>

			{{DEFAULTSORT:Big O In Probability Notation}}
			[[Category:Mathematical notation]]
			[[Category:Probability theory]]
			[[Category:Statistical terminology]]

en>Dfqppp: formula error fixed

2012-05-11T14:13:51Z

formula error fixed

New page

Travel can be a great educational experience for every member of the family. 

As long as you are careful, the [http://search.Huffingtonpost.com/search?q=developing&s_it=header_form_v1 developing] world [http://tinyurl.com/nqbkz6z ugg outlet] can be a good place to visit, and it can show your kids how life outside of their comfort zone is like, this is also a great time to teach them the value [http://tinyurl.com/nqbkz6z ugg boots] of money. You should get the National Park season pass if you go to parks often, these are great family outings, most only cost $50 and can be used all year round. For more about parks and places to visit check out online website. Don't forget to pack your child's car seats when you go on a road trip. Chances are [http://tinyurl.com/nqbkz6z ugg boots usa] that you will not get a high quality car seat with a rental car, [http://tinyurl.com/nqbkz6z http://tinyurl.com/nqbkz6z] the seats a rental car agency will provide will most likely be of low quality and very uncomfortable for your child. Keep in mind that many people stay in your hotel. Be courteous to others; keep your noise level down and close doors quietly. Even at mid-day others may be napping, someone may be napping or recovering from a bad case of jet lag. You don't have to take it to an extreme, but make sure a reasonable volume level is maintained. Sometimes you can choose the area or the actual seat that you want on the plane, but there are times when you have no choice. Once you have paid for your tickets, check the booking site often so that you can take advantage of this feature as soon as possible. Research your destination online to find reviews of hotels, resorts and attractions before deciding on your travel plans. When you are planning a vacation, you should think about your destinations [http://tinyurl.com/nqbkz6z ugg boots outlet] and recreational activities; then, and only pack what you need to fulfill that itinerary. Efficiency ensures a great trip that is both fun and enjoyable. Make sure your dog is easily accustomed to new places if you plan on traveling with him. They should be leashed as well as capable of travel where people are present. You can roll your [http://Search.Usa.gov/search?query=clothes clothes] or use a travel vacuum bag instead of folding them. This little tip saves you lots of space in your luggage. This should really cut down on the baggage amounts on your trip. This article has supplied you with good information for travel planning. These strategies were compiled to give new travelers the basics, as well as other great techniques that are easy to do when going on a trip.

Symmetric set - Revision history

en>Mgkrupa: /* References */ Added {{Functional Analysis}} footer

en>Dfqppp: formula error fixed