Derived demand: Difference between revisions
en>Cretog8 Undid revision 500769631 by 121.217.235.112 (talk) |
Removed holiday flights as example of derived demand. The good being consumed is the tourist destination, not likely the flight thereto. |
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{{Dablink|This article is about the method of moments in [[probability theory]]. See [[method of moments]] for other techniques bearing the same name.}} | |||
In [[probability theory]], the '''method of moments''' is a way of proving [[convergence in distribution]] by proving convergence of a sequence of [[moment (mathematics)|moment]] sequences.<ref>{{cite book|last=Prokhorov|first=A.V.|chapter=Moments, method of (in probability theory)|title=Encyclopaedia of Mathematics (online)|isbn=1-4020-0609-8|url=http://eom.springer.de/m/m064610.htm|mr=1375697|editor=M. Hazewinkel}}</ref> Suppose ''X'' is a [[random variable]] and that all of the moments | |||
:<math>\operatorname{E}(X^k)\,</math> | |||
exist. Further suppose the [[probability distribution]] of ''X'' is completely determined by its moments, i.e., there is no other probability distribution with the same sequence of moments | |||
(cf. the [[problem of moments]]). If | |||
:<math>\lim_{n\to\infty}\operatorname{E}(X_n^k) = \operatorname{E}(X^k)\,</math> | |||
for all values of ''k'', then the sequence {''X''<sub>''n''</sub>} converges to ''X'' in distribution. | |||
The method of moments was introduced by [[Pafnuty Chebyshev]] for proving the [[central limit theorem]]; Chebyshev cited earlier contributions by [[Irénée-Jules Bienaymé]].<ref>{{cite book|mr=2743162|last=Fischer|first=H.|title=A history of the central limit theorem. From classical to modern probability theory.|series= Sources and Studies in the History of Mathematics and Physical Sciences|publisher=Springer|location=New York|year=2011|isbn=978-0-387-87856-0|chapter=4. Chebyshev's and Markov's Contributions.}}</ref> More recently, it has been applied by [[Eugene Wigner]] to prove [[Wigner's semicircle law]], and has since found numerous applications in the [[random matrix theory|theory of random matrices]].<ref>{{cite book|last=Anderson|first=G.W.|last2=Guionnet|first2=A.|last3=Zeitouni|first3=O.|title=An introduction to random matrices.|year=2010|publisher=Cambridge University Press|location=Cambridge|isbn=978-0-521-19452-5|chapter=2.1}}</ref> | |||
==Notes== | |||
{{Reflist}} | |||
{{DEFAULTSORT:Method Of Moments (Probability Theory)}} | |||
[[Category:Probability theory]] | |||
Revision as of 15:10, 3 February 2014
In probability theory, the method of moments is a way of proving convergence in distribution by proving convergence of a sequence of moment sequences.[1] Suppose X is a random variable and that all of the moments
exist. Further suppose the probability distribution of X is completely determined by its moments, i.e., there is no other probability distribution with the same sequence of moments (cf. the problem of moments). If
for all values of k, then the sequence {Xn} converges to X in distribution.
The method of moments was introduced by Pafnuty Chebyshev for proving the central limit theorem; Chebyshev cited earlier contributions by Irénée-Jules Bienaymé.[2] More recently, it has been applied by Eugene Wigner to prove Wigner's semicircle law, and has since found numerous applications in the theory of random matrices.[3]
Notes
43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.
- ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534