|
|
| Line 1: |
Line 1: |
| {{Dablink|This article is about the method of moments in [[probability theory]]. See [[method of moments]] for other techniques bearing the same name.}}
| | Hello, my title is Andrew and my spouse doesn't like it at all. I am an invoicing officer and I'll be promoted quickly. What me and my family members love is bungee leaping but I've been taking on new things recently. Alaska is where I've usually been living.<br><br>Also visit my page - free psychic readings ([http://www.publicpledge.com/blogs/post/7034 click through the next document]) |
| | |
| 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]]
| |
Latest revision as of 22:00, 28 November 2014
Hello, my title is Andrew and my spouse doesn't like it at all. I am an invoicing officer and I'll be promoted quickly. What me and my family members love is bungee leaping but I've been taking on new things recently. Alaska is where I've usually been living.
Also visit my page - free psychic readings (click through the next document)