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| | They are typically a free website that are pre-designed for enabling businesses of every size in marking the presence on the internet and allows them in showcasing the product services and range through images, contents and various other elements. Good luck on continue learning how to make a wordpress website. This CMS has great flexibility to adapt various extensions and add-ons. Out of the various designs of photography identified these days, sports photography is preferred most, probably for the enjoyment and enjoyment associated with it. The top 4 reasons to use Business Word - Press Themes for a business website are:. <br><br> |
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| In mathematics, the '''second moment method''' is a technique used in [[probability theory]] and [[analysis]] to show that a [[random variable]] has positive probability of being positive. More generally, the "moment method" consists of bounding the probability that a random variable fluctuates far from its mean, by using its moments.<ref>{{cite web |url=http://terrytao.wordpress.com/2008/06/18/the-strong-law-of-large-numbers/ |title=The strong law of large numbers|author=[[Terence Tao]]|accessdate=2009-02-10 |work=What’s new?|date=2008-06-18 }}</ref>
| | These folders as well as files have to copied and the saved. After all, Word - Press is free, many of the enhancements for Word - Press like themes and plugins are also free, and there is plenty of free information online about how to use Word - Press. We also help to integrate various plug-ins to expand the functionalities of the web application. This is identical to doing a research as in depth above, nevertheless you can see various statistical details like the number of downloads and when the template was not long ago updated. Many times the camera is following Mia, taking in her point of view in almost every frame. <br><br>You can down load it here at this link: and utilize your FTP software program to upload it to your Word - Press Plugin folder. But if you are not willing to choose cost to the detriment of quality, originality and higher returns, then go for a self-hosted wordpress blog and increase the presence of your business in this new digital age. You can now search through the thousands of available plugins to add all kinds of functionality to your Word - Press site. User friendly features and flexibility that Word - Press has to offer is second to none. Socrates: (link to ) Originally developed for affiliate marketers, I've used this theme again and again to develop full-fledged web sites that include static pages, squeeze pages, and a blog. <br><br>You can add keywords but it is best to leave this alone. I have compiled a few tips on how you can start a food blog and hopefully the following information and tips can help you to get started on your food blogging creative journey. One of the great features of Wordpress is its ability to integrate SEO into your site. Fast Content Update - It's easy to edit or add posts with free Wordpress websites. Fortunately, Word - Press Customization Service is available these days, right from custom theme design, to plugin customization and modifying your website, you can take any bespoke service for your Word - Press development project. <br><br>This advice is critical because you don't want to waste too expensive time establishing your Word - Press blog the exact method. It can run as plugin and you can still get to that whole database just in circumstance your webhost does not have a c - Panel area. While deciding couple should consider the expertise of the doctor,clinics success rate,the costs of fertility treatment,including fertility tests and IVF costs and overall ones own financial budget. Word - Press is the most popular personal publishing platform which was launched in 2003. In case you have almost any issues with regards to where by in addition to the best way to use [http://peppa-piggames.com/profile/pibrantley wordpress backup plugin], you possibly can email us with the web page. However, if you're just starting out your blog site or business site, you can still search for an ideal theme for it without breaking your bank account. |
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| The method is often quantitative, in that one can often deduce a lower bound on the probability that the random variable is larger than some constant times its expectation. The method involves comparing the second [[moment (mathematics)|moment]] of random variables to the square of the first moment.
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| ==First moment method==
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| The first moment method is a simple application of [[Markov's inequality]] for integer-valued variables. For a '''non-negative''', '''integer-valued''' random variable ''X'', we may want to prove that ''X'' = 0 with high probability. To obtain an upper bound for P(''X'' > 0), and thus a lower bound for P(''X'' = 0), we first note that since ''X'' takes only integer values, P(''X'' > 0) = P(''X'' ≥ 1). Since ''X'' is non-negative we can now apply [[Markov's inequality]] to obtain P(''X'' ≥ 1) ≤ E[''X'']. Combining these we have P(''X'' > 0) ≤ E[''X'']; the first moment method is simply the use of this inequality.
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| ==General outline of the second moment method==
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| In the other direction, E[''X''] being "large" does not directly imply that P(''X'' = 0) is small. However, we can often use the second moment to derive such a conclusion, using some form of [[Chebyshev's inequality]].
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| Suppose that ''X<sub>n</sub>'' is a sequence of non-negative real-valued random variables which [[converge in law]] to a random variable ''X''. If there are finite positive constants ''c''<sub>1</sub>, ''c''<sub>2</sub> such that
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| :<math>E \left [X_n^2 \right ] \le c_1 E[X_n]^2</math> | |
| :<math>E \left [X_n \right ]\ge c_2</math>
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| hold for every ''n'', then it follows from the [[Paley–Zygmund inequality]] that for every ''n'' and θ in (0, 1)
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| :<math> P (X_n \geq c_2 \theta) \geq \frac{(1-\theta)^2}{c_1}.</math>
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| Consequently, the same inequality is satisfied by ''X''. In many situations, instead of using the Paley–Zygmund inequality, it is sufficient to use [[Cauchy–Schwarz inequality|Cauchy–Schwarz]].
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| ==Example application of method==
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| ===Setup of problem===
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| The [[Bernoulli bond percolation]] [[Glossary of graph theory#Subgraphs|subgraph]] of a graph ''G'' at parameter ''p'' is a random subgraph obtained from ''G'' by deleting every edge of ''G'' with probability 1−''p'', independently. The [[binary tree|infinite complete binary tree]] ''T'' is an infinite [[tree (graph theory)|tree]] where one vertex (called the root) has two neighbors and every other vertex has three neighbors. The second moment method can be used to show that at every parameter ''p'' ∈ (1/2, 1] with positive probability the connected component of the root in the percolation subgraph of ''T'' is infinite.
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| ===Application of method===
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| Let ''K'' be the percolation component of the root, and let ''T<sub>n</sub>'' be the set of vertices of ''T'' that are at distance ''n'' from the root. Let ''X<sub>n</sub>'' be the number of vertices in ''T<sub>n</sub>'' ∩ ''K''. To prove that ''K'' is infinite with positive probability, it is enough to show that <math>\limsup_{n\to\infty}1_{X_n>0}>0</math> with positive probability. By the [[reverse Fatou lemma]], it suffices to show that <math>\inf_{n\to\infty}P(X_n>0)>0</math>. The [[Cauchy–Schwarz inequality]] gives
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| :<math>E[X_n]^2\le E[X_n^2] \, E\left [(1_{X_n>0})^2\right ]=E[X_n^2]\,P(X_n>0).</math> | |
| Therefore, it is sufficient to show that
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| :<math> \inf_n \frac{E \left[ X_n \right ]^2}{E \left[ X_n^2 \right ]}>0\,,</math>
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| that is, that the second moment is bounded from above by a constant times the first moment squared (and both are nonzero). In many applications of the second moment method, one is not able to calculate the moments precisely, but can nevertheless establish this inequality.
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| In this particular application, these moments can be calculated. For every specific ''v'' in ''T<sub>n</sub>'',
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| :<math>P(v\in K)=p^n. \, </math>
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| Since <math>|T_n|=2^n</math>, it follows that
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| :<math>E[X_n]=2^n\,p^n</math>
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| which is the first moment. Now comes the second moment calculation.
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| :<math>E\!\left[X_n^2 \right ] = E\!\left[\sum_{v\in T_n} \sum_{u\in T_n}1_{v\in K}\,1_{u\in K}\right] = \sum_{v\in T_n} \sum_{u\in T_n} P(v,u\in K).</math>
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| For each pair ''v'', ''u'' in ''T<sub>n</sub>'' let ''w(v, u)'' denote the vertex in ''T'' that is farthest away from the root and lies on the simple path in ''T'' to each of the two vertices ''v'' and ''u'', and let ''k(v, u)'' denote the distance from ''w'' to the root. In order for ''v'', ''u'' to both be in ''K'', it is necessary and sufficient for the three simple paths from ''w(v, u)'' to ''v'', ''u'' and the root to be in ''K''. Since the number of edges contained in the union of these three paths is 2''n'' − ''k(v, u)'', we obtain
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| :<math>P(v,u\in K)= p^{2n-k(v,u)}.</math>
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| The number of pairs ''(v, u)'' such that ''k(v, u)'' = ''s'' is equal to <math>2^s\,2^{n-s}\,2^{n-s-1}=2^{2n-s-1}</math>, for ''s'' = 0, 1, ..., ''n''. Hence,
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| :<math>E\!\left[X_n^2 \right ] = \sum_{s=0}^n 2^{2n-s-1} p^{2n-s} = \frac 12\,(2p)^n \sum_{s=0}^n (2p)^s = \frac12\,(2p)^n \, \frac{(2p)^{n+1}-1}{2p-1} \le \frac p{2p-1} \,E[X_n]^2,</math>
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| which completes the proof.
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| ===Discussion===
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| *The choice of the random variables ''X''<sub>''n''</sub> was rather natural in this setup. In some more difficult applications of the method, some ingenuity might be required in order to choose the random variables ''X''<sub>''n''</sub> for which the argument can be carried through.
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| *The [[Paley–Zygmund inequality]] is sometimes used instead of the [[Cauchy–Schwarz inequality]] and may occasionally give more refined results.
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| *Under the (incorrect) assumption that the events ''v'', ''u'' in ''K'' are always independent, one has <math>P(v,u\in K)=P(v\in K)\,P(u\in K)</math>, and the second moment is equal to the first moment squared. The second moment method typically works in situations in which the corresponding events or random variables are “nearly independent".
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| *In this application, the random variables ''X''<sub>''n''</sub> are given as sums
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| :: <math>X_n=\sum_{v\in T_n}1_{v\in K}.</math>
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| : In other applications, the corresponding useful random variables are [[integral]]s
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| :: <math>X_n=\int f_n(t)\,d\mu(t),</math>
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| : where the functions ''f''<sub>''n''</sub> are random. In such a situation, one considers the product measure ''μ'' × ''μ'' and calculates
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| ::<math> \begin{align}
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| E \left[X_n^2 \right ] & = E\left[\int\int f_n(x)\,f_n(y)\,d\mu(x)\,d\mu(y)\right ] \\
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| & = E\left[ \int\int E\left[f_n(x)\,f_n(y)\right]\,d\mu(x)\,d\mu(y)\right ],
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| \end{align}</math>
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| :where the last step is typically justified using [[Fubini's theorem]].
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| ==References==
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| *{{Citation | last1=Burdzy | first1=Krzysztof | last2=Adelman | first2=Omer | last3=Pemantle | first3=Robin | title=Sets avoided by Brownian motion | id={{hdl|1773/2194}} | year=1998 | journal=Annals of Probability | volume=26 | issue=2 | pages=429–464 | doi=10.1214/aop/1022855639}}
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| *{{Citation | last1=Lyons | first1=Russell | title=Random walk, capacity, and percolation on trees | year=1992 | journal=Annals of Probability | volume=20 | issue=4 | pages=2043–2088 | doi=10.1214/aop/1176989540}}
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| *{{Citation | last1=Lyons | first1=Russell | last2=Peres | first2=Yuval | title=Probability on trees and networks | url=http://mypage.iu.edu/~rdlyons/prbtree/prbtree.html}}
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| <references/>
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| [[Category:Probabilistic inequalities]]
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| [[Category:Articles containing proofs]]
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They are typically a free website that are pre-designed for enabling businesses of every size in marking the presence on the internet and allows them in showcasing the product services and range through images, contents and various other elements. Good luck on continue learning how to make a wordpress website. This CMS has great flexibility to adapt various extensions and add-ons. Out of the various designs of photography identified these days, sports photography is preferred most, probably for the enjoyment and enjoyment associated with it. The top 4 reasons to use Business Word - Press Themes for a business website are:.
These folders as well as files have to copied and the saved. After all, Word - Press is free, many of the enhancements for Word - Press like themes and plugins are also free, and there is plenty of free information online about how to use Word - Press. We also help to integrate various plug-ins to expand the functionalities of the web application. This is identical to doing a research as in depth above, nevertheless you can see various statistical details like the number of downloads and when the template was not long ago updated. Many times the camera is following Mia, taking in her point of view in almost every frame.
You can down load it here at this link: and utilize your FTP software program to upload it to your Word - Press Plugin folder. But if you are not willing to choose cost to the detriment of quality, originality and higher returns, then go for a self-hosted wordpress blog and increase the presence of your business in this new digital age. You can now search through the thousands of available plugins to add all kinds of functionality to your Word - Press site. User friendly features and flexibility that Word - Press has to offer is second to none. Socrates: (link to ) Originally developed for affiliate marketers, I've used this theme again and again to develop full-fledged web sites that include static pages, squeeze pages, and a blog.
You can add keywords but it is best to leave this alone. I have compiled a few tips on how you can start a food blog and hopefully the following information and tips can help you to get started on your food blogging creative journey. One of the great features of Wordpress is its ability to integrate SEO into your site. Fast Content Update - It's easy to edit or add posts with free Wordpress websites. Fortunately, Word - Press Customization Service is available these days, right from custom theme design, to plugin customization and modifying your website, you can take any bespoke service for your Word - Press development project.
This advice is critical because you don't want to waste too expensive time establishing your Word - Press blog the exact method. It can run as plugin and you can still get to that whole database just in circumstance your webhost does not have a c - Panel area. While deciding couple should consider the expertise of the doctor,clinics success rate,the costs of fertility treatment,including fertility tests and IVF costs and overall ones own financial budget. Word - Press is the most popular personal publishing platform which was launched in 2003. In case you have almost any issues with regards to where by in addition to the best way to use wordpress backup plugin, you possibly can email us with the web page. However, if you're just starting out your blog site or business site, you can still search for an ideal theme for it without breaking your bank account.