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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.^[1]

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 of random variables to the square of the first moment.

First moment method

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.

General outline of the second moment method

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.

Suppose that X_n 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 c₁, c₂ such that

E [X_{n}^{2}] \leq c_{1} E [X_{n}]^{2}

E [X_{n}] \geq c_{2}

hold for every n, then it follows from the Paley–Zygmund inequality that for every n and θ in (0, 1)

P (X_{n} \geq c_{2} θ) \geq \frac{(1 - θ)^{2}}{c_{1}} .

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.

Example application of method

Setup of problem

The Bernoulli bond percolation 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 infinite complete binary tree T is an infinite 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.

Application of method

Let K be the percolation component of the root, and let T_n be the set of vertices of T that are at distance n from the root. Let X_n be the number of vertices in T_n ∩ K. To prove that K is infinite with positive probability, it is enough to show that ${lim sup}_{n \to \infty} 1_{X_{n} > 0} > 0$ with positive probability. By the reverse Fatou lemma, it suffices to show that $\inf_{n \to \infty} P (X_{n} > 0) > 0$ . The Cauchy–Schwarz inequality gives

E [X_{n}]^{2} \leq E [X_{n}^{2}] E [(1_{X_{n} > 0})^{2}] = E [X_{n}^{2}] P (X_{n} > 0) .

Therefore, it is sufficient to show that

\inf_{n} \frac{E {[X_{n}]}^{2}}{E [X_{n}^{2}]} > 0,

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.

In this particular application, these moments can be calculated. For every specific v in T_n,

P (v \in K) = p^{n} .

Since $| T_{n} | = 2^{n}$ , it follows that

E [X_{n}] = 2^{n} p^{n}

which is the first moment. Now comes the second moment calculation.

E [X_{n}^{2}] = E [\sum_{v \in T_{n}} \sum_{u \in T_{n}} 1_{v \in K} 1_{u \in K}] = \sum_{v \in T_{n}} \sum_{u \in T_{n}} P (v, u \in K) .

For each pair v, u in T_n 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 2n − k(v, u), we obtain

P (v, u \in K) = p^{2 n - k (v, u)} .

The number of pairs (v, u) such that k(v, u) = s is equal to $2^{s} 2^{n - s} 2^{n - s - 1} = 2^{2 n - s - 1}$ , for s = 0, 1, ..., n. Hence,

E [X_{n}^{2}] = \sum_{s = 0}^{n} 2^{2 n - s - 1} p^{2 n - s} = \frac{1}{2} (2 p)^{n} \sum_{s = 0}^{n} (2 p)^{s} = \frac{1}{2} (2 p)^{n} \frac{(2 p)^{n + 1} - 1}{2 p - 1} \leq \frac{p}{2 p - 1} E [X_{n}]^{2},

which completes the proof.

Discussion

The choice of the random variables X_n 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_n for which the argument can be carried through.
The Paley–Zygmund inequality is sometimes used instead of the Cauchy–Schwarz inequality and may occasionally give more refined results.
Under the (incorrect) assumption that the events v, u in K are always independent, one has $P (v, u \in K) = P (v \in K) P (u \in K)$ , 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".
In this application, the random variables X_n are given as sums

X_{n} = \sum_{v \in T_{n}} 1_{v \in K} .

In other applications, the corresponding useful random variables are integrals

X_{n} = \int f_{n} (t) d μ (t),

where the functions f_n are random. In such a situation, one considers the product measure μ × μ and calculates

\begin{aligned} E [X_{n}^{2}] & = E [\int \int f_{n} (x) f_{n} (y) d μ (x) d μ (y)] \\ = E [\int \int E [f_{n} (x) f_{n} (y)] d μ (x) d μ (y)], \end{aligned}

where the last step is typically justified using Fubini's theorem.

References

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Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

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+{{Cleanup|date=January 2009}}
+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>
+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.
+==First moment method==
+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.
+==General outline of the second moment method==
+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]].
+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
+:<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>
+hold for every ''n'', then it follows from the [[Paley–Zygmund inequality]] that for every ''n'' and θ in (0, 1)
+:<math> P (X_n \geq c_2 \theta) \geq \frac{(1-\theta)^2}{c_1}.</math>
+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]].
+==Example application of method==
+===Setup of problem===
+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.
+===Application of method===
+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
+:<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
+:<math> \inf_n \frac{E \left[ X_n \right ]^2}{E \left[ X_n^2 \right ]}>0\,,</math>
+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.
+In this particular application, these moments can be calculated. For every specific ''v'' in ''T<sub>n</sub>'',
+:<math>P(v\in K)=p^n. \, </math>
+Since <math>|T_n|=2^n</math>, it follows that
+:<math>E[X_n]=2^n\,p^n</math>
+which is the first moment. Now comes the second moment calculation.
+:<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>
+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
+:<math>P(v,u\in K)= p^{2n-k(v,u)}.</math>
+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,
+:<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>
+which completes the proof.
+===Discussion===
+*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.
+*The [[Paley–Zygmund inequality]] is sometimes used instead of the [[Cauchy–Schwarz inequality]] and may occasionally give more refined results.
+*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".
+*In this application, the random variables ''X''<sub>''n''</sub> are given as sums
+:: <math>X_n=\sum_{v\in T_n}1_{v\in K}.</math>
+: In other applications, the corresponding useful random variables are [[integral]]s
+:: <math>X_n=\int f_n(t)\,d\mu(t),</math>
+: where the functions ''f''<sub>''n''</sub> are random. In such a situation, one considers the product measure ''&mu;''&nbsp;&times;&nbsp;''&mu;'' and calculates
+::<math> \begin{align}
+E \left[X_n^2 \right ] & = E\left[\int\int f_n(x)\,f_n(y)\,d\mu(x)\,d\mu(y)\right ] \\
+& = E\left[ \int\int E\left[f_n(x)\,f_n(y)\right]\,d\mu(x)\,d\mu(y)\right ],
+\end{align}</math>
+:where the last step is typically justified using [[Fubini's theorem]].
+==References==
+*{{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}}
+*{{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}}
+*{{Citation | last1=Lyons | first1=Russell | last2=Peres | first2=Yuval | title=Probability on trees and networks | url=http://mypage.iu.edu/~rdlyons/prbtree/prbtree.html}}
+<references/>
+[[Category:Probabilistic inequalities]]
+[[Category:Articles containing proofs]]