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78-523: There are two closely related variants of the Erdős–Rényi random graph model. The behavior of random graphs are often studied in the case where n {\displaystyle n} , the number of vertices, tends to infinity. Although p {\displaystyle p} and M {\displaystyle M} can be fixed in this case, they can also be functions depending on n {\displaystyle n} . For example,

156-454: A fair coin toss is a Bernoulli trial. When a fair coin is flipped once, the theoretical probability that the outcome will be heads is equal to 1 ⁄ 2 . Therefore, according to the law of large numbers, the proportion of heads in a "large" number of coin flips "should be" roughly 1 ⁄ 2 . In particular, the proportion of heads after n flips will almost surely converge to 1 ⁄ 2 as n approaches infinity. Although

234-679: A (possibly non-linear) operator T : X → X ∗ {\displaystyle T:X\rightarrow X^{*}} is said to be a monotone operator if ( T u − T v , u − v ) ≥ 0 ∀ u , v ∈ X . {\displaystyle (Tu-Tv,u-v)\geq 0\quad \forall u,v\in X.} Kachurovskii's theorem shows that convex functions on Banach spaces have monotonic operators as their derivatives. A subset G {\displaystyle G} of X × X ∗ {\displaystyle X\times X^{*}}

312-567: A collection of independent and identically distributed (iid) samples from a random variable with finite mean, the sample mean converges in probability to the expected value That is, for any positive number ε , lim n → ∞ Pr ( | X ¯ n − μ | < ε ) = 1. {\displaystyle \lim _{n\to \infty }\Pr \!\left(\,|{\overline {X}}_{n}-\mu |<\varepsilon \,\right)=1.} Interpreting this result,

390-833: A function f {\displaystyle f} defined on a subset of the real numbers with real values is called monotonic if it is either entirely non-decreasing, or entirely non-increasing. That is, as per Fig. 1, a function that increases monotonically does not exclusively have to increase, it simply must not decrease. A function is termed monotonically increasing (also increasing or non-decreasing ) if for all x {\displaystyle x} and y {\displaystyle y} such that x ≤ y {\displaystyle x\leq y} one has f ( x ) ≤ f ( y ) {\displaystyle f\!\left(x\right)\leq f\!\left(y\right)} , so f {\displaystyle f} preserves

468-488: A generalization of real numbers. The above definition of monotonicity is relevant in these cases as well. However, the terms "increasing" and "decreasing" are avoided, since their conventional pictorial representation does not apply to orders that are not total . Furthermore, the strict relations < {\displaystyle <} and > {\displaystyle >} are of little use in many non-total orders and hence no additional terminology

546-414: A graph is NP-complete , the size of the largest clique in a "typical" graph (according to this model) is very well understood. Edge-dual graphs of Erdos-Renyi graphs are graphs with nearly the same degree distribution, but with degree correlations and a significantly higher clustering coefficient . In percolation theory one examines a finite or infinite graph and removes edges (or links) randomly. Thus

624-457: A growth and rewiring model, respectively. Another alternative family of random graph models, capable of reproducing many real-life phenomena, are exponential random graph models . The G ( n ,  p ) model was first introduced by Edgar Gilbert in a 1959 paper studying the connectivity threshold mentioned above. The G ( n , M ) model was introduced by Erdős and Rényi in their 1959 paper. As with Gilbert, their first investigations were as to

702-597: A law of large numbers. A special form of the LLN (for a binary random variable) was first proved by Jacob Bernoulli . It took him over 20 years to develop a sufficiently rigorous mathematical proof which was published in his Ars Conjectandi ( The Art of Conjecturing ) in 1713. He named this his "Golden Theorem" but it became generally known as " Bernoulli's theorem ". This should not be confused with Bernoulli's principle , named after Jacob Bernoulli's nephew Daniel Bernoulli . In 1837, S. D. Poisson further described it under

780-486: A monotonic transform (see also monotone preferences ). In this context, the term "monotonic transformation" refers to a positive monotonic transformation and is intended to distinguish it from a "negative monotonic transformation," which reverses the order of the numbers. The following properties are true for a monotonic function f : R → R {\displaystyle f\colon \mathbb {R} \to \mathbb {R} } : These properties are

858-451: A one-to-one mapping from their range to their domain. However, functions that are only weakly monotone are not invertible because they are constant on some interval (and therefore are not one-to-one). A function may be strictly monotonic over a limited a range of values and thus have an inverse on that range even though it is not strictly monotonic everywhere. For example, if y = g ( x ) {\displaystyle y=g(x)}

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936-402: A random variable that does not have a finite variance under some other weaker assumption, and Khinchin showed in 1929 that if the series consists of independent identically distributed random variables, it suffices that the expected value exists for the weak law of large numbers to be true. These further studies have given rise to two prominent forms of the LLN. One is called the "weak" law and

1014-479: A rough heuristic is that if pn → ∞, then G ( n , p ) should behave similarly to G ( n , M ) with M = ( n 2 ) p {\displaystyle M={\tbinom {n}{2}}p} as n increases. For many graph properties, this is the case. If P is any graph property which is monotone with respect to the subgraph ordering (meaning that if A is a subgraph of B and B satisfies P , then A will satisfy P as well), then

1092-505: A sequence of random infinite graphs of decreasing sizes: ( Γ i ) i ∈ N {\displaystyle (\Gamma _{i})_{i\in \mathbb {N} }} . The theorem states that this graph corresponds in a certain sense to the limit object of G n {\displaystyle G_{n}} as n → + ∞ {\displaystyle n\to +\infty } . Law of large numbers In probability theory ,

1170-422: A source may state that all monotonic functions are invertible when they really mean that all strictly monotonic functions are invertible. The term monotonic transformation (or monotone transformation ) may also cause confusion because it refers to a transformation by a strictly increasing function. This is the case in economics with respect to the ordinal properties of a utility function being preserved across

1248-468: Is a monotonically increasing function. A function is unimodal if it is monotonically increasing up to some point (the mode ) and then monotonically decreasing. When f {\displaystyle f} is a strictly monotonic function, then f {\displaystyle f} is injective on its domain, and if T {\displaystyle T} is the range of f {\displaystyle f} , then there

1326-409: Is a sharp threshold for the connectedness of G ( n , p ). Further properties of the graph can be described almost precisely as n tends to infinity. For example, there is a k ( n ) (approximately equal to 2log 2 ( n )) such that the largest clique in G ( n , 0.5) has almost surely either size k ( n ) or k ( n ) + 1. Thus, even though finding the size of the largest clique in

1404-479: Is also admissible , monotonicity is a stricter requirement than admissibility. Some heuristic algorithms such as A* can be proven optimal provided that the heuristic they use is monotonic. In Boolean algebra , a monotonic function is one such that for all a i and b i in {0,1} , if a 1 ≤ b 1 , a 2 ≤ b 2 , ..., a n ≤ b n (i.e. the Cartesian product {0, 1}

1482-439: Is also monotone. The dual notion is often called antitone , anti-monotone , or order-reversing . Hence, an antitone function f satisfies the property x ≤ y ⟹ f ( y ) ≤ f ( x ) , {\displaystyle x\leq y\implies f(y)\leq f(x),} for all x and y in its domain. A constant function is both monotone and antitone; conversely, if f

1560-472: Is an inverse function on T {\displaystyle T} for f {\displaystyle f} . In contrast, each constant function is monotonic, but not injective, and hence cannot have an inverse. The graphic shows six monotonic functions. Their simplest forms are shown in the plot area and the expressions used to create them are shown on the y -axis. A map f : X → Y {\displaystyle f:X\to Y}

1638-604: Is both monotone and antitone, and if the domain of f is a lattice , then f must be constant. Monotone functions are central in order theory. They appear in most articles on the subject and examples from special applications are found in these places. Some notable special monotone functions are order embeddings (functions for which x ≤ y {\displaystyle x\leq y} if and only if f ( x ) ≤ f ( y ) ) {\displaystyle f(x)\leq f(y))} and order isomorphisms ( surjective order embeddings). In

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1716-826: Is called strictly monotone . Functions that are strictly monotone are one-to-one (because for x {\displaystyle x} not equal to y {\displaystyle y} , either x < y {\displaystyle x<y} or x > y {\displaystyle x>y} and so, by monotonicity, either f ( x ) < f ( y ) {\displaystyle f\!\left(x\right)<f\!\left(y\right)} or f ( x ) > f ( y ) {\displaystyle f\!\left(x\right)>f\!\left(y\right)} , thus f ( x ) ≠ f ( y ) {\displaystyle f\!\left(x\right)\neq f\!\left(y\right)} .) To avoid ambiguity,

1794-414: Is connected tends to 1 {\displaystyle 1} . The expected number of edges in G ( n , p ) is ( n 2 ) p {\displaystyle {\tbinom {n}{2}}p} , and by the law of large numbers any graph in G ( n , p ) will almost surely have approximately this many edges (provided the expected number of edges tends to infinity). Therefore,

1872-523: Is difficult or impossible to use other approaches. The average of the results obtained from a large number of trials may fail to converge in some cases. For instance, the average of n results taken from the Cauchy distribution or some Pareto distributions (α<1) will not converge as n becomes larger; the reason is heavy tails . The Cauchy distribution and the Pareto distribution represent two cases:

1950-491: Is introduced for them. Letting ≤ {\displaystyle \leq } denote the partial order relation of any partially ordered set, a monotone function, also called isotone , or order-preserving , satisfies the property x ≤ y ⟹ f ( x ) ≤ f ( y ) {\displaystyle x\leq y\implies f(x)\leq f(y)} for all x and y in its domain. The composite of two monotone mappings

2028-929: Is known as Kolmogorov's strong law , see e.g. Sen & Singer (1993 , Theorem 2.3.10). The weak law states that for a specified large n , the average X ¯ n {\displaystyle {\overline {X}}_{n}} is likely to be near μ . Thus, it leaves open the possibility that | X ¯ n − μ | > ε {\displaystyle |{\overline {X}}_{n}-\mu |>\varepsilon } happens an infinite number of times, although at infrequent intervals. (Not necessarily | X ¯ n − μ | ≠ 0 {\displaystyle |{\overline {X}}_{n}-\mu |\neq 0} for all n ). The strong law shows that this almost surely will not occur. It does not imply that with probability 1, we have that for any ε > 0

2106-487: Is neither non-decreasing nor non-increasing. A function f {\displaystyle f} is said to be absolutely monotonic over an interval ( a , b ) {\displaystyle \left(a,b\right)} if the derivatives of all orders of f {\displaystyle f} are nonnegative or all nonpositive at all points on the interval. All strictly monotonic functions are invertible because they are guaranteed to have

2184-404: Is no principle that a small number of observations will coincide with the expected value or that a streak of one value will immediately be "balanced" by the others (see the gambler's fallacy ). The LLN only applies to the average of the results obtained from repeated trials and claims that this average converges to the expected value; it does not claim that the sum of n results gets close to

2262-524: Is ordered coordinatewise ), then f( a 1 , ..., a n ) ≤ f( b 1 , ..., b n ) . In other words, a Boolean function is monotonic if, for every combination of inputs, switching one of the inputs from false to true can only cause the output to switch from false to true and not from true to false. Graphically, this means that an n -ary Boolean function is monotonic when its representation as an n -cube labelled with truth values has no upward edge from true to false . (This labelled Hasse diagram

2340-403: Is said to be maximal monotone if it is maximal among all monotone sets in the sense of set inclusion. The graph of a monotone operator G ( T ) {\displaystyle G(T)} is a monotone set. A monotone operator is said to be maximal monotone if its graph is a maximal monotone set . Order theory deals with arbitrary partially ordered sets and preordered sets as

2418-472: Is said to be monotone if each of its fibers is connected ; that is, for each element y ∈ Y , {\displaystyle y\in Y,} the (possibly empty) set f − 1 ( y ) {\displaystyle f^{-1}(y)} is a connected subspace of X . {\displaystyle X.} In functional analysis on a topological vector space X {\displaystyle X} ,

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2496-553: Is said to be a monotone set if for every pair [ u 1 , w 1 ] {\displaystyle [u_{1},w_{1}]} and [ u 2 , w 2 ] {\displaystyle [u_{2},w_{2}]} in G {\displaystyle G} , ( w 1 − w 2 , u 1 − u 2 ) ≥ 0. {\displaystyle (w_{1}-w_{2},u_{1}-u_{2})\geq 0.} G {\displaystyle G}

2574-403: Is strictly increasing on the range [ a , b ] {\displaystyle [a,b]} , then it has an inverse x = h ( y ) {\displaystyle x=h(y)} on the range [ g ( a ) , g ( b ) ] {\displaystyle [g(a),g(b)]} . The term monotonic is sometimes used in place of strictly monotonic , so

2652-418: Is that the probability that, as the number of trials n goes to infinity, the average of the observations converges to the expected value, is equal to one. The modern proof of the strong law is more complex than that of the weak law, and relies on passing to an appropriate subsequence. The strong law of large numbers can itself be seen as a special case of the pointwise ergodic theorem . This view justifies

2730-417: Is the dual of the function's labelled Venn diagram , which is the more common representation for n ≤ 3 .) The monotonic Boolean functions are precisely those that can be defined by an expression combining the inputs (which may appear more than once) using only the operators and and or (in particular not is forbidden). For instance "at least two of a , b , c hold" is a monotonic function of

2808-405: Is useful to derive consistency of a large class of estimators (see Extremum estimator ). Monotonic function In mathematics , a monotonic function (or monotone function ) is a function between ordered sets that preserves or reverses the given order . This concept first arose in calculus , and was later generalized to the more abstract setting of order theory . In calculus ,

2886-436: The degree of any particular vertex is binomial : where n is the total number of vertices in the graph. Since this distribution is Poisson for large n and np = const. In a 1960 paper, Erdős and Rényi described the behavior of G ( n ,  p ) very precisely for various values of p . Their results included that: Thus ln ⁡ n n {\displaystyle {\tfrac {\ln n}{n}}}

2964-431: The law of large numbers ( LLN ) is a mathematical law that states that the average of the results obtained from a large number of independent random samples converges to the true value, if it exists. More formally, the LLN states that given a sample of independent and identically distributed values, the sample mean converges to the true mean . The LLN is important because it guarantees stable long-term results for

3042-667: The Cauchy distribution does not have an expectation, whereas the expectation of the Pareto distribution ( α <1) is infinite. One way to generate the Cauchy-distributed example is where the random numbers equal the tangent of an angle uniformly distributed between −90° and +90°. The median is zero, but the expected value does not exist, and indeed the average of n such variables have the same distribution as one such variable. It does not converge in probability toward zero (or any other value) as n goes to infinity. And if

3120-505: The Erdős–Rényi process is in fact unweighted link percolation on the complete graph . (One refers to percolation in which nodes and/or links are removed with heterogeneous weights as weighted percolation). As percolation theory has much of its roots in physics , much of the research done was on the lattices in Euclidean spaces. The transition at np = 1 from giant component to small component has analogs for these graphs, but for lattices

3198-437: The average of a set of normally distributed variables). The variance of the sum is equal to the sum of the variances, which is asymptotic to n 2 / log ⁡ n {\displaystyle n^{2}/\log n} . The variance of the average is therefore asymptotic to 1 / log ⁡ n {\displaystyle 1/\log n} and goes to zero. There are also examples of

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3276-434: The average of the first n values goes to zero as n goes to infinity. As an example, assume that each random variable in the series follows a Gaussian distribution (normal distribution) with mean zero, but with variance equal to 2 n / log ⁡ ( n + 1 ) {\displaystyle 2n/\log(n+1)} , which is not bounded. At each stage, the average will be normally distributed (as

3354-664: The average to converge almost surely on something (this can be considered another statement of the strong law), it is necessary that they have an expected value (and then of course the average will converge almost surely on that). If the summands are independent but not identically distributed, then provided that each X k has a finite second moment and ∑ k = 1 ∞ 1 k 2 Var ⁡ [ X k ] < ∞ . {\displaystyle \sum _{k=1}^{\infty }{\frac {1}{k^{2}}}\operatorname {Var} [X_{k}]<\infty .} This statement

3432-424: The averages of some random events . For example, while a casino may lose money in a single spin of the roulette wheel, its earnings will tend towards a predictable percentage over a large number of spins. Any winning streak by a player will eventually be overcome by the parameters of the game. Importantly, the law applies (as the name indicates) only when a large number of observations are considered. There

3510-512: The case where X 1 , X 2 , ... is an infinite sequence of independent and identically distributed (i.i.d.) Lebesgue integrable random variables with expected value E( X 1 ) = E( X 2 ) = ... = μ , both versions of the law state that the sample average X ¯ n = 1 n ( X 1 + ⋯ + X n ) {\displaystyle {\overline {X}}_{n}={\frac {1}{n}}(X_{1}+\cdots +X_{n})} converges to

3588-712: The connectivity of G ( n ,  M ), with the more detailed analysis following in 1960. A continuum limit of the graph was obtained when p {\displaystyle p} is of order 1 / n {\displaystyle 1/n} . Specifically, consider the sequence of graphs G n := G ( n , 1 / n + λ n − 4 3 ) {\displaystyle G_{n}:=G(n,1/n+\lambda n^{-{\frac {4}{3}}})} for λ ∈ R {\displaystyle \lambda \in \mathbb {R} } . The limit object can be constructed as follows: Applying this procedure, one obtains

3666-402: The context of search algorithms monotonicity (also called consistency) is a condition applied to heuristic functions . A heuristic h ( n ) {\displaystyle h(n)} is monotonic if, for every node n and every successor n' of n generated by any action a , the estimated cost of reaching the goal from n is no greater than the step cost of getting to n' plus

3744-414: The convergence is only weak (in probability). See differences between the weak law and the strong law . The strong law applies to independent identically distributed random variables having an expected value (like the weak law). This was proved by Kolmogorov in 1930. It can also apply in other cases. Kolmogorov also showed, in 1933, that if the variables are independent and identically distributed, then for

3822-418: The estimated cost of reaching the goal from n' , h ( n ) ≤ c ( n , a , n ′ ) + h ( n ′ ) . {\displaystyle h(n)\leq c\left(n,a,n'\right)+h\left(n'\right).} This is a form of triangle inequality , with n , n' , and the goal G n closest to n . Because every monotonic heuristic

3900-483: The expected difference grows, but at a slower rate than the number of flips. Another good example of the LLN is the Monte Carlo method . These methods are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The larger the number of repetitions, the better the approximation tends to be. The reason that this method is important is mainly that, sometimes, it

3978-407: The expected value times n as n increases. Throughout its history, many mathematicians have refined this law. Today, the LLN is used in many fields including statistics, probability theory, economics, and insurance. For example, a single roll of a fair, six-sided die produces one of the numbers 1, 2, 3, 4, 5, or 6, each with equal probability . Therefore, the expected value of the average of

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4056-639: The expected value: (Lebesgue integrability of X j means that the expected value E( X j ) exists according to Lebesgue integration and is finite. It does not mean that the associated probability measure is absolutely continuous with respect to Lebesgue measure .) Introductory probability texts often additionally assume identical finite variance Var ⁡ ( X i ) = σ 2 {\displaystyle \operatorname {Var} (X_{i})=\sigma ^{2}} (for all i {\displaystyle i} ) and no correlation between random variables. In that case,

4134-472: The giant component, P ∞ , is given by Both of the two major assumptions of the G ( n , p ) model (that edges are independent and that each edge is equally likely) may be inappropriate for modeling certain real-life phenomena. Erdős–Rényi graphs have low clustering, unlike many social networks. Some modeling alternatives include Barabási–Albert model and Watts and Strogatz model . These alternative models are not percolation processes, but instead represent

4212-415: The inequality | X ¯ n − μ | < ε {\displaystyle |{\overline {X}}_{n}-\mu |<\varepsilon } holds for all large enough n , since the convergence is not necessarily uniform on the set where it holds. The strong law does not hold in the following cases, but the weak law does. There are extensions of

4290-411: The intuitive interpretation of the expected value (for Lebesgue integration only) of a random variable when sampled repeatedly as the "long-term average". Law 3 is called the strong law because random variables which converge strongly (almost surely) are guaranteed to converge weakly (in probability). However the weak law is known to hold in certain conditions where the strong law does not hold and then

4368-406: The law of large numbers that the empirical probability of success in a series of Bernoulli trials will converge to the theoretical probability. For a Bernoulli random variable , the expected value is the theoretical probability of success, and the average of n such variables (assuming they are independent and identically distributed (i.i.d.) ) is precisely the relative frequency. For example,

4446-411: The law of large numbers to collections of estimators, where the convergence is uniform over the collection; thus the name uniform law of large numbers . Suppose f ( x , θ ) is some function defined for θ ∈ Θ, and continuous in θ . Then for any fixed θ , the sequence { f ( X 1 , θ ), f ( X 2 , θ ), ...} will be a sequence of independent and identically distributed random variables, such that

4524-417: The name "la loi des grands nombres" ("the law of large numbers"). Thereafter, it was known under both names, but the "law of large numbers" is most frequently used. After Bernoulli and Poisson published their efforts, other mathematicians also contributed to refinement of the law, including Chebyshev , Markov , Borel , Cantelli , Kolmogorov and Khinchin . Markov showed that the law can apply to

4602-424: The network. There exists a critical percolation threshold p c ′ = 1 ⟨ k ⟩ {\displaystyle p'_{c}={\tfrac {1}{\langle k\rangle }}} below which the network becomes fragmented while above p c ′ {\displaystyle p'_{c}} a giant connected component of order n exists. The relative size of

4680-463: The order ≤ {\displaystyle \leq } in the definition of monotonicity is replaced by the strict order < {\displaystyle <} , one obtains a stronger requirement. A function with this property is called strictly increasing (also increasing ). Again, by inverting the order symbol, one finds a corresponding concept called strictly decreasing (also decreasing ). A function with either property

4758-406: The order (see Figure 1). Likewise, a function is called monotonically decreasing (also decreasing or non-increasing ) if, whenever x ≤ y {\displaystyle x\leq y} , then f ( x ) ≥ f ( y ) {\displaystyle f\!\left(x\right)\geq f\!\left(y\right)} , so it reverses the order (see Figure 2). If

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4836-399: The other the "strong" law, in reference to two different modes of convergence of the cumulative sample means to the expected value; in particular, as explained below, the strong form implies the weak. There are two different versions of the law of large numbers that are described below. They are called the strong law of large numbers and the weak law of large numbers . Stated for

4914-573: The proofs. This assumption of finite variance is not necessary . Large or infinite variance will make the convergence slower, but the LLN holds anyway. Mutual independence of the random variables can be replaced by pairwise independence or exchangeability in both versions of the law. The difference between the strong and the weak version is concerned with the mode of convergence being asserted. For interpretation of these modes, see Convergence of random variables . The weak law of large numbers (also called Khinchin 's law) states that given

4992-402: The property of having an even number of edges). In practice, the G ( n , p ) model is the one more commonly used today, in part due to the ease of analysis allowed by the independence of the edges. With the notation above, a graph in G ( n , p ) has on average ( n 2 ) p {\displaystyle {\tbinom {n}{2}}p} edges. The distribution of

5070-432: The proportion of heads (and tails) approaches 1 ⁄ 2 , almost surely the absolute difference in the number of heads and tails will become large as the number of flips becomes large. That is, the probability that the absolute difference is a small number approaches zero as the number of flips becomes large. Also, almost surely the ratio of the absolute difference to the number of flips will approach zero. Intuitively,

5148-511: The reason why monotonic functions are useful in technical work in analysis . Other important properties of these functions include: An important application of monotonic functions is in probability theory . If X {\displaystyle X} is a random variable , its cumulative distribution function F X ( x ) = Prob ( X ≤ x ) {\displaystyle F_{X}\!\left(x\right)={\text{Prob}}\!\left(X\leq x\right)}

5226-442: The robustness of the graph, viewed as a communication network. Given a random graph of n  ≫ 1 nodes with an average degree  ⟨ k ⟩ {\displaystyle \langle k\rangle } . Remove randomly a fraction 1 − p ′ {\displaystyle 1-p'} of nodes and leave only a fraction p ′ {\displaystyle p'} from

5304-413: The rolls is: 1 + 2 + 3 + 4 + 5 + 6 6 = 3.5 {\displaystyle {\frac {1+2+3+4+5+6}{6}}=3.5} According to the law of large numbers, if a large number of six-sided dice are rolled, the average of their values (sometimes called the sample mean ) will approach 3.5, with the precision increasing as more dice are rolled. It follows from

5382-857: The sample mean of this sequence converges in probability to E[ f ( X , θ )]. This is the pointwise (in θ ) convergence. A particular example of a uniform law of large numbers states the conditions under which the convergence happens uniformly in θ . If Then E[ f ( X , θ )] is continuous in θ , and sup θ ∈ Θ ‖ 1 n ∑ i = 1 n f ( X i , θ ) − E ⁡ [ f ( X , θ ) ] ‖ → P   0. {\displaystyle \sup _{\theta \in \Theta }\left\|{\frac {1}{n}}\sum _{i=1}^{n}f(X_{i},\theta )-\operatorname {E} [f(X,\theta )]\right\|{\overset {\mathrm {P} }{\rightarrow }}\ 0.} This result

5460-412: The series, keeping the expected value constant. If the variances are bounded, then the law applies, as shown by Chebyshev as early as 1867. (If the expected values change during the series, then we can simply apply the law to the average deviation from the respective expected values. The law then states that this converges in probability to zero.) In fact, Chebyshev's proof works so long as the variance of

5538-468: The statement that almost every graph in G ( n , 2 ln ⁡ ( n ) / n ) {\displaystyle G(n,2\ln(n)/n)} is connected means that, as n {\displaystyle n} tends to infinity, the probability that a graph on n {\displaystyle n} vertices with edge probability 2 ln ⁡ ( n ) / n {\displaystyle 2\ln(n)/n}

5616-490: The statements " P holds for almost all graphs in G ( n ,  p )" and " P holds for almost all graphs in G ( n , ( n 2 ) p ) {\displaystyle G(n,{\tbinom {n}{2}}p)} " are equivalent (provided pn → ∞). For example, this holds if P is the property of being connected , or if P is the property of containing a Hamiltonian cycle . However, this will not necessarily hold for non-monotone properties (e.g.

5694-442: The terms weakly monotone , weakly increasing and weakly decreasing are often used to refer to non-strict monotonicity. The terms "non-decreasing" and "non-increasing" should not be confused with the (much weaker) negative qualifications "not decreasing" and "not increasing". For example, the non-monotonic function shown in figure 3 first falls, then rises, then falls again. It is therefore not decreasing and not increasing, but it

5772-408: The transition point is difficult to determine. Physicists often refer to study of the complete graph as a mean field theory . Thus the Erdős–Rényi process is the mean-field case of percolation. Some significant work was also done on percolation on random graphs. From a physicist's point of view this would still be a mean-field model, so the justification of the research is often formulated in terms of

5850-409: The trials embed a selection bias , typical in human economic/rational behaviour, the law of large numbers does not help in solving the bias. Even if the number of trials is increased the selection bias remains. The Italian mathematician Gerolamo Cardano (1501–1576) stated without proof that the accuracies of empirical statistics tend to improve with the number of trials. This was then formalized as

5928-789: The variance of the average of n random variables is Var ⁡ ( X ¯ n ) = Var ⁡ ( 1 n ( X 1 + ⋯ + X n ) ) = 1 n 2 Var ⁡ ( X 1 + ⋯ + X n ) = n σ 2 n 2 = σ 2 n . {\displaystyle \operatorname {Var} ({\overline {X}}_{n})=\operatorname {Var} ({\tfrac {1}{n}}(X_{1}+\cdots +X_{n}))={\frac {1}{n^{2}}}\operatorname {Var} (X_{1}+\cdots +X_{n})={\frac {n\sigma ^{2}}{n^{2}}}={\frac {\sigma ^{2}}{n}}.} which can be used to shorten and simplify

6006-494: The weak law applying even though the expected value does not exist. The strong law of large numbers (also called Kolmogorov 's law) states that the sample average converges almost surely to the expected value That is, Pr ( lim n → ∞ X ¯ n = μ ) = 1. {\displaystyle \Pr \!\left(\lim _{n\to \infty }{\overline {X}}_{n}=\mu \right)=1.} What this means

6084-455: The weak law states that for any nonzero margin specified ( ε ), no matter how small, with a sufficiently large sample there will be a very high probability that the average of the observations will be close to the expected value; that is, within the margin. As mentioned earlier, the weak law applies in the case of i.i.d. random variables, but it also applies in some other cases. For example, the variance may be different for each random variable in

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