query: the law of large numbers says that the mean outcome in many repetitions eventually gets close to

pos: Law of large numbers the sample mean converges in probability to the derivative of the characteristic function at the origin, as long as the latter exists. Law of large numbers In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials should be close to the expected value, and will tend to become closer as more trials are performed. The LLN is important because it guarantees stable long-term results for the averages of

pos: Expected value Expected value In probability theory, the expected value of a random variable, intuitively, is the long-run average value of repetitions of the experiment it represents. For example, the expected value in rolling a six-sided die is 3.5, because the average of all the numbers that come up in an extremely large number of rolls is close to 3.5. Less roughly, the law of large numbers states that the arithmetic mean of the values almost surely converges to the expected value as the number of repetitions approaches infinity. The expected value is also known as the expectation, mathematical expectation, EV, average,

pos: Law of large numbers the sample mean converges in probability to the derivative of the characteristic function at the origin, as long as the latter exists. Law of large numbers In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials should be close to the expected value, and will tend to become closer as more trials are performed. The LLN is important because it guarantees stable long-term results for the averages of

pos: Law of large numbers 3, 4, 5, or 6, each with equal probability. Therefore, the expected value of a single die roll is 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) is likely to be close to 3.5, with the precision increasing as more dice are rolled. It follows from 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

pos: Law of large numbers Law of large numbers In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials should be close to the expected value, and will tend to become closer as more trials are performed. The LLN is important because it guarantees stable long-term results for 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

pos: Law of large numbers 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 the sample average converges in probability towards the expected value That is, for any positive number "ε", Interpreting this result, the weak law states that for any nonzero margin specified, 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

pos: Regression toward the mean event, the next random event is likely to be less extreme. In no sense does the future event "compensate for" or "even out" the previous event, though this is assumed in the gambler's fallacy (and the variant law of averages). Similarly, the law of large numbers states that in the long term, the average will tend towards the expected value, but makes no statement about individual trials. For example, following a run of 10 heads on a flip of a fair coin (a rare, extreme event), regression to the mean states that the next run of heads will likely be

pos: Law of large numbers 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. It is important to remember that the law only applies (as the name indicates) when a "large number" of observations is considered. There 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). For example, a single roll of a fair, six-sided dice produces one of the numbers 1, 2,

pos: Monte Carlo method problem having a probabilistic interpretation. By the law of large numbers, integrals described by the expected value of some random variable can be approximated by taking the empirical mean (a.k.a. the sample mean) of independent samples of the variable. When the probability distribution of the variable is parametrized, mathematicians often use a Markov chain Monte Carlo (MCMC) sampler. The central idea is to design a judicious Markov chain model with a prescribed stationary probability distribution. That is, in the limit, the samples being generated by the MCMC method will be samples from the desired (target) distribution. By the ergodic theorem,

neg: Probability interpretations of experiment will generate given outcome types at persistent rates, which are known as propensities or chances. Frequentists are unable to take this approach, since relative frequencies do not exist for single tosses of a coin, but only for large ensembles or collectives (see "single case possible" in the table above). In contrast, a propensitist is able to use the law of large numbers to explain the behaviour of long-run frequencies. This law, which is a consequence of the axioms of probability, says that if (for example) a coin is tossed repeatedly many times, in such a way that its

neg: Central limit theorem mean or variance are not defined. The scaling factor may be proportional to , for any ; it may also be multiplied by a slowly varying function of . The law of the iterated logarithm specifies what is happening "in between" the law of large numbers and the central limit theorem. Specifically it says that the normalizing function , intermediate in size between of the law of large numbers and of the central limit theorem, provides a non-trivial limiting behavior. The density of the sum of two or more independent variables is the convolution of their densities (if these densities

neg: Generic property is an event that occurs almost surely, meaning that it occurs with probability 1. For example, the law of large numbers states that the sample mean converges almost surely to the population mean. This is the definition in the measure theory case specialized to a probability space. In discrete mathematics, one uses the term almost all to mean cofinite (all but finitely many), cocountable (all but countably many), for sufficiently large numbers, or, sometimes, asymptotically almost surely. The concept is particularly important in the study of random graphs. In topology and algebraic geometry, a generic property is one that holds

neg: Riemann integral develop this definition now, with a proof of equivalence following. Our new definition says that the Riemann integral of equals if the following condition holds: For all , there exists a tagged partition and such that for any tagged partition and which is a refinement of and , we have Both of these mean that eventually, the Riemann sum of with respect to any partition gets trapped close to . Since this is true no matter how close we demand the sums be trapped, we say that the Riemann sums converge to . These definitions are actually a special case

neg: Pafnuty Chebyshev the probability that the outcome of formula_1 is no less than formula_3 away from its mean is no more than formula_4: The Chebyshev inequality is used to prove the Weak Law of Large Numbers. The Bertrand–Chebyshev theorem (1845,1852) states that for any formula_6, there exists a prime number formula_7 such that formula_8. This is a consequence of the Chebyshev inequalities for the number formula_9 of prime numbers less than formula_10, which state that formula_9 is of the order of formula_12. A more precise form is given by the celebrated prime number theorem: the "quotient" of the two expressions approaches 1.0

neg: Law of large numbers This result is useful to derive consistency of a large class of estimators (see Extremum estimator). Borel's law of large numbers, named after Émile Borel, states that if an experiment is repeated a large number of times, independently under identical conditions, then the proportion of times that any specified event occurs approximately equals the probability of the event's occurrence on any particular trial; the larger the number of repetitions, the better the approximation tends to be. More precisely, if "E" denotes the event in question, "p" its probability of occurrence, and "N"("E") the number of times "E" occurs in the

neg: Taguchi methods among all mean-unbiased estimators. The emphasis on comparisons of means also draws (limiting) comfort from the law of large numbers, according to which the sample means converge to the true mean. Fisher's textbook on the design of experiments emphasized comparisons of treatment means. However, loss functions were avoided by Ronald A. Fisher. Taguchi knew statistical theory mainly from the followers of Ronald A. Fisher, who also avoided loss functions. Reacting to Fisher's methods in the design of experiments, Taguchi interpreted Fisher's methods as being adapted for seeking to improve the mean outcome of a process. Indeed, Fisher's work had been

neg: Mean especially for small samples. The law of large numbers dictates that the larger the size of the sample, the more likely it is that the sample mean will be close to the population mean. Outside probability and statistics, a wide range of other notions of "mean" are often used in geometry and analysis; examples are given below. The "arithmetic mean" (or simply "mean") of a sample formula_6, usually denoted by formula_1, is the sum of the sampled values divided by the number of items in the example For example, the arithmetic mean of five values: 4, 36, 45, 50, 75

neg: Statistical regularity Statistical regularity Statistical regularity is a notion in statistics and probability theory that random events exhibit regularity when repeated enough times or that enough sufficiently similar random events exhibit regularity. It is an umbrella term that covers the law of large numbers, all central limit theorems and ergodic theorems. If one throws a die once, it is difficult to predict the outcome, but if we repeat this experiment many times, we will see that the number of times each result occurs divided by the number of throws will eventually stabilize towards a specific value. Repeating a series of trials will

neg: Statistical regularity because the latter considers the whole rather than individual cases. Statistical regularity Statistical regularity is a notion in statistics and probability theory that random events exhibit regularity when repeated enough times or that enough sufficiently similar random events exhibit regularity. It is an umbrella term that covers the law of large numbers, all central limit theorems and ergodic theorems. If one throws a die once, it is difficult to predict the outcome, but if we repeat this experiment many times, we will see that the number of times each result occurs divided by the number of throws will eventually stabilize

neg: Propensity probability is the law of large numbers. This law, which is a consequence of the axioms of probability, says that if (for example) a coin is tossed repeatedly many times, in such a way that its probability of landing heads is the same on each toss, and the outcomes are probabilistically independent, then the relative frequency of heads will (with high probability) be close to the probability of heads on each single toss. This law suggests that stable long-run frequencies are a manifestation of invariant "single-case" probabilities. Frequentists are unable to take this approach, since relative frequencies do not exist for

neg: 2009 New Zealand citizens-initiated referendum be a vote for the status quo, while a 'No' vote is seen to be a vote against the 2007 amendment. "In this context, we recommend a 'Yes' vote, as we believe the status quo is close to the position that we recommended to the Select Committee. However, the wording of the question is so ambiguous, many New Zealanders who support efforts to reduce violence against children, may in good conscience still feel obliged to vote 'No'. It will be hard to understand what the outcome of the referendum may mean," says Mr Smith. He says Caritas will be writing

neg: Large deviations theory of certain kinds of extreme or "tail" events. Consider a sequence of independent tosses of a fair coin. The possible outcomes could be heads or tails. Let us denote the possible outcome of the i-th trial by formula_1, where we encode head as 1 and tail as 0. Now let formula_2 denote the mean value after formula_3 trials, namely Then formula_2 lies between 0 and 1. From the law of large numbers (and also from our experience) we know that as N grows, the distribution of formula_2 converges to formula_7 (the expectation value of a single coin toss) almost surely.

neg: Chemical law says that pure chemicals are composed of elements in a definite formulation; we now know that the structural arrangement of these elements is also important. Dalton's law of multiple proportions says that these chemicals will present themselves in proportions that are small whole numbers (i.e. 1:2 O:H in water); although in many systems (notably biomacromolecules and minerals) the ratios tend to require large numbers, and are frequently represented as a fraction. Such compounds are known as non-stoichiometric compounds More modern laws of chemistry define the relationship between energy and transformations. Chemical law Chemical laws are those laws of nature relevant

neg: Law of large numbers average The weak law of large numbers states: This proof uses the assumption of finite variance formula_30 (for all formula_31). The independence of the random variables implies no correlation between them, and we have that The common mean μ of the sequence is the mean of the sample average: Using Chebyshev's inequality on formula_34 results in This may be used to obtain the following: As "n" approaches infinity, the expression approaches 1. And by definition of convergence in probability, we have obtained By Taylor's theorem for complex functions, the characteristic function of any random variable, "X", with finite mean μ,

neg: Law of averages a rare event occurring at least once by carrying out more trials. For example, a job seeker might argue, "If I send my résumé to enough places, the law of averages says that someone will eventually hire me." Assuming a non-zero probability, it is true that conducting more trials increases the overall likelihood of the desired outcome. However, there is no particular number of trials that guarantees that outcome; rather, the probability that it will already have occurred approaches but never quite reaches 100%. The Steve Goodman song "A Dying Cub Fan's Last Request" mentions the Law of Averages in

neg: Stochastic simulation is an estimation procedure. The main idea is that if it is necessary to know the average value of some random variable and its distribution cannot be stated, and if it is possible to take samples from the distribution, we can estimate it by taking the samples, independently, and averaging them. If there are sufficient samples, then the law of large numbers says the average must be close to the true value. The central limit theorem says that the average has a Gaussian distribution around the true value. As a simple example, suppose we need to measure area of a

neg: Mahmoud Ahmadinejad said "that Israel is an unnatural creature, it will not survive. They didn't say, 'We'll wipe it out,' you're right, but, 'It will not survive.'" adding "If Iran says this, and continues to pile up uranium that they enrich, and build missiles in big numbers, and have a nuclear military plan—if you put all this together, you can't say, they don't really mean it." "The Washington Post"s fact-checker editor Glenn Kessler says the interpretation gets murkier when Ahmadinejad's quote is set against other Iranian propaganda. Karim Sadjadpour, an Iranian specialist at the Carnegie Endowment for International Peace, cites proof that

neg: Doctor Zhivago (novel) wife, children, and father-in-law fled the estate and returned to Moscow. From Lara, he learns that Tonya delivered a daughter after he left. Lara assisted at the birth and she and Tonya became close friends. Yuri gets a job and stays with Lara and her daughter for a few months. Eventually, a townsperson delivers a letter to Yuri from Tonya, which Tonya wrote five months before and which has passed through innumerable hands to reach Yuri. In the letter, Tonya informs him that she, the children, and her father are being deported, probably to Paris. She says "The whole trouble

neg: Law of large numbers 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, 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 the proportion of heads (and tails)

neg: Gibrat's law magnitude ranging from zero, while eventually falling off at unreasonably large outcome sizes. In the study of the firms (business), the scholars do not agree that the foundation and the outcome of Gibrat's law are empirically correct. Gibrat's law Gibrat's law (sometimes called Gibrat's rule of proportionate growth or the law of proportionate effect) is a rule defined by Robert Gibrat (1904–1980) in 1931 stating that the proportional rate of growth of a firm is independent of its absolute size. The law of proportionate growth gives rise to a distribution that is log-normal. Gibrat's law is also applied to cities

neg: Benford's law gets multiplied by a randomly chosen factor between 0.99 and 1.01, then over an extended period the probability distribution of its price satisfies Benford's law with higher and higher accuracy. The reason is that the "logarithm" of the stock price is undergoing a random walk, so over time its probability distribution will get more and more broad and smooth (see above). (More technically, the central limit theorem says that multiplying more and more random variables will create a log-normal distribution with larger and larger variance, so eventually it covers many orders of magnitude almost uniformly.) To be sure of approximate

neg: Tweedie distribution held hypothesis is based on Self-organized criticality where dynamical systems close to a critical point are thought to manifest scale-invariant spatial and/or temporal behavior. In this subsection a mathematical connection between 1/"f" noise and the Tweedie variance-to-mean power law will be described. To begin, we first need to introduce self-similar processes: For the sequence of numbers with mean deviations variance and autocorrelation function with lag "k", if the autocorrelation of this sequence has the long range behavior as "k" and where "L"("k") is a slowly varying function at large values of "k", this sequence is called a self-similar process. The

neg: Sod's law upbeat—it was a reminder to the engineers and team members to be cautious and make sure everything was accounted for, to let no stone be left unturned—not an acceptance of an uncaring uninfluenceable fate. According to David J. Hand, emeritus professor of mathematics and senior research investigator at Imperial College London, Sod's law is a more extreme version of Murphy's law. While Murphy's law says that anything that can go wrong, will go wrong (eventually), Sod's law requires that it "always" goes wrong with the worst possible outcome. Hand suggests that belief in Sod's law is a combination of the

neg: Probability theory independent and identically distributed random variables formula_59 converges towards their common expectation formula_60, provided that the expectation of formula_61 is finite. It is in the different forms of convergence of random variables that separates the "weak" and the "strong" law of large numbers It follows from the LLN that if an event of probability "p" is observed repeatedly during independent experiments, the ratio of the observed frequency of that event to the total number of repetitions converges towards "p". For example, if formula_66 are independent Bernoulli random variables taking values 1 with probability "p" and 0 with probability 1-"p", then

neg: Ek Veer Ki Ardaas...Veera Baldev is released, he and Veera elope and get married in a mandir. Ranvi gets mad at Veera and does not talk to her. Baldev and Veera have a fight, in which Baldev starts to act mean with Veera, and she tries to hide this from her brother. Eventually, all becomes well with Baldev and Veera. Then enters Baldev & Gunjan's aunt named, Manjeet Singh (Usha Bachani), with her widowed daughter-in-law, Geet Singh (Keerti Nagpure), and granddaughter, Deepu Singh (Vrudhi Jakra), to Baldev's house. Geet starts to become close with Ranvi, who reminds him of a younger Veera and still

neg: Byari (film) continues her content married life with her child, she gets divorced because of a dispute between her husband and her father. Nadira is made to spend her life at her father’s place devoid of any contact with the outside world. In the mean time, her husband takes the child away. Eventually bypassing many hurdles, it turns out that Nadira can spend her life with her husband yet again. However, for that as per the dictates of the law, she must marry and divorce some other man. The film was Directed by Suveeran and produced by T.H. Althaf Hussain under the

neg: 2015 CrossFit Games 50-foot unbroken increments. Ben Smith won the event for the men and Elisabeth Akinwale won for the women. Each athlete gets two 20-second windows in which to attempt their heaviest clean and jerk for one repetition. Aaron Hanna won the event with a 350-pound lift for the men and Brooke Ence won with a 242-pound lift for the women. 15–10–6 repetitions of: Named "Triangle Couplet" because the repetitions per round are all triangular numbers. A thruster in CrossFit is a combination of a front squat and push press in one continuous movement and the weights were 165-pounds for the men

neg: 1982 California gubernatorial election be a close race but as the campaign went on, Bradley slowly increased his lead in the polls. Despite this, the outcome was still in doubt as election night wore on. A large late surge of absentee ballots (many from conservative Orange County and Deukmejian's home area of Long Beach) helped tip the balance in his favor. The incorrect polling numbers led to the theory — later dubbed "the Bradley effect" — that a statistically significant number of voters had given inaccurate responses when questioned by pollsters. 1982 California gubernatorial election The 1982 California gubernatorial election occurred on November 2,

neg: Pluto of the mean-motion resonance is due to phase protection. If Pluto's period is slightly shorter than 3/2 of Neptune, its orbit relative to Neptune will drift, causing it to make closer approaches behind Neptune's orbit. The strong gravitational pull between the two causes angular momentum to be transferred to Pluto, at Neptune's expense. This moves Pluto into a slightly larger orbit, where it travels slightly more slowly, according to Kepler's third law. After many such repetitions, Pluto is sufficiently slowed, and Neptune sufficiently sped up, that Pluto orbit relative to Neptune drifts in the opposite direction until the process is

neg: Francis Sargent shift towards public transportation." I mean it was thrilling. It was thrilling for us that had worked hard on it, but also, in fairness to Sargent how often do you see a public official who gets up and says, "I was wrong"? I mean it was an incredibly courageous thing for Frank Sargent to do, and I'm a Democrat. I don't say many good things about republicans. But he was a great man. I mean he had worked for this program. He always had an environmentalist bent to him. [A] lot of people do political analysis as to why he

neg: Gambler's fallacy particular outcome is due after a long streak of another outcome. Type two gambler's fallacy, as defined by Gideon Keren and Charles Lewis, occurs when a gambler underestimates how many observations are needed to detect a favorable outcome, such as watching a roulette wheel for a length of time and then betting on the numbers that appear most often. For events with a high degree of randomness, detecting a bias that will lead to a favorable outcome takes an impractically large amount of time and is very difficult, if not impossible, to do. The two types differ in that type

neg: Asymptotic theory (statistics) infinitely, i.e. . Under the assumption, many results can be obtained that are unavailable for samples of finite size. An example is the law of large numbers. The law states that for a sequence of independent and identically distributed (IID) random variables , if one value is drawn from each random variable and the average of the first values is computed as , then the converge in probability to the population mean as . In asymptotic theory, the standard approach is . For some statistical models, slightly different approaches of asymptotics may be used. For example, with panel data, it

neg: Main Teri Parchhain Hoon the kids give her presents. However, when Sachin gives his gift it is so picture of Jaya and Aanchal which on Jaya's photo it says 'maa' and on Aanchal's it says 'teacher maa'. The whole family is shocked to see this especially Siddharth's step mother and sister in law, Chanchal who do not like Aanchal because of her close bond to the kids and family members. She and Lalita conspire against Aanchal on many occasions. Ayush gets really angry and starts to hate Aanchal as he thinks she's replacing his mother and he convinces Juhi the same which she begins

neg: Rethinking Madness Rethinking Madness Rethinking Madness: Towards a Paradigm Shift In Our Understanding and Treatment of Psychosis (Sky’s Edge Publishing, 2012) is a book by the psychologist Paris Williams which explores creative ways of dealing with madness (psychosis). Williams says that psychotic experiences, which cause significant distress, typically occur deep within the mind. Given suitable conditions, this process can often result in a positive outcome, but Williams avoids the romantic notion that psychosis is always beneficial. Much of what Williams says is in close accord with a recovery approach. Williams says that the term "psychosis" has many meanings, and the definitions that

neg: Names of large numbers of large numbers including lakhs and crores. English also has many words, such as "zillion", used informally to mean large but unspecified amounts; see indefinite and fictitious numbers. Apart from "million", the words in this list ending with -"illion" are all derived by adding prefixes ("bi"-, "tri"-, etc., derived from Latin) to the stem -"illion". "Centillion" appears to be the highest name ending in -"illion" that is included in these dictionaries. "Trigintillion", often cited as a word in discussions of names of large numbers, is not included in any of them, nor are any of the names that can easily

neg: Rethinking Madness problems. Rethinking Madness Rethinking Madness: Towards a Paradigm Shift In Our Understanding and Treatment of Psychosis (Sky’s Edge Publishing, 2012) is a book by the psychologist Paris Williams which explores creative ways of dealing with madness (psychosis). Williams says that psychotic experiences, which cause significant distress, typically occur deep within the mind. Given suitable conditions, this process can often result in a positive outcome, but Williams avoids the romantic notion that psychosis is always beneficial. Much of what Williams says is in close accord with a recovery approach. Williams says that the term "psychosis" has many meanings, and the definitions

neg: Doris Salcedo where some minor crimes have become felonies. I’m really shocked by the sheer numbers of people being thrown into jails. And also I think it’s amazing how this system, being in jail and then going out, has so many collateral effects that a fairly large portion of the population are not allowed to be alive. The idea of having a large portion of the population excluded from civil rights, from many, many possibilities, implies that you have people that can almost be considered socially dead. What does it mean to be socially dead? What does it mean to be alive

neg: Sod's law law of truly large numbers and the psychological effect of the law of selection. The former says we should expect things to go wrong now and then, and the latter says we remember the exceptional events where something went wrong, but the great number of mundane events where nothing exceptional happened are forgotten. Both Dawkins and Hand offered these examples as part of a critique of the thinking that leads to belief in such laws, rather than examples supporting the existence of the law. Both went on to explain the effect in terms of selection bias and the law of

neg: Central limit theorem gets larger, the distribution of the difference between the sample average and its limit , when multiplied by the factor (that is ), approximates the normal distribution with mean 0 and variance . For large enough , the distribution of is close to the normal distribution with mean and variance /. The usefulness of the theorem is that the distribution of approaches normality regardless of the shape of the distribution of the individual . Formally, the theorem can be stated as follows: Lindeberg–Lévy CLT. Suppose } is a sequence of i.i.d. random variables with and . Then as approaches infinity,

neg: Law of large numbers absolutely convergent, but using conditional convergence, we can say: 3. If the cumulative distribution function of a random variable is Suppose "f"("x","θ") is some function defined for "θ" ∈ Θ, and continuous in "θ". Then for any fixed "θ", the sequence {"f"("X","θ"), "f"("X","θ"), …} will be a sequence of independent and identically distributed random variables, such that the sample mean of this sequence converges in probability to E["f"("X","θ")]. This is the "pointwise" (in "θ") convergence. The 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

neg: Junkyard tornado that the entity could appear by chance is exceedingly low, ignoring the key process involved, natural selection. Sometimes, arguments invoking the junkyard tornado analogy also invoke Borel's Law, which claims that highly improbable events do not occur. The usual argument against Borel's "Law" is that if "all" possible outcomes of a natural process are highly improbable when taken individually, then a highly improbable outcome is certain. The true law being referenced is actually the Strong Law of large numbers, but creationists have taken a simple statement made by Borel in books written late in his life concerning probability theory and

neg: The Bear (play) of guns her husband owned. Luka overhears their conversation, gets frightened for his mistress, and goes off to find someone to help put an end to their feud before anyone gets hurt. Meanwhile, Smirnov says to himself how impressed he is by Popova's audacity and slowly realizes that he has actually fallen in love with her and her dimpled cheeks. When Popova returns with the pistols, Smirnov makes his love confession. Popova oscillates between refusing him and ordering him to leave and telling him to stay. Eventually, the two get close and kiss each other just as Luka returns with

neg: John Edmund Kerrich Law of Large Numbers using a coin which they tossed 10,000 times. By recording the number of heads obtained as the trials continued, Kerrich was able to demonstrate that the proportion of heads obtained asymptotically approached the theoretical value of 50 percent (the precise number obtained was 5,067, which is 1.34 standard deviations above the mean for a "fair" coin thrown that many times). Kerrich and Christensen also performed experiments using a "biased coin", made from a wooden disk partly coated in lead, to show that it too tended towards a stable asymptotic state with probability of approximately 70 percent.

neg: Weapons of Class Destruction Ben and Norris talk about guns and bombs, leading Veronica to track Ben, eventually seeing him buying fertilizer and taking out a rifle. Ben gets into the car and orders her to drive to the Camelot motel. There, Logan appears and beats him up before it is revealed that the "suspect" is actually an ATF agent. Ben says that he is investigating Norris, but needs her help: Norris has a crush on Veronica, and Ben asks her to get close to him. As she leaves the motel, Veronica and Logan kiss unexpectedly. Veronica gets an invitation to Norris's house, but

neg: Prime version though some later versions can be argued to improve on the originals. In many periods the later "repetitions" were often produced by the workshop of the master, with varying degrees of supervision and direct attention from him. This was especially the case with official portraits of monarchs and politicians, which in the Early Modern period were often ordered in large numbers of versions from the court artist as diplomatic gifts. Sometimes "reduced versions" that are considerably smaller than the prime one are made. In the case of official portraits the later creation of replica versions was likely to have been

neg: Persistence of a number of candidates for "n"-digit numbers with record-breaking persistence is only proportional to the square of "n", a tiny fraction of all "n"-digit numbers. However, any number that is missing from the sequence above would have multiplicative persistence > 11; such numbers are believed not to exist, and would need to have over 200 digits if they do exist. The additive persistence of a number, however, can become arbitrarily large (proof: For a given number formula_1, the persistence of the number consisting of formula_1 repetitions of the digit 1 is 1 higher than that of formula_1). The smallest numbers of additive

neg: (ε, δ)-definition of limit "function "f" approaches the limit "L" near "a" (symbolically, formula_19) if we can make "f(x)" as close as we like to "L" by requiring that "x" be sufficiently close to, but unequal to, "a"." When we say that two things are close (such as "f(x)" and "L" or "x" and "a") we mean that the distance between them is small. When "f(x)", "L", "x", and "a" are real numbers, the distance between two numbers is the absolute value of the difference of the two. Thus, when we say "f(x)" is close to "L" we mean formula_20 is small. When we

neg: History of Mexico City large numbers still migrate from rural areas to the city in search of work and the other economic. Many of these immigrants settle illegally in the urban fringe with the hope that the government will eventually provide public services. The provision of water and wastewater service for the growing population of Mexico City is the problem air pollution was in the 1970s and 1980s. Such growth rates and patterns mean substandard potable water supplies and waste water treatment, if they exist at all. Over 70% of Mexico City's potable water from the aquifer below it, which is being overexploited, causing

neg: Baseball park the last chance a new style gets for some time." The numbers mean the number of feet from home plate to the wall of that part of the field. Left and Right Field normally refer to the distances along the foul lines. Left Center and Right Center are the approximate power alley figures. Center Field could mean straightaway center field or it could mean to the deepest part of the center field area. Backstop refers to the distance behind home plate to the backstop screen. These numbers are one researcher's opinion of the true values and may differ from the

neg: DNA microarray probes are unmasked in preparation for a different nucleotide exposure. After many repetitions, the sequences of every probe become fully constructed. More recently, Maskless Array Synthesis from NimbleGen Systems has combined flexibility with large numbers of probes. "Two-color microarrays" or "two-channel microarrays" are typically hybridized with cDNA prepared from two samples to be compared (e.g. diseased tissue versus healthy tissue) and that are labeled with two different fluorophores. Fluorescent dyes commonly used for cDNA labeling include Cy3, which has a fluorescence emission wavelength of 570 nm (corresponding to the green part of the light spectrum), and Cy5 with a fluorescence

neg: Info-metrics develop new models/theories, validate these models using all available information, and test statistical hypotheses about the model. "Inference based on information resulting from repeated independent experiments." The following example is attributed to Boltzmann and was further popularized by Jaynes. Consider a six-sided , where tossing the is the event and the distinct outcomes are the numbers 1 through 6 on the upper face of the . The experiment is the independent repetitions of tossing the same . Suppose you only observe the empirical mean value, y, of N tosses of a six-sided . Given that information, you want to infer

neg: At Close Quarters was commissioned by the Ministry of Defence in Madrid. In it, he recommended the closure of Camp San Marcial as an efficiency measure. This highlighted the growing cleavage between his own generation and style of troop which were dedicated to peace-keeping and winning wars with technology; vis-a-vis the older, Francoists, who still believed in numbers and yearned for a reinstatement of the recently abolished compulsory military service. When he gets shot there is an internal inquiry which results in a verdict of suicide. Marina, the Major's daughter, is not satisfied with this outcome and hires a private detective, Ricardo Cupido.

neg: Irish Americans Colson that "[t]he Irish have certain — for example, the Irish can't drink. What you always have to remember with the Irish is they get mean. Virtually every Irish I've known gets mean when he drinks. Particularly the real Irish." Discrimination against Irish Americans differed depending on gender. For example, Irish women were sometimes stereotyped as "reckless breeders" because some American Protestants feared high Catholic birth rates would eventually result in a Protestant minority. Many native-born Americans claimed that "their incessant childbearing [would] ensure an Irish political takeover of American cities [and that] Catholicism would become the reigning faith of

neg: Nevermore: The Final Maximum Ride Adventure her flock and tells them to stay. Only Angel hears her. She says they have to go. Max disagrees. Angel says that Max should listen to her because Max always listens to the Voice. She reveals that she is the Voice. Max says that she will go die with the humans. Fang tells Max that whatever the outcome, they'll face it together. Before Angel came Dylan also tells Max that after he gets everyone to safety, he will come back to her, even if it means that he will die with her; he tells Max that the only way he

neg: Stand Up (Ludacris song) Stand Up (Ludacris song) "Stand Up" is a song by Ludacris, released as the second official single in 2003, and taken from his fourth album, "Chicken-n-Beer". It was his first number-one single, with production by Kanye West and co-production by Ludacris himself. The chorus consists of three repetitions of Ludacris rapping, "When I move, you move" and Shawnna responding, "Just like that?" After three repetitions, Ludacris says, "Hell, yeah! Hey, DJ, bring that back." Then Shawnna says, "When I move, you move" and Ludacris says "Just like that?" and they trade parts of the chorus. The song topped the "Billboard"

neg: Empirical measure empirical processes provide rates of this convergence. Let formula_5 be a sequence of independent identically distributed random variables with values in the state space "S" with probability distribution "P". Definition Properties Definition To generalize this notion further, observe that the empirical measure formula_15 maps measurable functions formula_16 to their "empirical mean", In particular, the empirical measure of "A" is simply the empirical mean of the indicator function, "P"("A") = "P" "I". For a fixed measurable function formula_18, formula_19 is a random variable with mean formula_20 and variance formula_21. By the strong law of large numbers, "P"("A") converges to "P"("A") almost

neg: Epistemic possibility meanings. The contrast is best explained by example. Consider the two statements: Although these two statements are often confused with one another, they mean two different things: the first says something "true" about the vagaries of war; the second says something that is certainly "false". The difference comes from the fact that the first statement—a statement of "subjunctive" possibility—says something about how things "might have been" under counterfactual conditions, whereas the second—a statement of "epistemic" possibility—says something about the relation between a particular outcome (a victory by Japan) and "our knowledge about the actual world" (since, as it happens, we

neg: Mathematical coincidence or, perhaps, to encourage new mathematical learners at an elementary level. Sometimes simple rational approximations are exceptionally close to interesting irrational values. These are explainable in terms of large terms in the continued fraction representation of the irrational value, but further insight into why such improbably large terms occur is often not available. Rational approximants (convergents of continued fractions) to ratios of logs of different numbers are often invoked as well, making coincidences between the powers of those numbers. Many other coincidences are combinations of numbers that put them into the form that such rational approximants provide close relationships. Some

neg: Hsu–Robbins–Erdős theorem is also a necessary condition for formula_6 to hold. Two years later, the famed mathematician Paul Erdős proved the conjecture. Since then, many authors extended this result in several directions. Hsu–Robbins–Erdős theorem In the mathematical theory of probability, the Hsu–Robbins–Erdős theorem states that if formula_1 is a sequence of i.i.d. random variables with zero mean and finite variance and then for every formula_4. The result was proved by Pao-Lu Hsu and Herbert Robbins in 1947. This is an interesting strengthening of the classical strong law of large numbers in the direction of the Borel–Cantelli lemma. The idea of such a

neg: Japamala grain is placed in a bowl for each 108 repetitions. Each time a full mala of repetitions has been completed, one grain of rice is removed from the bowl. Often, practitioners add extra counters to their malas, usually in strings of ten. These may be positioned differently depending on the tradition; for example some traditions place these strings after every 10th bead. This is an alternative way to keep track of large numbers, sometimes going into the hundreds of thousands, and even millions. The 109th bead on a mala is called the sumeru, bindu, stupa, or guru bead. Counting should

neg: Glassy Mountain (South Carolina) Glassy Mountain (South Carolina) Glassy Mountain is a mountain close to Pickens, South Carolina, USA, in Northwestern South Carolina, with a summit elevation of 1647 feet (502 m) above mean sea level. While it is an isolated monadnock, it is still close to the Blue Ridge, providing views of nearby summits such as Table Rock and Caesar's Head as well as Greenville. The mountain gets its name from a large, bare, granite face on its North side. At the top of this rock face are small springs that trickle down the mountain, giving it a "glassy" appearance in the sun.

neg: Regression toward the mean less than 10, while the law of large numbers states that in the long term, this event will likely average out, and the average fraction of heads will tend to 1/2. By contrast, the gambler's fallacy incorrectly assumes that the coin is now "due" for a run of tails to balance out. This is the definition of regression toward the mean that closely follows Sir Francis Galton's original usage. Suppose there are "n" data points {"y", "x"}, where "i" = 1, 2, …, "n". We want to find the equation of the regression line, "i.e." the straight line which would

neg: Asymptotic equipartition property are individual outcomes which have a higher probability than any outcome in this set, the vast number of outcomes in the set almost guarantees that the outcome will come from the set. One way of intuitively understanding the property is through Cramér's large deviation theorem, which states that the probability of a large deviation from mean decays exponentially with the number of samples. Such results are studied in large deviations theory; intuitively, it is the large deviations that would violate equipartition, but these are unlikely. In the field of pseudorandom number generation, a candidate generator of undetermined quality whose output

neg: Internet censorship and surveillance by country to the internet filtering legislation proposed by Minister Stephen Conroy. Regardless, and the outcome of the 2010 election, it would be highly unlikely for the filter to pass the Senate if proposed due to the close numbers of seats held by Labor and the Coalition, who Joe Hockey says do not support it. In June 2011 two Australian ISPs, Telstra and Optus, confirmed they would voluntary block access to a list of child abuse websites provided by the Australian Communications and Media Authority and more websites on a list compiled by unnamed international organizations from mid-year. In May 2013, Senator

neg: Pithoprakta Pithoprakta Pithoprakta (1955–56) is a piece by Iannis Xenakis for string orchestra (with 46 separate solo parts), two trombones, xylophone, and wood block, premièred by conductor Hermann Scherchen in Munich on March 1957. A typical performance of the piece lasts about 10 minutes. The word "Pithoprakta" translates to "actions through probability". This relates to Jacob Bernoulli's law of large numbers which states that as the number of occurrences of a chance event increases, the more the average outcome approaches a determinate end. The piece is based on the statistical mechanics of gases, Gauss's law, or Brownian motion. Each instrument is

neg: Stumbling on Happiness the shortcomings of imagination. The Subjectivity Section addresses the meaning of happiness and emphasizes that happiness is a subjective feeling. Gilbert says, “Evaluating people’s claims about their own happiness is an exceptionally thorny business.” No perfectly reliable tool exists to measure a person’s happiness. Instead researchers must rely primarily on the “honest, real-time report of the attentive individual”. Finally, given the imperfections in self-reported feelings of happiness, scientists must rely on the law of large numbers, namely, to ask many people the same question and compare their answers. The Realism Section explains that imagination suffers from shortcomings. The first shortcoming

neg: MyLifeIsAverage MLIT." MyLifeIsAverage MyLifeIsAverage (MLIA) is a website where people may submit stories and tales of their day, about average occurrences and average people. MLIA is a spin-off from the website FMyLife. The website's purpose is to demonstrate to its readers that, like them, there are many average people in the world. MLIA seeks to show that a person being average does not necessarily mean that he or she is boring. MLIA was co-founded by UCLA students Guru Khalsa and Enrico Mills. Khalsa says that MLIA seeks to "bring to light how much stupid and boring stuff gets posted on the

neg: MyLifeIsAverage MyLifeIsAverage MyLifeIsAverage (MLIA) is a website where people may submit stories and tales of their day, about average occurrences and average people. MLIA is a spin-off from the website FMyLife. The website's purpose is to demonstrate to its readers that, like them, there are many average people in the world. MLIA seeks to show that a person being average does not necessarily mean that he or she is boring. MLIA was co-founded by UCLA students Guru Khalsa and Enrico Mills. Khalsa says that MLIA seeks to "bring to light how much stupid and boring stuff gets posted on the Internet."

neg: Weight training are brief but very intense, and many people are unaccustomed to this level of effort. The expression "no pain, no gain" refers to working through the discomfort expected from such vigorous effort, rather than to willfully ignore extreme pain, which may indicate serious soft tissue injuries. The focus must be proper form, not the amount of weight lifted. Discomfort can arise from other factors. Individuals who perform large numbers of repetitions, sets, and exercises for each muscle group may experience a burning sensation in their muscles. These individuals may also experience a swelling sensation in their muscles from increased blood

neg: Concentration inequality Concentration inequality In probability theory, concentration inequalities provide bounds on how a random variable deviates from some value (typically, its expected value). The laws of large numbers of classical probability theory state that sums of independent random variables are, under very mild conditions, close to their expectation with a large probability. Such sums are the most basic examples of random variables concentrated around their mean. Recent results show that such behavior is shared by other functions of independent random variables. Concentration inequalities can be sorted according to how much information about the random variable is needed in order to use

neg: History of the Jews in Germany that what makes Judaism unique is its divine revelation of a code of law. Mendelssohn's commitment to Judaism lead to tensions even with some of those who subscribed to Enlightenment philosophy. Faithful Christians who were less opposed to his rationalistic ideas than to his adherence to Judaism found it difficult to accept this "Juif de Berlin." In most of Western Europe, the "Haskalah" ended with large numbers of Jews assimilating. Many Jews stopped adhering to Jewish law, and the struggle for emancipation in Germany awakened some doubts about the future of Jews in Europe and eventually led to both immigrations

neg: Ideal free distribution ratio is taken into consideration. These two matching relationships are assessed by a regression of the log ratio of the numbers at each site against the log ratio of resources at the site. The results they found do not support IFD predictions and some take this outcome to mean that the current model is too simple. Animal behaviorists have proposed a modification to the model that denotes an ultimate outcome of a population always having more individuals on the least profitable site and less on the more resource abundant site. Knowledge of the competitive interactions, effects of travel between sites,

neg: Disney Junior – Live on Stage! everyone wants to do, after which Lambie suggests that everyone should dance with her; they start to "boogie." After the dance, Stuffy gets out of control and knocks over Lambie, causing her tutu to rip. Lambie then says she knows Stuffy didn't mean to rip her tutu, and she asks Doc to fix her. The whole group then sing "Time for Your Check Up," before Doc says that she is diagnosing Lambie with the "rippies." Hallie then says they need to look in the Big Book of Boo Boos. Doc agrees, but also says that she needs her magnifying glass.

neg: Prime number for numbers of special forms, such as Mersenne numbers. the largest known prime number has 23,249,425 decimal digits. There are infinitely many primes, as demonstrated by Euclid around 300 BC. No known simple formula separates prime numbers from composite numbers. However, the distribution of primes within the natural numbers in the large can be statistically modelled. The first result in that direction is the prime number theorem, proven at the end of the 19th century, which says that the probability of a randomly chosen number being prime is inversely proportional to its number of digits, that is, to its logarithm.

neg: Volkswagen Karmann Ghia Segre and Virgil Exner became close professionally and personally, eventually traveling Europe together. Peter Grist wrote in his 2007 Exner biography that when Exner in 1955 eventually saw the Karmann Ghia, which cribbed heavily from his Chrysler d'Elegance, "he was pleased with the outcome and glad that one of his designs had made it into large-scale production.” Chris Voss, a stylist in Exner's office, reported in 1993, that Exner considered the Karmann Ghia the ultimate form of flattery. Segre in turn sent Exner the first production Karmann Ghia imported into the state of Michigan, in gratitude. After Volkswagen approved the

neg: Riemann series theorem Riemann series theorem In mathematics, the Riemann series theorem (also called the Riemann rearrangement theorem), named after 19th-century German mathematician Bernhard Riemann, says that if an infinite series of real numbers is conditionally convergent, then its terms can be arranged in a permutation so that the new series converges to an arbitrary real number, or diverges. As an example, the series 1 – 1 + 1/2 – 1/2 + 1/3 – 1/3 + ... converges to 0 (for a sufficiently large number of terms, the partial sum gets arbitrarily near to 0); but replacing all terms with their absolute values

neg: Riemann series theorem sums of converging rearranged series is an affine subspace of "E". Riemann series theorem In mathematics, the Riemann series theorem (also called the Riemann rearrangement theorem), named after 19th-century German mathematician Bernhard Riemann, says that if an infinite series of real numbers is conditionally convergent, then its terms can be arranged in a permutation so that the new series converges to an arbitrary real number, or diverges. As an example, the series 1 – 1 + 1/2 – 1/2 + 1/3 – 1/3 + ... converges to 0 (for a sufficiently large number of terms, the partial sum gets arbitrarily

neg: Monte Carlo integration deterministic approach. Monte Carlo integration, on the other hand, employs a non-deterministic approach: each realization provides a different outcome. In Monte Carlo, the final outcome is an approximation of the correct value with respective error bars, and the correct value is likely to be within those error bars. The problem Monte Carlo integration addresses is the computation of a multidimensional definite integral where Ω, a subset of R, has volume The naive Monte Carlo approach is to sample points uniformly on Ω: given "N" uniform samples, "I" can be approximated by This is because the law of large numbers ensures

neg: Hiro Nakamura considers another time-travel attempt, the possibility of being eaten by a dinosaur dissuades him. Hiro tries to warn Charlie and protect her from the "very bad man", but ends up forming a close relationship with her after his attempts to convince her fail. Eventually, he asks her to come with him to Japan. As the date of their departure gets closer, Charlie reveals that she has a blood clot in her brain and will die no matter what. She says she was going to give up before Hiro came and that she has fallen in love with him. Just as

neg: Bank run time; that is, by the law of large numbers, banks can expect only a small percentage of accounts withdrawn on any one day because individual expenditure needs are largely uncorrelated. A bank can make loans over a long horizon, while keeping only relatively small amounts of cash on hand to pay any depositors who may demand withdrawals. However, if many depositors withdraw all at once, the bank itself (as opposed to individual investors) may run short of liquidity, and depositors will rush to withdraw their money, forcing the bank to liquidate many of its assets at a loss, and eventually

neg: Price's Raid Boonville, Price offered Union General Rosecrans time he might not otherwise have had to organize an effective response. Furthermore, he says, Price's insistence on guarding an ever-expanding wagon train of looted military supplies and other items ultimately became "an albatross to [his] withdrawal". Price, wrote Davis, "ought" to have used Confederate bushwhackers to harass Federal formations, forcing the Unionists to disperse large numbers of troops to pursue them over wide ranges of territory. This in turn would have reduced the number of effectives available to fight against Price's main force. Instead, Price kept many guerrillas close to his army and

neg: Arbitrarily large Arbitrarily large In mathematics, the phrases arbitrarily large, arbitrarily small, and arbitrarily long are used in statements such as: which is shorthand for: "Arbitrarily large" is not equivalent to "sufficiently large". For instance, while it is true that prime numbers can be arbitrarily large since there are an infinite number of them, it is not true that all sufficiently large numbers are prime. "Arbitrarily large" does not mean "infinitely large" because although prime numbers can be arbitrarily large, an infinitely large prime does not exist since all prime numbers (as well as all other integers) are finite. In some cases,

neg: Empirical distribution function formula_6 The mean of the empirical distribution is an unbiased estimator of the mean of the population distribution. The variance of the empirical distribution times formula_7 is an unbiased estimator of the variance of the population distribution. Since the ratio approaches 1 as goes to infinity, the asymptotic properties of the two definitions that are given above are the same. By the strong law of large numbers, the estimator formula_8 converges to as almost surely, for every value of : thus the estimator formula_8 is consistent. This expression asserts the pointwise convergence of the empirical distribution function to the true

neg: Peter Cammarano Hoboken's coordinator of the John Kerry presidential campaign, 2004, and in 2006 was the New Jersey legal director in Senator Robert Menendez's campaign. In a 2005 run-off election, he was elected Councilman-at-Large in Hoboken. At the time he was an associate attorney at Genova, Burns & Vernoia, a top election law firm. On June 9, 2009, Cammarano won the Hoboken Mayor's race in a runoff election, beating Dawn Zimmer by 161 votes. Observers credited his victory to absentee and provisional ballots in combination with hiring many residents from districts that eventually voted in large numbers for him. Zimmer's three running

neg: Objects in the Rear View Mirror May Appear Closer Than They Are tragedies in his life yet still feels haunted by their memory." BBC.co.uk also says that the song about a "melancholy middle-aged man reminiscing about his youth... is in many ways uncomfortably close to home, dealing as it does with episodes uncannily similar to events in his own life." Steinman says that it was "the hardest song to write and get across." It's a very passionate song. It's really, I think maybe, the most passionate one on the record. I mean, I'm really proud of it because that's really one that goes over-the-top in the sense that it's got images -

neg: Hamburg Historic District (Davenport, Iowa) 1850, that number rose to close to 3,000, or 20% of the city's population. German immigration remained strong through the 1880s. The Iowa census of 1890 showed that a quarter of Scott County's residents were natives of Germany. A disproportionate number of those immigrants came from Schleswig-Holstein, which was in a border and personal rights dispute with Denmark in the 1840s. German “free thinkers” from the two states were leaving in large numbers and many came to Davenport. Their political and philosophical thinking tended to be anti-clerical and secular. It would eventually dominate Davenport politics and set the city apart

neg: Central limit theorem Theorem. Let a martingale satisfy then converges in distribution to as . "Caution:" The restricted expectation should not be confused with the conditional expectation . The central limit theorem has a simple proof using characteristic functions. It is similar to the proof of the (weak) law of large numbers. Assume } are independent and identically distributed random variables, each with mean and finite variance . The sum has mean and variance . Consider the random variable where in the last step we defined the new random variables , each with zero mean and unit variance (). The characteristic function of

neg: Expected value the squares of the residuals (the sum of the squared differences between the observations and the estimate). The law of large numbers demonstrates (under fairly mild conditions) that, as the size of the sample gets larger, the variance of this estimate gets smaller. This property is often exploited in a wide variety of applications, including general problems of statistical estimation and machine learning, to estimate (probabilistic) quantities of interest via Monte Carlo methods, since most quantities of interest can be written in terms of expectation, e.g. formula_306, where formula_307 is the indicator function of the set formula_308. Expected values can

neg: Integer overflow for very large input data sets, which are less likely to be used in validation tests. Taking the arithmetic mean of two numbers by adding them and dividing by two, as done in many search algorithms, causes error if the sum (although not the resulting mean) is too large to be represented and hence overflows. An unhandled arithmetic overflow in the engine steering software was the primary cause of the crash of the 1996 maiden flight of the Ariane 5 rocket. The software had been considered bug-free since it had been used in many previous flights, but those used smaller

neg: UK Immigration Service immigration law so controversial. The practicalities of assessing large numbers of people arriving at ports mean that a great deal of discretion is given to the immigration officer, (although less now than was once the case). While the officer must obtain the approval of a Chief Immigration Officer or Inspector to refuse someone entry the way in which interviews are conducted and recorded are based on a lower level of proof than those in other areas of law enforcement. The immigration rules for visitors say that the immigration officer must be "satisfied" that the person is a genuine visitor not

neg: Chebyshev's inequality it can be applied to any probability distribution in which the mean and variance are defined. For example, it can be used to prove the weak law of large numbers. In practical usage, in contrast to the 68–95–99.7 rule, which applies to normal distributions, Chebyshev's inequality is weaker, stating that a minimum of just 75% of values must lie within two standard deviations of the mean and 89% within three standard deviations. The term "Chebyshev's inequality" also refer to Markov's inequality, especially in the context of analysis. They are closely related, and some authors refer to Markov's inequality as "Chebyshev's