Talk:Law of large numbers

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will be ignoring the "bad roll". What else would you expect them to do? If the squirrel eats the die, then maybe they would count it as "rolling a 10 trillion"?? No! That's silly, no one would ever think to do that. The only sensible thing to do is to re-roll. If they did not have a spare die, they would say "I was not able to successfully roll the die, go ask someone else." People understand that "rolling a die" is not complete until you have an answer which is either 1,2,3,4,5,6.

1495:"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 LLN as stated might as well be Bernoulli's original LLN from 1713. There seems to be no reason to attribute *that* statement to Borel. Compare e.g. 1220:). They mention that "C is uniformly learnable if and only if the VC dimension of C is finite," where "a learning function for C is a function that, given a large enough randomly drawn sample of any target concept in C, returns a region in E (a hypothesis) that is with high probability a good approximation to the target concept." My understanding of "concept" here is function. Perhaps my understanding of "concept" is wrong or the LLN has limitations. 263: 1302:

the variance would need to exist): in fact your source seems to be using this in its proof. In this case your source is stating a sufficient condition for the law to hold ... it can and does hold under weaker conditions. The "law" is the description of the behaviour of the mean, not really any one statement of conditions under which it can be said/shown to hold. However, the article could do with better citations.

1076:(a run of heads is compensated by a run of tails, so difference between heads and tails approaches 0, which is false). The way it's presented here now is gambler's fallacy. And gambler's fallacy seems somewhat more common online, but I didn't look enough to be sure. Therefore my initial impression is that this page should be a disambiguation or short article sending interested readers to learn more at either 1982:"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 success, and the average of n such variables (assuming they are independent and identically distributed (i.i.d.)) is precisely the relative frequency." 1486:"Convergence in probability is also called weak convergence of random variables". I don't think this is standard or fortunate. Convergence in distribution is already called weak convergence. The MSC (Mathematics Subject Classification) category 60F05 is "Central limit and other weak theorems", meaning theorems with convergence in distribution, not convergence in probability (as far as I know). 22: 846:. A sequence of sample means won't converge, because the average of n samples drawn from the Cauchy distribution has *exactly* the same distribution as the samples. I think the article definitely needs a section about this misconception with examples and a neat graph of diverging sequence of averages, but as you might see, my English is too bad for writing it myself. -- 1518:(eg tossing a coin or betting on roulette) tends to give a pattern over a large number of incidences? Why is it that we can anticipate (roughly) what the average will be, rather than its being completely random and not able to be anticipated? Surely there's a place for that issue in the article, if there is some literature on it. Thanks. 372:"The strong law implies the weak law but not vice versa, when the strong law conditions hold the variable converges both strongly (almost surely) and weakly (in probability). However the weak law may hold in conditions where the strong law does not hold and then the convergence is only weak (in probability)." 2121:

and i think it would be good, to underline the fact that only the average converges and not the sum of the outcome minus the theoretical outcome, to add a paragraph about this, and also a diagram showing a dice throwing experiment (with excel, with the number of throws on the abscissa, and the sum of

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There are two requirements for the term "black swan events" to be technically applicable: The events are (1) rare, and (2) sufficiently consequential to affect long-term average behavior. For example, the performance of a stockmarket investor, even averaged over 15 years, may be significantly altered

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First of all, you removed information without explaining in an edit summary -- twice. That makes the edits subject to revert. Secondly, it might help the rest of us who can't read your mind to explain what you are referring to with your comment "lead to Nasim Taleb's vanity page". And finally, give a

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Why on earth does an article on the law of large numbers lead to Nasim Taleb's vanity page? I have also deleted the rest of the sentence, which was unencyclopedic, and unnecessary. If you disagree, could you please show a reference from the serious lln literature that mentions lightning or references

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OK, I'll be up front and admit that the math on this page is beyond me. I looked up the Law of Large Numbers to try and find out why it happens. (I mean why it happens, not how it happens.) So can someone explain in plain (or even complicated) English why something that is random each time you do it

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Looking through the references currently given for the uniform law of large numbers, I notice a technical issue in the current statement. One of the references (Newey & McFadden 1994) gives a formulation of the uniform LLN which allows for the function f to be continuous almost everywhere, but

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There is a proof of the Strong Law of Large Numbers that is accessible to students with an undergraduate study of measure theory, its established by applying the dominated convergence theorem to the limit of indicator functions, and then using the Weak Law of Large Numbers on the resulting limit of

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The section on the strong law gets excessively wordy describing exactly what it means to be strong (in an unclear way, since a theorem can be strong when the hypothesis is weaker, (so that it implies the weak one and applies to more cases) or when both the hypothesis and conclusion are stronger, as

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I don't know if this page is active, but it's a classic mistake to think that since by increasing the number of trials, the empirical average gets closer to its theoretical value, it's the same for the sum of the outcome, that would get closer to the sum of the average outcome, while in fact, it's

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allowing a set of measure 0 on which discontinuities may occur), but gives the stronger mode of a.s. convergence. Is there an obvious synthesis of these two statements which yields the hybrid given in the article (a.e. continuous function f as well as a.s. convergence) or is there a reference out

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There are citations separately (slightly later) for the weak and strong forms of the law and when they hold. The article is quite specific about what these laws mean, but it may be that your source says the "law" means something else, such as the variance of the mean descreasing to zero (for which

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Generally I agree. But in this instance I don't. We are talking about convergence. The sample average converges towards the mean, this should simply be interpreted as the distance between the two grows smaller as the number in the sample increases and tend to infinity. That we are dealing with two

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Let's say we do coin tosses and let's say we assume P(head) = 0.1 and P(tail) = 0.9 as probabilities. That's a legitimate probability function according to Kolmogorov. But now the LLN becomes obviously false. So there must be some premise of LLN that forbids this constellations. Which one is it?

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integral over and came out with an answer that was very close to 9. When I switched back to the presented function I did not get answers consistent with the article. Concerned that I was doing something wrong in SAS, I also carried out the same process in R and got the same answers I got in SAS.

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This whole section about "Lebesgue integrable" random variables makes *no mathematical sense whatsoever*. The person who wrote this clearly does not know the slightest thing about math. You cannot just take any random variable and start "integrating" it with respect to the Lebesgue measure. First

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Go find a random person on the street and ask him to roll five dice and calculate the average. Suppose that one of the five dice lands on a corner, or is eaten by a squirrel. The person will naturally and immediately pick up a die and re-roll it to get the fifth number to average. In effect, they

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Stable relative frequencies in the real world are discovered empirically and are not conclusions from any mathematical theorem. Ascribing "P()'s" to events with a frequency interpretation in mind is the same as already assuming the relative frequencies of those events converge in the limit of an

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This confuses conclusions from the mathematical theorem proven from Kolmogorov's axioms (of which there is very little for the axioms are very weak and do not provide a definition or constraints strong enough for a meaningful interpretation of probability), from its intuitive interpretation that

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By the way, I was motivated to make one myself, so I just put in the animated gif of red and blue balls. I think my animated one and your non-animated one are complementary and both should be in the article...different readers may respond better to one or the other. Let me know if you have any

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The section on application that uses Monte Carlo simulation and the Law of Large Numbers to approximate an integral may be erroneous. I wrote a program in SAS to carry out this approximation and was getting strange results. I changed the function to x^2 and used my program to approximate this

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It is true that it is simply an explanation of convergence in probability in words. However, this may be very insightful to those who are not well versed in probability theory or even mathematical formalism. The section that includes this paragraph would be substantially poorer without this

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The six numbers on a die are interchangeable with any other set of symbols - is an integer mean relevant? My first guess is that the result would be 3, if I'm looking at random integers , rather than 7/2 which seems like part of a different concept, or an artifact of the way dice are labeled

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This section also gets wordy on another count, as it appears some are arguing as to whether the strong and weak forms are possibly equivalent. There are examples of probability distributions for which the weak law applies, but not the strong law. As such, I suggest the following be removed:

1150:, for which we already have a fairly developed article; the second example should be dubbed “idiot’s fallacy” or something like that — really, is there a person who would think that out of 99 coin tosses, exactly 49.5 of them should be heads?; the third example is just a corollary from the 2125:

I don't do the modification myself, because I'm not too used to the wikipedia codes (and I already got a slap on the wrist for doing that, but, as I have the impression that this page doesn't seem to be very active, if nobody reacts after a month or so, I'll do the modification myself.

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usage is more common and the focus of the article.) The difficulty in preventing students from conflating the so-called-"law" of averages/Gambler's fallacy with the law of large numbers is an extremely common problem for introductory probability and statistics instructors. I agree that

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Interpreting this result, the weak law essentially 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 the

1489:"Differences between the weak law and the strong law". It may be interesting to add here that the Weak Law may hold even if the expected value does not exist (see e.g. Feller's book). This underlines that, in their full generality, none of the laws follows directly from the other. 1492:"Uniform law of large numbers". The uniform LLN holds under quite weaker hypotheses. This is definitely uninteresting to the average reader, but a reference to the Blum-DeHardt LLN or the Glivenko-Cantelli problem might be very valuable to a small fraction of readers. 837:

From reading this article many can get the wrong impression that a sequence of averages almost surely converges, and converges to the expected value. But in reality the law of large numbers only works when expected value of the distribution exists, and there are many

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the results on the ordinate, so that we can see that the curve of the results doesn't converge towards the theoretical curve), and next to it the curve of the empirical average (of the same experiment) which we would see converging towards the theoretical line

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by the amount he lost in a single hour during a market crash. So that's a black swan event. When you are rolling dice, a black swan event is impossible because the distribution of possible numerical results is so restricted: 1,2,3,4,5,6. It is never 10 billion!

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such as a die landing on edge or being struck by lightning mid-roll are not possible or ignored if they do occur" is "unencyclopedic and unnecessary"; it is linked to a page with well sourced explanations as to why it is enyclopedic. Additionally, if you think

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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

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and with source code available. It also looks a little different and has different data (new data may be generated by anyone with the inclination using my provided source code (or their own)). I would like to propose that we switch to my image,

342:→I don't understand your message, but why would the LLN "become obviously false"? if you toss your coin an high number of times, the number of tail divided by the number of toss may lean toward 0.9, i don't understand your problem with that? 1318:

The Cauchy distribution is a bad example, it does not have a mean, and hence no finite variance or higher moments. And the proof using characteristic functions does not seem to use the assumption of finite variance. I'll add a citetation.

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Yes; if the VC dimension is infinite the learning function (with high probability) still converges pointwise to the correct characteristic function. But the convergence won't be uniform. Unfortunately, this is hardly a "quick" answer....

1988:"According to the law of large numbers, if a large number of six-sided die 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." 2112:

add a category to recall the fact that despite the deviation to the mean decreases with increasing numbers, the standard deviation (so the raw deviation between theoretical output and the actual one) increases when the number of trial

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forms of convergence and that thier exact definitions are different from oneanother is not of interest to the non-mathematician. And if the reader is interested in the exact difference between the two then there's an article about

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this article (meaning Law of Averages), with redirect to LLN. The cited source uses the term “law of averages” as a synonym for LLN, and does not provide the interpretation given in this article. The examples section looks like

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I believe my programs are producing the correct answers, which is why I am concerned that the example on the LLN page may be erroneous. Can someone please verify my calculations or tell me where the error is in my code?

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The problem with this is to make it clear exactly what "Borel's law of large numbers" is in the context of the larger article, since presumably Borel's law of large numbers is notable enough to br mention specifically.

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all the relative frequencies involved in the given reasoning are stable in the first place, the difference from a finite number of trails between the measured and "ideal" mean is likely to be less than so and so.

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of observations are 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

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to clarify that it was the margin to which the paragraph referred. I did so because, despite knowing the LLN and its mathematical formulation, it wasn't immediately clear to me what margin was being discussed.

2826:* Set up function for integration manually *; /* fi = xi**2; Simple Example for process checking */ fi = cos(xi)*cos(xi)*sqrt((xi*xi*xi)+1); /* Knowledge Example 1 */ 2683: 2372: 1413:

There is a universal common-sense intuitive understanding of what it means to "roll a die". According to that understanding, black swan events are irrelevant and the simple sentence is completely correct.

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It looks to me very much like the two articles are covering the same ground, so yes, a merger makes sense to me. I can't see any material in that article that needs to be copied into this one. —

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I agree, that the statement is wrong. However your explanation makes no sense to me. The strong law uses almost sure pointwise convergence. The wrong statement corresponds to uniform convergence.

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I'm looking for a quick answer, trying to resolve a certain issue. Does the LLN hold even when we're collecting samples from and for a model/function that has infinite VC dimension?

1158:; the last example is not even funny — people don't think that in the longrun a good team and a bad team would perform equally, that would contradict the mere notion of “skill”. 1878:

that the empirical mean deviates by 1/2 from the theoretical expectation ; the strong LLN only states that for these bad initial n tosses the error will a.s. be corrected later.

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Is false, the difference in absolute value will be unbounded, but it will also be 0 an infinite amount of times. The lim inf will be zero, and the lim sup will be infinity.

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there which gives the stronger statement? If not, it may be worth revising the statement to more accurately reflect the references. Gillespie 22:17, 21 March 2021 (UTC)

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Whitt, Ward (2002) Stochastic-Process Limits, An Introduction to Stochastic-Process Limits and their Application to Queues, Chapter 1: Experiencing Statistical Regularity

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provides only for uniform convergence in probability. The other (Jennrich 1969) gives a formulation which allows only for the function f to be continuous everywhere (

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I removed the annoying references/citations tag and added a few references. Should there be more citations? Should I have left the tag where it was? I dont think so.

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study math (and measure theory) before you come to Knowledge to "educate" people with your wisdom that is nothing more than pure ignorance and stupidity.

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I think any time you can add text interpretation to a math article it is hugely helpful, even if those who already know it all find it just extra words.

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as well as the clarification needed prior, and the citation needed after. Is the StackExchange conversation a sufficient reference to make such an edit?

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that might be of use? I added a {{mergeto}} tag to that article. If there is nothing worthwhile then perhaps simply replace it with a redirect?

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not the case (and on the contrary, the standard deviation increases), and this value converges only if we divide it by the number of trials.

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This is simply explaining in words what convergence in probability is. I don't consider it useful. I'll remove it shortly if no-one objects.

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The LLN is important because it guarantees stable long-term results for the averages of some random events. For example, while a

92: 69: 2090:, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section. 2919: 2410:

I hope that someone knolwedgeable about this topic can rewrite the introduction so that it is comprehensible to most readers.

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http://math.stackexchange.com/questions/777493/do-the-kolmogorovs-axioms-permit-speaking-of-frequencies-of-occurence-in-any-me

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The two ideas are not the same, and there's already a cross-reference in both articles' "See also" sections. I don't think an

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requires additional assumptions, equivalent to assuming the law itself true a priori. See a more elaborate explanation here:

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So again, black swan events should certainly not be mentioned in the context of rolling dice. It is an irrelevant tangent. --

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as an example where the variance is not finite, and the Law of Large Numbers does not hold (in the section 'Cauchy case').

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detailed explanation as to why you think the sentence "This assumes that all possible die roll outcomes are known and that

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and linked to from here (and perhaps any other place that uses the terms). I'll be adding a talk over there about that.

44: 2859:%mcintegral(-1,2,1000); *** Should be about 1 when using f(x)=cos^2(x)*sqrt(x^3 + 1) if Knowledge page is correct ***; 1151: 2823:

do i = 1 to &n; xi = &a+%sysevalf(&b-&a)*ranuni(0); * Generate U random values *;

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infinite number of trails to a definite number, this "P()". The only thing the theorem allows to conclude, is that

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should be part of a collection of non-mathematically-formal articles partly related to gambling, as in mention of

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To date it has not been possible to prove that the strong law conditions are the same as those of the weak law.

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that describes the result of performing the same experiment a large number of times. According to the law, the

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probabilities. Would this be appropriate for inclusion with the group of articles on the Laws of Large Numbers?

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And it is repetitive. Convergence in probability is stated in words twice. It would be neater if it just said

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other formulas that look similar are not verified, such as the raw deviation from "theoretical results":

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other formulas that look similar are not verified, such as the raw deviation from "theoretical results":

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required, without citation or justification. Every single other source I saw said that finite variance

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despite it being pointed out that discussion is here, so I have copied the immediately above to here.

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condition E(|X|)<inf is same as that the random variables x has Lebesgue integrateable expectation

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depends on the point in the set of convergence (the non-uniformity alluded to in the second comment).

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Disagree.... Looking through "google books" and the web, the term "Law of averages" refers to either

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x <- a + (b-a)*runif(n) ## iid Uniform Xs on f <- f_x(x) ## Apply the function

1030:. (Although some texts may use the term "law of averages" to refer to the law of large numbers, the 631:

I've read a bit on the subject and it would seem that's it's an umbrella term. I invite you to read

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These two comments seem correct to me : the mistake in the article stems from the fact that, given

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on Knowledge. If you would like to participate, please visit the project page, where you can join

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on Knowledge. If you would like to participate, please visit the project page, where you can join

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This seems like a fairly serious mistake to me so I'll immediately remove the wrong statement.

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What if we would use different legitimate integration methods for expectation definition as:

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Then we will shurely have random variables with finite expectation where L.L.N do not hold.

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It is also important to note that the LLN only applies to the average. Therefore, while

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It is also important to note that the LLN only applies to the average. Therefore, while

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Yao, Kai; Gao, Jinwu (2016). "Law of Large Numbers for Uncertain Random Variables".

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Preferably by someone who understands what writing an encyclopedia article entails.

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is here.) I think it would be better to rewrite most of that verbage, specifically:

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Last edited at 11:28, 29 July 2007 (UTC). Substituted at 20:03, 1 May 2016 (UTC)

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In my opinion, many statements expressed on this page are not correct, such as:

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and tends to become closer to the expected value as more trials are performed.

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I agree with deleting the sentence. It makes a simple thing sound complicated.

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I suggest instead a general explanation of strong vs weak theorems be made in

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idea to use a wholly unexplained symbol in the introductory paragraph ... or

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of the results obtained from a large number of trials should be close to the

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http://www.encyclopediaofmath.org/index.php/Borel_strong_law_of_large_numbers

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instead (maybe with a disambiguation note at the top for users arriving via

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set mc1; sum + fi; n = _n_; mean = %sysevalf(&b - &a)*sum/n;

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For example, for the flip of a fair coin, for any n there is a probability

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I have created and uploaded an image similar to the current image, but in

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BirthdaBirthday date problem issue sir Birthday date problem issue sir

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Agreed. In fact the other article should probably simply be removed.

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even specifically says that finite variance is needed, and uses the

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BUT ALSO: The introduction states the law of large numbers as an

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and anybody who didn't know the term could go there and learn.

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means ### Should return the AUC for f(x) from a to b ###

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http://www.math.vanderbilt.edu/~schectex/ccc/gauge/venn.gif

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which don't have an expected value. Take, for example, the

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means <- rep(0,n) ## Initialize vector of means

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f_x <- function(s){cos(s)*cos(s)*sqrt((s*s*s) + 1)}

1821:{\displaystyle |{\overline {X}}_{n}-\mu |<\varepsilon } 1597:{\displaystyle |{\overline {X}}_{n}-\mu |<\varepsilon } 1072:(ratio of heads to tails approaches 1, which is true), or 510:

correction of misdirected merger proposal from March2008.

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increases, but it tends to increase in absolute value as

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increases, but it tends to increase in absolute value as

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The entire introductory section is currently as follows:

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needs to be merged, but I think it should be merged into

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increases, but it tends to increase in absolute value as

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Difference between the Strong and the Weak Law - mistake

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get merged into this article and made a redirect page?

2624: 2532: 2313: 2221: 1843: 1778: 1750: 1627: 1554: 1185:: As above, the discussion was also occurring on the 974:. Please state your comments regarding this action. 201:, a collaborative effort to improve the coverage of 96:, a collaborative effort to improve the coverage of 1216:

I was reading some papers on statistical learning (

609:I think that there may have been an intention that 2677: 2602: 2366: 2291: 1870: 1820: 1762: 1680:. Indeed, with non-zero probability all the first 1670: 1596: 2862:Observed answer using 1000 random values: 1.5905 2534: 2517:The introductory section contains this passage: 2223: 1382:is unencyclopedic you need to make your case at 2940:Knowledge level-4 vital articles in Mathematics 2868:a <- -1 b <- 2 n <- 10000 2086:, and are posted here for posterity. Following 1924:which is much more general than Lebesgue one? 1671:{\displaystyle |{\overline {X}}_{n}-\mu |: --> 2689:not only does it not converge toward zero as 2378:not only does it not converge toward zero as 2080:The comment(s) below were originally left at 8: 2738:not only does it not converge toward zero as 2043:Possible issue with statement of uniform LLN 2806:Incorrect Example for "Application" section 1708:harvtxt error: no target: CITEREFRoss2009 ( 1109:, so I have copied this earlier comment by 2424:; it is an equality with probability one. 2127: 861:A Proof of the Strong Law of Large Numbers 392: 343: 147: 58: 2665: 2650: 2640: 2629: 2623: 2590: 2576: 2570: 2564: 2553: 2537: 2531: 2354: 2339: 2329: 2318: 2312: 2279: 2265: 2259: 2253: 2242: 2226: 2220: 1856: 1847: 1842: 1807: 1795: 1785: 1779: 1777: 1749: 1656: 1644: 1634: 1628: 1626: 1583: 1571: 1561: 1555: 1553: 1263:The article says that finite variance is 718:explanation. I suggest leaving it in. 635:. The first chapter is available online 2930:Knowledge vital articles in Mathematics 2767:2409:40C4:2011:3634:D491:55EC:936D:EF6B 2457:was invoked but never defined (see the 2443: 2191:may lose money in a single spin of the 1697: 1218:http://dl.acm.org/citation.cfm?id=76371 421: 149: 60: 19: 2945:C-Class vital articles in Mathematics 2898:2600:381:D080:1A8D:B10A:CE9B:93A:5DAB 2717:2601:200:C000:1A0:3938:3645:9394:290D 2426:2601:200:C000:1A0:808C:2579:CA92:A870 2134:2A01:CB11:88F:A800:6CDA:5A2C:D4F2:CDC 1620:there is some small probability that 350:2A01:CB11:88F:A800:6CDA:5A2C:D4F2:CDC 7: 2096:I'd love a proof of the strong law. 1704: 1271:in fact needed. This writing titled 195:This article is within the scope of 90:This article is within the scope of 2753:2A01:11:8A10:8690:9A7:D52C:C665:FDF 2734:One of the introductory sentences: 2449: 833:What if there is no expected value? 49:It is of interest to the following 2970:High-priority mathematics articles 2955:Top-importance Statistics articles 2544: 2475:IEEE Transactions on Fuzzy Systems 2233: 2083:Talk:Law of large numbers/Comments 1961:2A02:A466:58DC:1:D9:BECA:6E31:A8D7 14: 2088:several discussions in past years 1350:Otherwise it's not appropriate. 534:Is there any information over at 215:Knowledge:WikiProject Mathematics 2925:Knowledge level-4 vital articles 1763:{\displaystyle \varepsilon : --> 1084:. What do other people think? -- 261: 218:Template:WikiProject Mathematics 182: 172: 151: 110:Knowledge:WikiProject Statistics 83: 62: 29: 20: 2960:WikiProject Statistics articles 997:Makes no sense on the average. 942:comments or suggestions! :-) -- 753:convergence of random variables 235:This article has been rated as 130:This article has been rated as 113:Template:WikiProject Statistics 2935:C-Class level-4 vital articles 2780:Huh? Are you referring to the 2541: 2230: 1808: 1780: 1657: 1629: 1584: 1556: 1146:. The first one describes the 1: 2850:where n gt %eval(&n-10); 2761:23:31, 14 February 2023 (UTC) 1466:03:30, 1 September 2012 (UTC) 1452:01:13, 1 September 2012 (UTC) 1024:article describing a lay term 962:It has been suggested at the 952:08:54, 27 February 2010 (UTC) 856:12:53, 21 November 2009 (UTC) 411:22:49, 23 November 2016 (UTC) 337:19:47, 20 February 2021 (UTC) 209:and see a list of open tasks. 104:and see a list of open tasks. 2965:C-Class mathematics articles 2892:Returns a value of 1.611217 2670: 2595: 2359: 2284: 2142:11:21, 30 October 2021 (UTC) 2023:The first example is awkward 1951:20:23, 2 February 2014 (UTC) 1790: 1639: 1566: 1528:20:36, 25 January 2013 (UTC) 1342:Wiki allows advertising now? 1236:04:02, 18 October 2011 (UTC) 1166: 1163: 1159: 1016:article describing a theorem 935:06:39, 3 February 2010 (UTC) 920:03:11, 3 February 2010 (UTC) 876:21:17, 20 January 2010 (UTC) 591:I second that. Just replace 471:16:04, 10 January 2008 (UTC) 459:Borel's law of large numbers 358:11:32, 30 October 2021 (UTC) 2950:C-Class Statistics articles 2817:%macro mcintegral (a,b,n); 2725:01:52, 24 August 2022 (UTC) 1604:holds for all large enough 1508:02:30, 2 January 2013 (UTC) 1396:19:42, 31 August 2012 (UTC) 1366:19:29, 31 August 2012 (UTC) 1253:17:48, 29 August 2014 (UTC) 1209:When the LLN does not hold? 1156:second Borel-Cantelli lemma 1152:strong law of large numbers 808:05:22, 20 August 2021 (UTC) 771:Convergence in probability| 2986: 2487:10.1109/TFUZZ.2015.2466080 2038:03:34, 19 March 2016 (UTC) 1897:09:07, 25 April 2022 (UTC) 1125:21:22, 26 March 2010 (UTC) 1063:20:45, 26 March 2010 (UTC) 1026:and its common usage in a 1007:21:40, 12 March 2010 (UTC) 972:Law of large numbers (LLN) 964:Knowledge:Proposed mergers 840:heavy-tailed distributions 2906:16:38, 30 July 2024 (UTC) 2800:21:02, 29 July 2024 (UTC) 2775:13:54, 29 July 2024 (UTC) 2434:15:48, 15 July 2022 (UTC) 2101:11:28, 29 July 2007 (UTC) 2095: 1871:{\displaystyle 1/2^{n-1}} 1736:19:54, 6 March 2014 (UTC) 1335:08:25, 30 June 2012 (UTC) 1187:Law of averages talk page 1107:Law of averages talk page 1094:22:53, 3 March 2010 (UTC) 1022:should be merged with an 986:20:58, 3 March 2010 (UTC) 828:19:44, 24 July 2009 (UTC) 783:08:55, 25 July 2009 (UTC) 765:08:50, 25 July 2009 (UTC) 746:22:01, 24 July 2009 (UTC) 728:07:34, 24 July 2008 (UTC) 712:22:17, 23 July 2008 (UTC) 676:22:20, 16 June 2008 (UTC) 648:22:14, 16 June 2008 (UTC) 627:09:47, 16 June 2008 (UTC) 605:13:18, 12 June 2008 (UTC) 530:07:44, 21 May 2008 (UTC) 524:Talk:Law of Large Numbers 496:09:43, 16 June 2008 (UTC) 434:Mathematics StackExchange 234: 167: 129: 78: 57: 1312:12:44, 29 May 2012 (UTC) 1295:10:23, 29 May 2012 (UTC) 1199:09:41, 4 June 2010 (UTC) 1174:22:18, 1 June 2010 (UTC) 691:Consider the paragraph: 595:with a redirect to LLN. 586:15:28, 21 May 2008 (UTC) 570:07:44, 21 May 2008 (UTC) 520:13:11, 12 May 2008 (UTC) 480:07:47, 21 May 2008 (UTC) 241:project's priority scale 2883:means = (b-a)*sum(f)/i 2392:It is an astonishingly 2018:06:56, 2 May 2014 (UTC) 1975:Methodological mistakes 1969:13:41, 8 May 2022 (UTC) 1479:A few technical remarks 1047:who intend to find the 198:WikiProject Mathematics 2920:C-Class vital articles 2679: 2645: 2604: 2569: 2403:The introduction uses 2368: 2334: 2293: 2258: 1872: 1822: 1765: 1678:=\varepsilon }" /: --> 1673: 1598: 1384:Talk:Black swan theory 654:Statistical regularity 611:Statistical regularity 593:Statistical regularity 557:) 02:43, 30 March 2008 536:Statistical regularity 504:Statistical regularity 93:WikiProject Statistics 2847:proc print data=mc2; 2680: 2625: 2605: 2549: 2407:unexplained symbols. 2369: 2314: 2294: 2238: 1873: 1823: 1766: 1674: 1599: 896:File:Largenumbers.svg 652:And I've changed the 36:level-4 vital article 2622: 2530: 2453:The named reference 2311: 2219: 2163:law of large numbers 1841: 1776: 1748: 1625: 1624:=\varepsilon }": --> 1552: 1273:Law of Large Numbers 1078:law of large numbers 1070:law of large numbers 1049:Law of large numbers 1018:with an established 800:TryingToUnderstand11 522:-- and copying from 221:mathematics articles 2730:Incorrect statement 1277:Cauchy distribution 844:Cauchy distribution 116:Statistics articles 2675: 2600: 2548: 2364: 2289: 2237: 2159:probability theory 2076:Assessment comment 2010:Jarosław Rzeszótko 1868: 1818: 1772:, the n for which 1760: 1668: 1594: 1456:Agree with Steve. 889:format instead of 526:corrected again — 476:Yes, certainly. — 190:Mathematics portal 45:content assessment 2880:for (i in 1:n) { 2798: 2673: 2598: 2585: 2533: 2362: 2287: 2274: 2222: 2203:gambler's fallacy 2144: 2132:comment added by 2106: 2105: 2059:comment added by 1954: 1937:comment added by 1793: 1642: 1569: 1380:Black swan theory 1375:Black Swan events 1356:comment added by 1325:comment added by 1285:comment added by 1226:comment added by 1148:gambler's fallacy 1127: 1082:gambler's fallacy 1074:gambler's fallacy 1065: 1041:Gambler's fallacy 1032:Gambler's fallacy 988: 678: 662:comment added by 615:Gambler's fallacy 559: 545:comment added by 413: 397:comment added by 360: 348:comment added by 323:comment added by 309: 308: 290: 289: 255: 254: 251: 250: 247: 246: 146: 145: 142: 141: 2977: 2792: 2790: 2782:birthday problem 2684: 2682: 2681: 2676: 2674: 2666: 2655: 2654: 2644: 2639: 2609: 2607: 2606: 2601: 2599: 2591: 2586: 2581: 2580: 2571: 2568: 2563: 2547: 2505: 2504: 2470: 2464: 2463: 2462: 2456: 2448: 2373: 2371: 2370: 2365: 2363: 2355: 2344: 2343: 2333: 2328: 2298: 2296: 2295: 2290: 2288: 2280: 2275: 2270: 2269: 2260: 2257: 2252: 2236: 2148:Bad introduction 2093: 2092: 2085: 2072: 1953: 1931: 1877: 1875: 1874: 1869: 1867: 1866: 1851: 1827: 1825: 1824: 1819: 1811: 1800: 1799: 1794: 1786: 1783: 1771: 1768: 1767: 1761: 1714: 1713: 1702: 1679: 1676: 1675: 1669: 1660: 1649: 1648: 1643: 1635: 1632: 1615: 1603: 1601: 1600: 1595: 1587: 1576: 1575: 1570: 1562: 1559: 1547: 1368: 1347:the black swan? 1337: 1297: 1238: 1172: 1168: 1165: 1114: 1113:from that page. 1052: 975: 958:Merge suggestion 914: 909: 904: 657: 558: 539: 453:Possible merger? 445: 444: 442: 440: 426: 339: 304: 276: 275: 265: 257: 223: 222: 219: 216: 213: 192: 187: 186: 176: 169: 168: 163: 155: 148: 136:importance scale 118: 117: 114: 111: 108: 87: 80: 79: 74: 66: 59: 42: 33: 32: 25: 24: 16: 2985: 2984: 2980: 2979: 2978: 2976: 2975: 2974: 2910: 2909: 2884: 2851: 2842: 2833: 2832:end; drop i; 2824: 2808: 2786: 2732: 2713:even less clear 2646: 2620: 2619: 2572: 2528: 2527: 2515: 2510: 2509: 2508: 2472: 2471: 2467: 2454: 2452: 2450: 2445: 2422:not an equality 2405:infinitely many 2335: 2309: 2308: 2261: 2217: 2216: 2150: 2115: 2081: 2078: 2054: 2045: 2025: 1977: 1932: 1907: 1852: 1839: 1838: 1784: 1774: 1773: 1745: 1744: 1719: 1718: 1717: 1707: 1703: 1699: 1684:tosses are say 1633: 1622: 1621: 1616:then for every 1609: 1560: 1550: 1549: 1548:the inequality 1541: 1538: 1515: 1481: 1351: 1344: 1320: 1280: 1261: 1259:Finite variance 1228:150.135.222.152 1221: 1211: 1045:Law of averages 1037:Law of averages 970:be merged with 968:Law of averages 960: 912: 907: 902: 883: 863: 835: 816: 689: 540: 508: 455: 450: 449: 448: 438: 436: 428: 427: 423: 366: 318: 314: 305: 299: 270: 220: 217: 214: 211: 210: 188: 181: 161: 115: 112: 109: 106: 105: 72: 43:on Knowledge's 40: 30: 12: 11: 5: 2983: 2981: 2973: 2972: 2967: 2962: 2957: 2952: 2947: 2942: 2937: 2932: 2927: 2922: 2912: 2911: 2882: 2849: 2840: 2831: 2822: 2807: 2804: 2803: 2802: 2731: 2728: 2686: 2685: 2672: 2669: 2664: 2661: 2658: 2653: 2649: 2643: 2638: 2635: 2632: 2628: 2611: 2610: 2597: 2594: 2589: 2584: 2579: 2575: 2567: 2562: 2559: 2556: 2552: 2546: 2543: 2540: 2536: 2514: 2511: 2507: 2506: 2481:(3): 615–621. 2465: 2442: 2441: 2437: 2391: 2375: 2374: 2361: 2358: 2353: 2350: 2347: 2342: 2338: 2332: 2327: 2324: 2321: 2317: 2300: 2299: 2286: 2283: 2278: 2273: 2268: 2264: 2256: 2251: 2248: 2245: 2241: 2235: 2232: 2229: 2225: 2179:expected value 2149: 2146: 2114: 2110: 2104: 2103: 2077: 2074: 2044: 2041: 2024: 2021: 1976: 1973: 1972: 1971: 1922: 1921: 1916: 1906: 1903: 1902: 1901: 1900: 1899: 1882: 1881: 1880: 1879: 1865: 1862: 1859: 1855: 1850: 1846: 1832: 1831: 1830: 1829: 1817: 1814: 1810: 1806: 1803: 1798: 1792: 1789: 1782: 1759: 1756: 1753: 1724: 1723: 1716: 1715: 1696: 1695: 1691: 1672:=\varepsilon } 1667: 1664: 1659: 1655: 1652: 1647: 1641: 1638: 1631: 1593: 1590: 1586: 1582: 1579: 1574: 1568: 1565: 1558: 1537: 1534: 1532: 1514: 1511: 1480: 1477: 1475: 1473: 1472: 1471: 1470: 1469: 1468: 1437: 1436: 1435: 1434: 1427: 1426: 1425: 1424: 1417: 1416: 1415: 1414: 1408: 1407: 1406: 1405: 1399: 1398: 1343: 1340: 1339: 1338: 1315: 1314: 1260: 1257: 1256: 1255: 1210: 1207: 1206: 1205: 1204: 1203: 1202: 1201: 1177: 1176: 1154:, or from the 1133: 1132: 1131: 1130: 1129: 1128: 1097: 1096: 1066: 1009: 959: 956: 955: 954: 938: 937: 882: 879: 862: 859: 848:87.117.185.161 834: 831: 815: 812: 811: 810: 790: 789: 788: 787: 786: 785: 767: 731: 730: 700: 699: 688: 685: 684: 683: 682: 681: 680: 679: 607: 574: 573: 572: 571: 561: 560: 507: 500: 499: 498: 482: 481: 454: 451: 447: 446: 420: 419: 415: 365: 364:Strong vs Weak 362: 313: 310: 307: 306: 301: 297: 295: 292: 291: 288: 287: 282: 272: 271: 266: 260: 253: 252: 249: 248: 245: 244: 233: 227: 226: 224: 207:the discussion 194: 193: 177: 165: 164: 156: 144: 143: 140: 139: 132:Top-importance 128: 122: 121: 119: 102:the discussion 88: 76: 75: 73:Top‑importance 67: 55: 54: 48: 26: 13: 10: 9: 6: 4: 3: 2: 2982: 2971: 2968: 2966: 2963: 2961: 2958: 2956: 2953: 2951: 2948: 2946: 2943: 2941: 2938: 2936: 2933: 2931: 2928: 2926: 2923: 2921: 2918: 2917: 2915: 2908: 2907: 2903: 2899: 2893: 2890: 2887: 2881: 2878: 2875: 2872: 2869: 2866: 2863: 2860: 2857: 2854: 2848: 2845: 2839: 2836: 2830: 2827: 2821: 2818: 2815: 2814:SAS Program: 2812: 2805: 2801: 2796: 2791: 2789: 2788:Novem Linguae 2783: 2779: 2778: 2777: 2776: 2772: 2768: 2763: 2762: 2758: 2754: 2749: 2747: 2743: 2739: 2735: 2729: 2727: 2726: 2722: 2718: 2714: 2710: 2709:Unfortunately 2706: 2704: 2700: 2698: 2694: 2690: 2667: 2662: 2659: 2656: 2651: 2647: 2641: 2636: 2633: 2630: 2626: 2618: 2617: 2616: 2615: 2592: 2587: 2582: 2577: 2573: 2565: 2560: 2557: 2554: 2550: 2538: 2526: 2525: 2524: 2523: 2518: 2512: 2502: 2499: 2495: 2492: 2488: 2484: 2480: 2476: 2469: 2466: 2460: 2447: 2444: 2440: 2436: 2435: 2431: 2427: 2423: 2419: 2414: 2411: 2408: 2406: 2401: 2399: 2395: 2389: 2387: 2383: 2379: 2356: 2351: 2348: 2345: 2340: 2336: 2330: 2325: 2322: 2319: 2315: 2307: 2306: 2305: 2304: 2281: 2276: 2271: 2266: 2262: 2254: 2249: 2246: 2243: 2239: 2227: 2215: 2214: 2213: 2212: 2207: 2206: 2204: 2197: 2194: 2190: 2183: 2182: 2180: 2176: 2172: 2168: 2164: 2160: 2153: 2147: 2145: 2143: 2139: 2135: 2131: 2123: 2119: 2111: 2109: 2102: 2099: 2094: 2091: 2089: 2084: 2075: 2073: 2070: 2066: 2062: 2058: 2051: 2042: 2040: 2039: 2035: 2031: 2022: 2020: 2019: 2015: 2011: 2007: 2004: 1998: 1997: 1993: 1989: 1986: 1983: 1980: 1974: 1970: 1966: 1962: 1957: 1956: 1955: 1952: 1948: 1944: 1940: 1936: 1930: 1925: 1919: 1917: 1914: 1912: 1911: 1910: 1904: 1898: 1894: 1890: 1886: 1885: 1884: 1883: 1863: 1860: 1857: 1853: 1848: 1844: 1836: 1835: 1834: 1833: 1815: 1812: 1804: 1801: 1796: 1787: 1757: 1754: 1751: 1742: 1741: 1740: 1739: 1738: 1737: 1733: 1729: 1728:93.219.149.62 1721: 1720: 1711: 1706: 1701: 1698: 1694: 1690: 1688: 1683: 1665: 1661: 1653: 1650: 1645: 1636: 1619: 1612: 1607: 1591: 1588: 1580: 1577: 1572: 1563: 1544: 1535: 1533: 1530: 1529: 1525: 1521: 1512: 1510: 1509: 1505: 1501: 1500:93.156.35.219 1498: 1493: 1490: 1487: 1484: 1478: 1476: 1467: 1463: 1459: 1455: 1454: 1453: 1449: 1445: 1441: 1440: 1439: 1438: 1431: 1430: 1429: 1428: 1421: 1420: 1419: 1418: 1412: 1411: 1410: 1409: 1403: 1402: 1401: 1400: 1397: 1393: 1389: 1385: 1381: 1376: 1371: 1370: 1369: 1367: 1363: 1359: 1358:82.132.235.94 1355: 1348: 1341: 1336: 1332: 1328: 1324: 1317: 1316: 1313: 1309: 1305: 1300: 1299: 1298: 1296: 1292: 1288: 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