Talk:Law of large numbers/Archive 1

message: The long-run relative frequency of repeated independant events get closer and cloer to the true relative frequency as the number of trials increases 202.7.183.131 11:42, 18 January 2006 (UTC)
thread_title:  Another possible definition of the LLN 

message: I'm putting some notes on history here to remind myself to add a history section later.  Someone else can feel free to do so though, especially if you know more about it than I do:
*Throughout the 19th century, the "law of large numbers" simply meant the weak law (convergence in probability); I think Bernoulli proved this, but I have to look that up.
*Some interesting stuff happened to clarify/extend it in the late 19th and early 20th centuries
*Émile Borel proved a special case of the strong law (almost-sure convergence) in 1909, for Bernoulli trials (cite?)
*Francesco Cantelli provided the first relatively general proof of the strong law (which he called "uniform convergence in probability") in his 1917 paper: "Sulla probabilità come limite della frequenza." Atti Reale Accademia Nazionale Lincei 26: 39-45.
*Aleksandr Yakovlevich Khinchin coined the now-current term "strong law of large numbers" to describe what Cantelli had called "uniform convergence", in a short published letter of 1928: "Sur la loi forte des grands nombres." Comptes Rendus de l'Académie des Sciences 186: 285-87.
*Andrey Kolmogorov proved that the strong law holds in cases other than independent identically-distributed variables, subject to some other conditions.  I think this was in 1929 (cite?).
--Delirium 09:43, 4 November 2005 (UTC)
thread_title:  Some notes on history 

message: in the first sentence do the words 'average' and 'mean' alternate on purpose? Spencerk 04:16, 5 December 2005 (UTC)
thread_title: intro

message: The Law of Large Numbers is a fundamental concept in statistics and probability.  Stated in a formal style of language the law is described as follows:If an event of probability p is observed repeatedly during independent repetitions, the ratio of the observed frequency of that event to the total number of repetitions converges towards p as the number of repetitions becomes arbitrarily large.  This means that the more units of something that are measured, the closer that sample average  will be to the average of all of the units -- including those that were not measured.  (The term "average" is specifically "the arithmetic mean.)  
For example, the average weight of 10 apples taken from a barrel of 100 apples is probably closer to the "real" average weight than the average weight of 3 apples taken from that same barrel.  This is because the sample of 10 is a larger number than the sample of 3.  And then, if you took a sample of 99 apples out of 100 apples, the average would be almost exactly the same as the average for all 100 apples. 
While this rule may appear to be self-evident to many readers, the development and use of this law allows statisticians to draw conclusions or make forecasts that would not be possible otherwise.  In particular, it permits precise measurement of the likelihood that an estimate is close to the "right" number.  
There are two versions of the Law of Large Numbers, one version called the "weak" law and the other is called the "strong" law.  This article will describe both versions in technical detail, but in essence the two laws do not describe different actual laws but instead refer to different ways of describing the convergence of the sample mean with the population mean.  The weak law states that as the sample size grows larger, the difference between the sample mean and the population mean will approach zero.  The strong law states that as the sample size grows larger, the probability that the sample mean and the population mean will be exactly equal approaches 1.0. 
One of the most important applications of the Law of Large Numbers, is called the Central Limit Theorem which, generally, describes how sample means tend to occur in a Normal Distribution around the mean of the population regardless of the shape of the population  distribution, especially as sample sizes get larger.  (See the article Central Limit Theorem for details of this application, including some important limitations.) This helps statisticians evaluate the reliability of their results because they are able to make  assumptions about a sample and extrapolate their results or conclusions to the population from which the sample was derived with a certain degree of confidence. See Statistical hypothesis testing as an example.
The phrase "law of large numbers" is also sometimes used in a less technical way to refer to the principle that the probability of any possible event (even an unlikely one) occurring at least once in a series increases with the number of events in the series. For example, the odds that you will win the lottery are very low; however, the odds that someone will win the lottery are quite good, provided that a large enough number of people purchased lottery tickets.
The remainder of this article will assume the reader has a familiarity with mathematical concepts and  notation.
----
I offer this as a revision to the first paragraph to make the concept more accessable to readers who may not be familiar with statistics.  It may not be right yet, but with editing perhaps it can be incorporated into the article.
--Blue Tie 04:47, 11 July 2006 (UTC)
replies: text:  Oh... I also like the term "Miracle of Large Numbers" instead of "Law of Large Numbers"... :-) --Blue Tie 04:48, 11 July 2006 (UTC)
replies: text:  Well, not a bad point from the perspective of a mathematician.  However, I did not mean formal definition.  I meant that the language was more formal than ordinary conversation (and I think this is a typical useage of the term "formal" when discussing general mathematical concepts in English Words rather than describing "forms", "proofs" or other specific items.).  My audience is someone who does not know what the Law of Large Numbers is or even, perhaps, what statistics are or that there are such things as formal proofs. However, if you think that the language is wrong, perhaps it should be changed.  However, I would not agree to the word "informal" for the audience I was looking to speak to.  They would NOT agree!--Blue Tie 15:42, 11 July 2006 (UTC)

text: You are right, it does not distinguish between those two laws.  I will see if I can adjust it by adding something in that regard, because even though the target audience may not have a full understanding, the opening paragraph should be a reasonably complete summary and an introduction to the topic. --Blue Tie 02:38, 12 July 2006 (UTC)
thread_title:  Suggested First Paragraph revision 

message: I still think the article could use some simplification.  Thoughts on these first few paragraphs?
The law of large numbers is a fundamental concept in statistics and probability that describes how the average of a random sample from a large population approaches the average of the whole population.  
In formal language:
If an event of probability p is observed repeatedly during independent repetitions, the ratio of the observed frequency of that event to the total number of repetitions converges towards p as the number of repetitions becomes arbitrarily large.
In statistics, this says simply that as the size of a random sample increases, the average of the sample converges to the true average.
For example, consider flipping a fair coin (that is, it comes up heads 50% of the time).  It is certainly possible that if we flip the coin 10 times, we may end up with only 3 heads (30%).  The Law of Large Numbers shows that as you flip the coin more and more (say 10,000 times), the percentage of coin flips that are heads will come ever closer to 50%.  Alternatively, it becomes less probable to maintain a rate of 30% heads as more coins are flipped, as this is below the true average. 
While this rule may appear self-evident, it allows statisticians to draw conclusions or make forecasts that would not be possible otherwise. In particular, it permits precise measurement of the likelihood that an estimate is close to the "right" or true number.  
Note, however, that the value of a single observation cannot be predicted using previous observations of independent events.  (For example, if a coin has come up heads 30% of the time, it is incorrect to say that the coin is "due" to come up heads - see the Gambler's Fallacy.
--Topher0128 21:28, 25 October 2006 (UTC)
thread_title:  Further Suggested Revision