Talk:Binomial distribution

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of them for my reference. And lets not get into a Mathematical jargon slanging match; whoever you are, I would be confident in my own Mathemaical standing to stand before anyone and prove my Theorem/ lemma. And, if it is indeed in many textbooks, I'd urge you to publish a proof of my statement. I have it on good authority, from highly esteemed Mathematicians, that the Theorem I put online is indeed a new and may I add correct proposition. It wasn't a Theorem as such, hence why I referred to it as the Binomial Lemma. I trust you know what a Lemma is! In future, before you make such claims, ensure that the nature of your statements is true. Do that rather than correcting me. And in response to this, if you do indeed give one, I'd appreciate being referred to as Dr. William Kitchen.

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to a text book they happen to have read. I had it checked, along with a proof by a university Professor who specialises in the concpets of probability and statisitics. It then underwent a stage of 'gaining plausibility', and under futher rigorous proof. There was a work through proof, and a proof by induction which clearly shows that the NEW theorem works, for all the possible values it outlines. I think you'll find the quote you have from your book involves a different concept to what I outlined before. I'll tell you what : take a look at it, and as Fermat said before he published his last Theorem "prove me wrong": I've got a mortgage on it saying you can't!! All the best, Dr. William Kitchen

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hands up on that front, and I understand the elementary nature of my error. I can provide you with my proof for the Theorem as soon as I finish my textbook which is in finalisation at the moment. All my work is momentarily on hold becasue of that. I welcome any scrutiny of my work - I feel that Mathematics is best done when under pressure from other esteemed Mathematicians. The workings of Knowledge, however, are something I am not aware of, and I appreciate any guidelines you offer me to follow. Again, however, as I have already said, I know I can stand before any Mathematician and prove my Theorem. Regards Dr. William Kitchen

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name "Kitchen's theorem". The notation in which it is written includes the use of the same letter for two different random variables in the same equality. Near the bottom it has some notation that is less than correct and that includes some very clumsy language. Then there is a signature---appropriate for a talk page but not for an article. In includes "Dr. William Kitchen PhD (Psychology)", apparently identifying that person as the one who added this material. It looks like an attempt to name after himself a proposition found in innumerable textbooks since before the births of most (or all?) people now living.

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not "theorom" as you wrote above) and following standard Knowledge conventions (e.g. who wrote what is in the edit history, NOT in the article itself). However, it would be a lot more efficient for you simply to tell me where to find your published article in the library than for you to go on at length about the whole history of your writing the article. (Oh, and I trust when you mention the copyright, you mean copyright on the article you wrote rather than on the theorem itself.)

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but this is an incomplete solution at best. For example, the article doesn't provide any guidance about choosing between (a) a combination of direct summation and the normal approximation and the poisson approximation, (b) a method based on the incomplete beta distribution, and (c) something else. A discussion of computing the CDF would be useful to a lot of people. ATBS 22:28, 30 November 2009 (UTC)ATBS

5207: 3295: 4329:? Similar errors appear to be made with the mode, median, variance, etc. I think skewness, kurtosis, and entropy could also be affected, but I don't know. When we look at the definition of this distribution it is clearly between 0 and 1 for all allowed values of p, k, and n; and yet if the mean is truly given by p*n, then with p=0.5 and n=3 we get 1.5 which is outside all output values. 4566:, which we conveniently have an article on already. Maybe a brief mention could be made, but it's a fairly general method for any discrete distribution, so it probably wouldn't make much sense to go into great detail (and on second thought, I'm a bit skeptical that we should even be saying as much as we already are, given the generality of the process being described). – 22: 2193: 3781: 5219: 3037: 3500: 1612:

statement should be sourced. If appropriate a proof can be added. There where a number of problems however with your contribution and Michael rightly reversed it. The notation is problematic (using X twice, using r both as an index and a parameter). The proof doesn't add to this article since it doesn't actually

1557:...oh, and since you emphasize that it's your own result, you should not put it in the article unless you also cite some place where you've published it in a journal, since otherwise it would be original research being presented here for the first time. Original research is contrary to Knowledge policy. 1717:

I experienced, that the normal approximation is indeed a time-saver if e.g. computing many different binomial distributions. In my case -- using octave -- computation speeded up a lot, especially since I was using quite large n's and always had to sum up about n/2 distributions (for only one point in

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I deleted a new section on "Kitchen's theorem". It began by saying "...we can see by Kitchen's Theorem that..." without having first said what "Kitchen's theorem" is. That is not appropriate. Then, as far as I can tell, the theorem turned out to be a proposition found in many textbooks without the

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and thus more difficult to mentally latch onto. Furthermore, a lesser issue, 'blank' and 'black' are two very similar words possibly leading to misreading. Why not just simply use: "Roll a die ten times and count the number of sixes.", thus appealing to a general feeling of wishing to see the highest

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Spot on. I thought exactly the same and immediately looked at the discussion. All political correctness aside, I just don't think anyone would feel harassed if we wrote "assume 5% of the population carry a certain gene" or "are infected with a certain desease", while I am very sure that everyone with

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You mean CI of p, the success probability, as estimated from the data. If 70 successes in 100 trials, then p_est = 0.7, and your question is what is standard deviation of p_est. It is sqrt(p_est(1-p_est)/n_trials). The 95% confidence interval is +/- 2 standard deviations. My question is what happens

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I agree that an expression with successive divisions appears ambiguous. Most mathematicians I know do indeed consider division as the inverse of multiplication and many programming languages explicitly specify that multiplication and divisions are performed left to right. You are correct, it is not a

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Correct about cumulative distribution. The "close to one" refers to the fact that in practice it will be necessary to approximate the probabilities. The problem with n bernoulli trials is that you have to do them for each output value, while computing the cumulative probabilities only has to be done

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It's common in Knowledge math articles to discuss algorithms for computing quantities of interest. On this page it would be very helpful to have a discussion of computing the cumulative distribution function (CDF). The article does mention various methods that can be used in various circumstances,

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After giving the expectation as np, the article states "This fact is easily proven as follows. Suppose first that we have exactly one Bernoulli trial. We have two possible outcomes, 1 and 0, with the first having probability p and the second having probability 1 − p; the mean for this trial is given

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This section is my first contribution. I sincerely hope it's sensible to have done so and that it is a (potential) boon to readers. I'm honing it, adding links, references, improving text etc. Please give me a couple of days, I'll post it in one single edit. I'd appreciate any advice you have for me

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OK, I will go back and look carefully at what you added to the article. But if it is to be included, it should be written clearly, using standard notation (not, for example, using the same letter for two different random variables in the same breath), standard language and spelling (e.g. "theorem",

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What you quoted from this textbook isn't even the same as my Theorem. And do not quote Knowledge policy to me - take me to court, sue me, do whatever you wish. I have this Theorem in a journal, and have had it copyrighted to my name, so that scavengers on internet sites cannot attribute a novel idea

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from a group of candidates on the merit of their desireable properties. (Being such properties as the fact that curve is guaranteed to be contained within the convex hull of the control points, that reversing the control points does not change the curve, that the tangents at the endpoints consist of

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I'm new and do not know the etiquette. Hence I will not update the page. However, it just came to my attention that you do not have the formula for entropy of the binomial distribution. it is 2^-S= ((N-U)/N)^N * (U/(N-U))^U, where in your notation N=n and U=np. The unit of S is in bits. I hope this

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Well Michael, it's nice to see a fellow 'Mathematician' scrutinising my work, labellng it a 'proposition'. Given the fact that my Theorom has went under rigorous investigation within a university, I fail to see how you can ever have seen it in "innumerable" textbooks. Perhaps you could name a few

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Okay, maybe this is standard jargon somewhere, but I've never come across it until today. I guess "mass" makes sense by the physical analogy to density. Honestly, I think it's stupid language. Should we also speak of cumulative mass distribution functions? Be consistent! I'm not going to change it,

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says: "The lead ... should ... establish context, explain why the topic is notable..." and "avoid difficult-to-understand terminology" "....with the goal of making the lead section accessible to as broad an audience as possible. Where uncommon terms are essential, they should be placed in context,

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I took a quick look at other probability distribution pages on Knowledge, and I'm not seeing any derivations of the mgfs there. I suppose that doesn't necessarily disqualify us for adding the mgf to this page, but considering that this page doesn't even derive the mean or variance of the binomial,

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will show them) - but I wanted to write the functions in Javascript for a web page. I went ahead and wrote the web page using the Poisson distribution - but I still think that this article should give expressions that people can use in normal languages and spreadsheets! I feel I've done my bit for

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Yes, this is the kind of consideration I was shooting for with my original post. While I understand adequately how to create a hypothesis test for a one-sided alternative, I was hoping that someone would come forward with a good methodology for doing a two-sided alternative hypothesis, since this

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Another intuitive understanding of confidence intervals on binomial distribution is this: say you S success in N trials, now we don't know what p really is, but let's make a guess p_guess. You can use the binomial distribution to calculate the probability of seeing in N trials if p=p_guess. If

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I see now the problem. (1-p) was intended as a parenthetical definition. I guess N1/X!/(N-X)! worked in my programming codes so I couldn't see the ambiguity. How would you calculate N!/X!/(N-X)!? From right to left? On the other hand, Today is 02/20/2001, so I think your "requesting a change for a

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Also, I know that Knowledge has a unique problem with writing in plain language, but I would add a simple descriptive example that is easy to grasp: If we have a fair coin (p = 0.5) and two trials (n = 2), then if we want to get 2 heads (k = 2), there is only one possibility to achieve this (2! /

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Hmm, just a reader here, but I can't make a modern computer delay visually for any reasonable n (up to 9999999999) when using the exact solution. I advise my students to always use the exact test and that the normal approximation is a relic of a bygone era. However it is interesting and perhaps

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I feel like there could be a better example than picking 500 people out of a population "with replacement" and seeing how many were green-eyed. Perhaps a more sensical and applicable example could be: out of 50 web servers, each of which has a 1% chance of failing by the end of the day, how many

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I was looking for a pointer to quickly simulate a Binomial trial. That is, given a p and an n, I want to randomly select a result with a Binomial distribution. I know I can approximate this with a normal distribution, but I would prefer an exact result if it can be calculated quickly for n <

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Well, I appreciate that. Like all Mathematicians, I like recognition for my work. I had to have it rigorously checked and compared with similar Theorems and Lemmas, to ensure I wasn't putting my name to a piece of work that someone else had previously discovered. Notation is a blunder, I hold my

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If you want to attribute this result to yourself in a Knowledge article, may I suggest that you cite some published paper that you've written in which you state it? What was the nature of this "rigorous investigation"? Was it simply mathematicians confirming that the result is correct? If so,

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I think at its current state, the lead section is very opaque for a layperson. It's so full of technical jargons that only a person already familiar with these statistical terms would care to understand what this concept means. I think the lead should be much more accessible. It should use very

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In the section entitled Mean, variance and mode, it isn't clear to me how the expression given follows from "Using the definition of variance, we have..." Should I try to find this in the entry for variance, figure it out from the problem statement, or use the definition of variance given just

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In the case where the confidence interval gets close to 0 or 1, the normal approximation of the binomial distribution is not accurate and rules like your "2 standard deviations" that are derived from the normal distribution are not accurate either. Depending on the circumstances, one can use a

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an HIV-infection or someone who knows someone closely who is infected will at least feel strange on reading this paragraph. I am all against political correctness for its own sake, but if there's no need whatsoever to use a certain formulation that might be considered inappropriate, why use it?

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Hello Dr. William Kitchen, please try to relax a bit, nobody is trying to discredit your work. But we are talking about cross purposes. What one wants for an encyclopedia article on the binomial distribution is the fact that it is related to the negative Binomial distribution. Ideally such a

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would imply some means of parameterizing the asymmetry of the distribution. I had this come up in a real-world scenario, where the question was whether or not we had a statistically significant result, and how close it might actually be, but the interval was not so critical or well-defined.

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Would it be possible to write an introductory section that gives just a conceptual description of what the binomial distribution is about, before we enter the maths? Like tossing a coin, or drawing marbles from a box, and replacing the drawn marble each time (and mixing the box up again)?

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I can see that it would be easier. But it would require more starting knowledge. I'd guess that anybody who knows about generating functions does not need to look up the derivation of the mean in wikipedia. Therefore I think the derivations should be kept as elementary as possible.

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I was looking for information about confidence intervals on a binomial distribution, but was surprised not to find it here. I know this case isn't quite as simple as for normal distributions, but it would be nice to have here, if somebody would like to contribute the information.

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and was unknown before you introduced it? If so, I would find that surprising and I would dispute it. Or was it a professor saying he did not happen to have seen it before? If he's not a probabilist, that's not too surprising and is not the same as saying that it is novel.

2445:{\displaystyle {\text{mode}}={\begin{cases}\lfloor (n+1)\,p\rfloor &{\text{if }}(n+1)p{\text{ is 0 or a noninteger}},\\\lfloor (n+1)\,p\rfloor {\text{ and }}\lfloor (n+1)\,p\rfloor -1&{\text{if }}(n+1)p\in \{1,\dots ,n\},\\n&{\text{if }}(n+1)p=n+1.\end{cases}}} 4287:

simple language or have very simple language definition for each opaque jargon-y term. It should also contain concrete examples, preferably historically relevant examples, associated with the concept, to provide a very basic, graspable understanding of the topic.

4878: 3290:{\displaystyle F(x)=\left\{{\begin{array}{*{20}{l}}0&{x<0}\\{\sum \limits _{j=0}^{k-1}{\left({\begin{array}{*{20}{c}}n\\k\end{array}}\right)}\,\,{p^{k}}{{(1-p)}^{n-k}}}&{k-1\leq x<k,\,\,k\in \{1,2,...,n\}}\\1&{x\geq n}\end{array}}\right.} 859:

Why is it necessary to express the CDF's upper bound of summation in terms of the floor function? The binomial distribution support already indicates that the random variable must take on positive integer values, with the exception of zero (0, 1, ... n).

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I think the given CDF is really merely an introduction of notation. Perhaps there is no simple closed-form expression for the CDF, although there is an obvious algorithm for computing its values (just add up the appropriate values of the mass function).

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Multiplication is associative over the reals. If you look at division as the inverse operation of multplication, i.e. 2/4/12=2*4^1*12^1=1/24 you are okay. If you look at division in the ordinary sense, you must specify the order of operations.RoseParks

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So if you make a sequence of the probabilities values (easy), you can easily make this into a sequence of n+2 points and then draw segments as in the above graphs. Having worked this out a gazillion times for my kiddies, I finally wrote it down.

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for this be inefficient. Second, we have closed form expressions for the pmf that always add up to one, so if it does not, you miscalculated. Third, using the cumulative probability distribution would probably work better than the pmf.

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Is it me, or should the "A typical example is the following: assume 5% of the population is HIV-positive." part in the second paragraph be changed to something a little less... you know... The HIV part is just not encyclopedia-ish...

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True, but there's also an initial cost of generating the table, which might be prohibitively high depending on the number of values needed. This section seems a little iffy; I'll try to look into it a bit more if I get a chance.

337:=p_guess. Similarly if S~0. For example if you did a trial with 45 samples (N=45) and all of them were successful (S=45), if p_guess=.95 there is only a 0.099 chance of seeing all 45 successful, so there you can be 1-0.099 =: --> 3919:

title says it all, see for example Ching-Hui Chang et. al.: "A note on Improved Approximation of the Binomiual Distribution by the Skew-Normal distribution" the American Statistician. Kjetil B Halvorsen 13:34, 8 June 2015 (UTC)

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Thank you, fair point. Since MGF are very basic for using these objects in various settings, I think this type of information should be available somewhere on Knowledge. a) would you agree? b) if so - where do you think would it

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Might it not be a lot easier to demonstrate this proof using generating functions? I can easily do it this way, unless anyone can spot a good reason not to (it requires a lot less algebra...but does requires some GF results)

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Is it possible to give a reference for that formula given that it is not standard textbook knowledge. Also it is unclear, what the operator/function O of 1 over n stands for. Thanks. (Sorry for all breaches of etiquette.)

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The example with a die is OK in principle. Most people, I reckon, will have seen and used dice. But why change the well known configuration ( 1 thru 6 dots) with "5 blank and 1 black side"? This now makes a familiar object

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I had already given this formula in my MK wikipedia page and was wanting to add an iterative "computer graphing formula" and so checked to see if there was one on this page and became confused with the above formula.

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is very large the exact computation can still be onerous. Perhaps more important is that the normal approximation means that a great many statistical tests designed for the normal distribution (such as the Student

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I'm deleting this since no justification has been offered. Zillions of things are "inspired" by the binomial distribution anyway and I don't see why this one is important enough to single out even if it is true.

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The article gives the following example: "A typical example is the following: assume 5% of the population is green-eyed. You pick 500 people randomly. How likely is it that you get 30 or more green-eyed people?".

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But shouldn't the mean listed on the binomial distribution page refer to the mean of the distribution and not the mean of a binomial random variable, which has a different definition, causing this confusion?

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In my view the statement related the two distribution can stay in the article. But the proof you supplied should either be replaced by a proper proof or by a reference to a published book or peer reviewed

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This is a CDF example. Unfortunately, the expression given for CDF is not very clear to me. How about giving a worked example with the green-eyed people given in the article as a good example, please? --

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Would someone care to source that statement? It seems rather dubious to me, but if it's true it's worthy of a proper explanation and not the vague description of being "inspired by". Certainly the

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I added two references to the article which describe binomial random variate generation. A modern C implementation of Kachitvichyanukul and Schmeiser's BTPE algorithm is available as part of the

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In lieu of misusing a hypergeometric distribution as an example of a binomial distribution, perhaps add a section detailing the relationship and how they are similar and yet different?

3776:{\displaystyle {\begin{array}{*{20}{l}}{{F_{0}}(x)=0}&{x<0}\\{{F_{k+1}}(x)={F_{k}}(x)+f(k)}&{k\leq x<k+1,\,\,k=0,1,2,...,n-1}\\{{F_{n+1}}(x)=1}&{x\geq n}\end{array}}} 4764: 1813:

It can be done both ways. In the Knowledge article, the normal distribution is defined in terms of the variance, so to be consistent, its probably best to do it that way here too.

4389: 3495:{\displaystyle F(x)=\left\{{\begin{array}{*{20}{l}}0&{x<0}\\{\sum \limits _{j=0}^{k-1}f(j)}&{k-1\leq x<k,\,\,k\in \{1,2,...,n\}}\\1&{x\geq n}\end{array}}\right.} 1670:

The exact calculations are only onerous if one doesn't have a computer. Considering that virtually all statistics is done over computers these days the above seems unimportant.

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2). And the formula for the numbers of ways of picking X items out of N items was: N!/X!/(N-X)!. This is plain wrong. Yes, after requesting a change for a week, I changed it.

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I agree with both of you. How about simply the "toss a coin..."? Hackneyed, perhaps, for us, but surely we want the general reader to "get the picture" as easily as possible?

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You can compute this in terms of the incomplete Beta function, as indicated in the article, using your favorite numerical software. For example, in Mathematica this becomes

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One way to generate random samples from a binomial distribution is to use an inversion algorithm. To do so, one must calculate the probability that P(X=k) for all values

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the binomial becomes normal-ish. Also, I thought the ability to use the normal approximation was based on np not n - with a low enough p, even a huge n will be skewed.

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From reading about Bézier curves I've always had the impression that the decision to use Bersteins as their parametrization wasn't 'inspired' by anything, but merely

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I would really like to know what you think, to me it looks very cool as I use expected value and sum of binomial series and I didn't see anything like it anywhere.

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to generate samples uniformly between 0 and 1, one can transform the calculated samples U into discrete numbers by using the probabilities calculated in step one.

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I came up with a way to add variance to the binomial distribution, for this purpose I consider the history of success compare to the expected value. Here is my

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the theorem, it only give some basic definition and a referral. And finally, the theorem quoted is unknown to mathematicians, so it doesn't help one at all.

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I agree. The current example suffers from the need to do sampling with replacement, which will seem unnatural to people unaccustomed to sampling theory. --

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The figures have unlabeled axes, making them pretty much useless. Can someone either introduce new figures, or edit the existing ones to have axis labels?

230: 5432: 141: 5374:(2!*0!) = 1), and since we need to get heads twice, it means that P = 0.5*0.5 = 0.25 (which is the same by our formula: P = 1*0.5^2*0.5^(2-2) = 0.25). 2111:

As another example, assume 5% of a very large population to be green-eyed. You pick 100 people randomly. The number of green-eyed people you pick is a

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if the CI range is outside the allowed 0 to 1 range for a probability. This can happen if p_est is ~1 or ~0. The CI has to be assymetric. Any ideas?

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I hope somebody could help me in finding derivations or how to derive the skewness and kurtosis, even link to other sites will be much appreciated.

1133:. Direct summation is likely going to be less numerically stable than a carefully designed subroutine for evaluating the incomplete Beta function. -- 5447: 5412: 5295: 1289: 342: 5355:

The binomial distribution is concerned with the probability of obtaining any of these sequences, meaning the probability of obtaining one of them

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linked and briefly defined. The subject should be placed in a context familiar to a normal reader." I don't think that is the case here at all.

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The second rule of thumb for normal approximations looks suspicious. It can be written in the following form: use normal approximation whenever

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Actually, come to think of it, neither have I. But there's no mathematical proof that says thats the way it has to be done, that's what I meant.

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From my training (and looking at the graphs on THIS page), we should be defining F on the real numbers and writing < and not ≤, that is:

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Thanks very much for your response. I agree with you - and as it happens, I do use Maxima, which has a shed-load of distribution functions (

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Good idea. The lead sort of introduces it, but there should be room for a more detailed overview. Sources shouldn't be too tricky to find.

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Your end result for the binomial approximation is incorrect. It should be N(np , (np(1-p))^1/2). You currently have N( np , (np(1-p))).

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For a derivation of the Fisher information, see example 2.10 of this book, and for a derivation of how taking the expectation leads to

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So it makes sense for the mean, median, and mode to fall somewhere within this range. The values listed in the article are correct. –

4017: 2857: 1759: 1719: 1301: 1245:. But binomial coefficients exist all over the place. It doesn't necessarily imply that they have much at all to do with the Binomial 1827:

In words, one can say “a normal distribution with mean xxx and standard deviation yyy”. But when writing a formula, it is always the

1370:-- Why does the article use a biased coin in the introductory example? Why not a normal coin that people encounter in a normal life? 5299: 4206:

When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.

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i=0 prob=0.117649 i=1 prob=0.050421 i=2 prob=0.021609 i=3 prob=0.009261 i=4 prob=0.003969 i=5 prob=0.001701 i=6 prob=0.000729

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The proof of the mode doesn't define a_k. It's relatively clear that f(k) is meant, but that should be defined (or f used)

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That would really be a whole lot more to the point than telling us how confident you are that everything about it is sound.

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pmf is fairly standard. It is linked there now. No, cumulative mass distribution function is not a phrase I have heard. --

4475:. (These probabilities should sum to a value close to one, in order to encompass the entire sample space.) Then by using a 4476: 4914:

Fisher information, i.e. expectation of the Fisher information (where the expectation is taken with respect to the data)

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Is it my imagination or are only the first and last probabilities for the biased coin correct? I have run that in SAS

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aha - that's the answer I was looking for! In that case, why not say something like, "The value can be computed with..."

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data _null_ ; p = 0.3 ; do i = 0 to 6 ; prob= (p**i) * ((1-p)**(6-i)) ; put i= prob= ; end ; run ;

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I think some parts should be more accented in the first sentences of the interpretation section. I mean the sentence:

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BTW: Here is the iterative formula I was getting ready to add. Any suggestions here to make this clearer how to use?

2127: 2081: 1356:--Why not start with a coin example? Isn't this the most straight-forward? The one that everyone did in 4th grade??-- 2084:

describes how it is used for sampling without replacement and states that Binomial Distribution is used for sampling

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week" is a bit off. Today is only the 20th by my calendar. In any case, the criticism has led to something better.

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This probability formula means the probability of obtaining k successes in n trials for all possible combinations.

4725: 3505:(I also switched to j in place of i since so many applications now assume i is the corresponding complex number.) 4873:{\displaystyle F(k;n,{\frac {1}{2}})\geq {\frac {1}{15}}\exp \left({\frac {-16({\frac {n}{2}}-k)^{2}}{n}}\right)} 4128: 3961: 4227:

to delete these "External links modified" talk page sections if they want to de-clutter talk pages, but see the

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to delete these "External links modified" talk page sections if they want to de-clutter talk pages, but see the

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Confirming the error. The closest formula in the source is in Proposition 7.3.2, page 46, and it provides the

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But normal distributions aren't given by mean and variance, they're given by mean and standard distribution.

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different approximation (such as the Poisson distribution) or the exact values of the binomial distribution.

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Dr. Kitchen, could you tell us the title of the paper and the name of the journal and which issue it's in?

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that's hardly surprising. Was it mathematicians with expertise in probability theory saying the result is

1413: 281:---- In answer to your question on how you evaluate, N!/X!/(X-N)!, this is ambiguous. In any easy example. 5271: 4197:

https://web.archive.org/web/20150113082307/http://psych.stanford.edu/~jlm/pdfs/Wison27SingleProportion.pdf

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Please label the axes on the graphs, and state the allowable range for parameter n. Does n include zero?

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If you have discovered URLs which were erroneously considered dead by the bot, you can report them with

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If you have discovered URLs which were erroneously considered dead by the bot, you can report them with

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i=0 x=0.117649 i=1 x=0.302526 i=2 x=0.324135 i=3 x=0.18522 i=4 x=0.059535 i=5 x=0.010206 i=6 x=0.000729

2681: 861: 50: 4347: 4177:. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit 4008:. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit 3897: 3853: 2805: 1293: 94: 5290: 5112: 2841:

data _null_ ; do i = 0 to 6 ; x = pmf('Binomial',i,.3,6) ; put i= x= ; end ; run ;

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If you go to previous versions and look at the first one, 02/15/2001, which is yours?, you will see :

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https://web.archive.org/web/20160303182353/http://www3.stat.sinica.edu.tw/statistica/oldpdf/A3n23.pdf

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Yes, it is your imagination. Seriously, you have forgotten to include the combinatorial coefficient.

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It doesn't make any sense for the parameter n to be 0. Most texts limit n to be a natural number.

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https://web.archive.org/web/20140515145146/http://www.mbastats.net/Content/Basic_Prob/Binomial.html

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universal convention. In addition, the vertical placement of numerator and denominator is clearer.

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

116:

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

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This article was the subject of a Wiki Education Foundation-supported course assignment, between

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before doing mass systematic removals. This message is updated dynamically through the template

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before doing mass systematic removals. This message is updated dynamically through the template

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S~N (p~1), there will be a value for p_guess so that there is only a 5% chance of getting S: -->

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But you might have been better off using the incomplete beta function result that is included.

5183: 5149: 4689:{\displaystyle {\frac {1}{2}}\log _{2}\left(2\pi enp(1-p)\right)+O\left({\frac {1}{n}}\right)} 2584: 1877: 1794: 4910:

Hi all, it seems to me that what is described here as the Fisher information is actually the

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This approximation is a huge time-saver (exact calculations with large n are very onerous);

5227: 4524: 3982: 3928: 2721: 2112: 2093: 336:=S_observed. At this value for p_guess, you would say you are 95% confident that p: --> 5231: 5223: 4495: 4213:, "External links modified" talk page sections are no longer generated or monitored by 4054:, "External links modified" talk page sections are no longer generated or monitored by 3874: 3833: 3790: 2540: 1956: 1632: 1347: 1218: 1134: 770: 387: 4253:

If you found an error with any archives or the URLs themselves, you can fix them with

4094:

If you found an error with any archives or the URLs themselves, you can fix them with

1205:

How about putting it here, then with attention on it someone may come up with better.

5391: 5313: 5243: 5169: 5135: 2820: 2698: 2506: 2138: 1898: 1818: 1736: 1628: 1693:-test) can also be used for the binomial distribution under the right conditions. -- 1631:

is established wikipedia policy, and this is not the place to put it to discussion.

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I really dislike the "nmemonic" section. If anyone else agrees, please delete it.

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Just wanted to add - thanks for your help in getting to this article improvement! --

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by μ = p." This is not a proof. These sentences should really just be removed.

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Currently the random variate section has this paragraph, which is kind of weird:

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regarding content choice, style etc. Thank you. Thanks already to Michael Hardy.

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which is important, so should be almost repeated at the beginning, for example:

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In particular that rule claims normal approximation should not be used for any

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The expression for the mode is incorrect. Imagine a Binomial distribution with

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For one, you can just do n bernoulli trials. You would need an absurdly large

4219:. No special action is required regarding these talk page notices, other than 4060:. No special action is required regarding these talk page notices, other than 3978: 2089: 1694: 1310: 1269: 1206: 1186: 324: 202: 90: 2835:

Oh yes, thanks for that. That's what I get for trying to do it long hand.

2164:= 2, the expression for the model will return 3 while the true value is 2. 3892:

I'm going to say no. This is material for a textbook, not an encyclopedia.

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the line between the endpoint and the neighboring control point, etc). --

614:{\displaystyle cdf(k;n,p)=\sum _{k=1}^{n}{n \choose k}p^{k}(1-p)^{n-k}\,} 5294:

This article was the subject of an educational assignment supported by

4127:

Is the "Spoilers!" in the conditional binomial proof section a joke? :)

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If it isn't strictly a binomial distribution, then it is a bad example.

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The formula for Bézier curves was inspired by the binomial distribution.

2636:)). Please someone fix it or explain what I do not understand there. 433:

but a mathematician should. At the very least link it to the pmf page.

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The formula as given in the article is correct. It matches the mean (

1118:{\displaystyle =1-I_{0.95}(471,30)=I_{0.05}(30,471)\approx 17.647\%.} 758:{\displaystyle F(k;n,p)=\sum _{j=0}^{k}{n \choose j}p^{j}(1-p)^{n-j}} 5008:{\displaystyle g_{n}(p)={\frac {x}{p^{2}}}+{\frac {n-x}{(1-p)^{2}}}} 2088:

replacement. How is Binomial Distribution method used for sampling?

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10,000. I'm sure others have come here looking as well. Thanks.

1430:(1994) by Sheldon Ross, page 181, exercise 26, quoted verbatim: 785:

Good corection - I have added this expression to the article! --

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http://psych.stanford.edu/~jlm/pdfs/Wison27SingleProportion.pdf

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Firstly, I want I am wondering about the definition of the CDF

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The cdf of a discrete distribution must be piecewise constant.

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article, mostly written by me, I did my best to make it clear

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http://www.le.ac.uk/users/dsgp1/COURSES/MATHSTAT/5binomgf.pdf

2576:= ½. So the sign should probably be reversed, and the factor 1195:

I agree. The mnemonic section is laughable. I'm deleting it.

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When you have finished reviewing my changes, please set the

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the plot). So thank you for mentioning it in the article! --

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After looking a bit, I found a paper which talks about the

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funny, but it is recently-added vandalism so I removed it.

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As one final point, please do not make legal threats. Also

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Something in Firefox? Found it: text color must be black.

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http://www3.stat.sinica.edu.tw/statistica/oldpdf/A3n23.pdf

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for additional information. I made the following changes:

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for additional information. I made the following changes:

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which approximately follows a binomial distribution with

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http://www.mbastats.net/Content/Basic_Prob/Binomial.html

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01:34, 25 April 2007 (UTC). OK, a couple of weeks. It's

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Hi, isn't the standard deviation calculated as : sqrt((

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be a negative binomial random variable with parameters

1215: 3118: 1217:. But I agree with deleting it, it is unencyclopedic. 5201:

Wiki Education Foundation-supported course assignment

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once followed by (say) a binary search of cost O(log

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We should add a section on skew-normal approximation

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then I don't see adding this material to this page.

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Computing the cumulative distribution function (CDF)

201:, a collaborative effort to improve the coverage of 112:, a collaborative effort to improve the coverage of 4223:using the archive tool instructions below. Editors 4064:using the archive tool instructions below. Editors 5095:{\displaystyle {\text{E}}_{X}={\frac {n}{p(1-p)}}} 5094: 5007: 4872: 4688: 4383: 3775: 3494: 3289: 3020: 2444: 1865: 1522: 1284:failed servers do you have at the end of the day? 1117: 1025: 757: 613: 2974: 2961: 711: 698: 565: 552: 4739:The current tail bound given for p=1/2, n/2: --> 2910: 2869:Question about Cummulative Distribution Function 1866:{\displaystyle {\mathcal {N}}(\mu ,\sigma ^{2})} 1680:It is less important than it used to be, but if 972: 945: 272:3).There were also wording problems. RoseParks. 5418:Knowledge level-4 vital articles in Mathematics 5258:Consistent capitalization required on same page 2042:above? In any case I don't see how it follows. 4209:This message was posted before February 2018. 4050:This message was posted before February 2018. 1451:be a binomial random variable with parameters 4344:The support of a binomial random variable is 1428:A First Course in Probability, Fourth Edition 8: 5369:The formula can be understood as follows ... 4375: 4351: 4321:Mean, Mode, Median seem off by a factor of n 3823:We should add the moment generating function 3461: 3431: 3256: 3226: 2953: 2947: 2755:)) ? In the article it is written as: sqrt(( 2385: 2367: 2330: 2307: 2299: 2276: 2236: 2213: 1346:I agree I like the original better as well. 900:This is the binomial distribution - how can 2789:Cumulative distribution function -- Example 1530:n\right\}=P\left\{Y<r\right\}\,}" /: --> 4881: 4699: 4489: 4000:I have just modified one external link on 1026:{\displaystyle \Pr=1-\Pr=1-F(29;500,0.05)} 163: 58: 5285:India Education Program course assignment 5266:(not at start of sentence) and in others 5240:Template:Dashboard.wikiedu.org assignment 5065: 5044: 5031: 5026: 5023: 4996: 4966: 4955: 4946: 4928: 4922: 4854: 4834: 4822: 4802: 4786: 4766: 4672: 4613: 4599: 4597: 4349: 4169:I have just modified 2 external links on 3758: 3727: 3722: 3721: 3642: 3608: 3603: 3578: 3573: 3572: 3557: 3532: 3527: 3526: 3522: 3520: 3474: 3396: 3369: 3358: 3353: 3338: 3329: 3308: 3269: 3191: 3175: 3158: 3156: 3149: 3144: 3117: 3112: 3100: 3089: 3084: 3069: 3060: 3039: 3006: 2984: 2973: 2960: 2958: 2946: 2935: 2881: 2400: 2341: 2302: 2264: 2241: 2205: 2197: 2195: 1917:Explicit derivations of mean and variance 1854: 1835: 1834: 1832: 1469: 1467:n\right\}=P\left\{Y<r\right\}\,}": --> 1082: 1054: 1039: 943: 904:not be either 0 or a positive integer? -- 743: 721: 710: 697: 695: 689: 678: 645: 597: 575: 564: 551: 549: 543: 532: 493: 5453:India Education Program student projects 5105:Should we change that in the page? Best 4325:Isn't the mean not the sum, but the sum 5408:Knowledge vital articles in Mathematics 5328:Why are equations suddenly not showing? 5306:The above message was substituted from 5238:Above undated message substituted from 5126: 3669: 3667: 3423: 3421: 3218: 3216: 3142: 3140: 2325: 2294: 2231: 1518: 609: 411:That might depend on which population. 165: 60: 19: 4303:Knowledge:Manual of Style/Lead section 2838:The SAS functions exist for a reason! 2616:There is an error in an example: σ = ( 5423:B-Class vital articles in Mathematics 4039:to let others know (documentation at 7: 1873:, and I've never seen it otherwise. 195:This article is within the scope of 106:This article is within the scope of 5168:Held, L., Sabanés Bové, D. (2014). 5134:Held, L., Sabanés Bové, D. (2014). 5102:see example 4.1 of the same book. 4917:The actual Fisher information is: 4459:The random variate section is weird 3355: 3086: 1524:n\right\}=P\left\{Y<r\right\}\,} 49:It is of interest to the following 5443:Top-importance Statistics articles 5215: 5211: 4735:Issue with one of the tail bounds? 4592:Entropy is given as entropy = 4384:{\displaystyle \{0,1,\ldots ,n\}.} 2965: 1109: 936:In that specific example, we have 702: 638:Actually, in this case the CDF is 556: 14: 5433:Mid-priority mathematics articles 4435:No, those are the same things. – 4173:. Please take a moment to review 4004:. Please take a moment to review 2669:(I hope it is OK to have a link) 1160:Knowledge maths clarity - in the 818:is an integer, and only when 0 ≤ 126:Knowledge:WikiProject Mathematics 5403:Knowledge level-4 vital articles 5289: 5262:In some places the article uses 5218:. Further details are available 5205: 215:Knowledge:WikiProject Statistics 188: 167: 129:Template:WikiProject Mathematics 93: 83: 62: 29: 20: 5448:WikiProject Statistics articles 4902:Fisher information is actually 1934:01:31, 11 May 2007 (UTC) Done. 235:This article has been rated as 218:Template:WikiProject Statistics 146:This article has been rated as 5413:B-Class level-4 vital articles 5342:19:09, 20 September 2023 (UTC) 5174:. Springer Berlin Heidelberg. 5140:. Springer Berlin Heidelberg. 5086: 5074: 5059: 5056: 5050: 5037: 4993: 4980: 4940: 4934: 4851: 4831: 4796: 4771: 4654: 4642: 4316:08:09, 14 September 2017 (UTC) 4297:08:05, 14 September 2017 (UTC) 3746: 3740: 3636: 3630: 3621: 3615: 3597: 3591: 3545: 3539: 3390: 3384: 3319: 3313: 3171: 3159: 3050: 3044: 3003: 2990: 2925: 2913: 2904: 2886: 2844:And yes, there's the sequence 2417: 2405: 2358: 2346: 2322: 2310: 2291: 2279: 2258: 2246: 2228: 2216: 1975:15:13, 21 September 2007 (UTC) 1860: 1841: 1745:15:53, 18 September 2009 (UTC) 1728:10:13, 18 September 2009 (UTC) 1231:The article currently states: 1227:Relationship to Bezier curves? 1168:how to do each calculation! -- 1100: 1088: 1072: 1060: 1020: 1002: 987: 975: 960: 948: 740: 727: 668: 650: 594: 581: 522: 504: 1: 5367:Then you can leave the rest: 5171:Applied Statistical Inference 5137:Applied Statistical Inference 5117:18:44, 14 November 2021 (UTC) 4477:pseudorandom number generator 3945:Mode proof doesn't define a_k 3818:19:11, 14 December 2014 (UTC) 3799:09:06, 22 November 2013 (UTC) 2596:22:41, 30 November 2009 (UTC) 2515:10:36, 24 November 2009 (UTC) 2493:21:03, 23 November 2009 (UTC) 2468:06:17, 25 November 2009 (UTC) 2071:01:30, 13 November 2008 (UTC) 2031:19:59, 16 February 2008 (UTC) 1662:Not sure about the statement 1523:{\displaystyle P\left\{X: --> 1384: 1314:05:52, 29 November 2006 (UTC) 1262:14:48, 3 September 2006 (UTC) 209:and see a list of open tasks. 120:and see a list of open tasks. 5428:B-Class mathematics articles 5322:19:51, 1 February 2023 (UTC) 5252:15:44, 16 January 2022 (UTC) 4896:00:08, 6 November 2019 (UTC) 4580:14:46, 1 February 2019 (UTC) 4552:14:35, 1 February 2019 (UTC) 4529:07:18, 1 February 2019 (UTC) 4508:05:49, 1 February 2019 (UTC) 4118:21:04, 2 November 2016 (UTC) 3827:Here is a reference to use: 2850:16:07, 16 April 2013 (UTC) 2180:21:52, 1 November 2009 (UTC) 2147:00:42, 13 October 2009 (UTC) 2004:19:29, 16 January 2008 (UTC) 1903:16:01, 26 January 2010 (UTC) 1889:09:49, 26 January 2010 (UTC) 1823:16:00, 25 January 2010 (UTC) 1806:22:42, 24 January 2010 (UTC) 1768:21:59, 24 January 2010 (UTC) 1713:04:45, 1 November 2007 (UTC) 1380:16:49, 19 January 2017 (UTC) 1273:04:31, 28 October 2006 (UTC) 1222:13:58, 31 October 2007 (UTC) 1210:03:44, 31 October 2007 (UTC) 442:08:25, 6 February 2006 (UTC) 416:19:42, 22 October 2005 (UTC) 328:06:36, 27 October 2006 (UTC) 5438:B-Class Statistics articles 4753:14:28, 12 August 2019 (UTC) 4730:10:12, 25 August 2019 (UTC) 4449:12:26, 31 August 2018 (UTC) 4423:12:25, 31 August 2018 (UTC) 4405:11:44, 31 August 2018 (UTC) 4339:11:37, 31 August 2018 (UTC) 2128:hypergeometric distribution 2082:Hypergeometric distribution 1786:)) of the binomial and the 396:04:08, 8 October 2005 (UTC) 265:1). q (1-p), maybe a typo? 5469: 4720:helps, Jens Adler Nielsen 4240:(last update: 5 June 2024) 4166:Hello fellow Wikipedians, 4081:(last update: 5 June 2024) 3997:Hello fellow Wikipedians, 3944: 3902:11:54, 26 March 2015 (UTC) 3883:22:28, 25 March 2015 (UTC) 3858:21:45, 25 March 2015 (UTC) 3842:17:04, 25 March 2015 (UTC) 2829:21:51, 15 April 2013 (UTC) 2814:21:46, 15 April 2013 (UTC) 2783:06:31, 15 March 2012 (UTC) 2052:11:43, 14 March 2008 (UTC) 1675:23:55, 31 March 2007 (UTC) 1653:20:31, 30 March 2007 (UTC) 1636:12:09, 30 March 2007 (UTC) 1601:23:34, 29 March 2007 (UTC) 1562:21:57, 29 March 2007 (UTC) 1553:20:14, 29 March 2007 (UTC) 1395:20:11, 23 March 2007 (UTC) 814:That is correct only when 428:Probability mass function? 355:17:14, 11 April 2007 (UTC) 285:2/4/12 is ambiguous since 5384:16:36, 15 July 2024 (UTC) 5180:10.1007/978-3-642-37887-4 5146:10.1007/978-3-642-37887-4 4714:14:56, 12 July 2019 (UTC) 4519:) for each output value. 4277:17:08, 20 July 2017 (UTC) 3987:03:40, 3 March 2016 (UTC) 3970:00:21, 3 March 2016 (UTC) 2686:16:31, 28 June 2011 (UTC) 2152:Mode Expression Incorrect 1698:05:38, 1 April 2007 (UTC) 1366:20:34, 9 March 2008 (UTC) 1200:14:44, 18 June 2006 (UTC) 1190:14:55, 11 June 2006 (UTC) 338:90% confident that p: --> 289:(2/4)/12= 2/48=1/24 while 234: 183: 145: 78: 57: 5280:14:01, 28 May 2022 (UTC) 2730:04:20, 3 July 2011 (UTC) 2715:02:18, 3 July 2011 (UTC) 2658:Controlling the variance 2611:18:29, 14 May 2010 (UTC) 2186:I've changed it to this: 2098:18:11, 2 July 2009 (UTC) 1960:13:20, 5 June 2007 (UTC) 1949:20:45, 31 May 2007 (UTC) 1939:01:28, 16 May 2007 (UTC) 1351:09:53, 21 May 2007 (UTC) 1341:01:41, 21 May 2007 (UTC) 1325:01:41, 21 May 2007 (UTC) 918:16:34, 10 May 2006 (UTC) 909:08:30, 10 May 2006 (UTC) 870:16:52, 5 June 2010 (UTC) 152:project's priority scale 5300:India Education Program 4162:External links modified 4157:02:31, 2 May 2017 (UTC) 4139:15:38, 1 May 2017 (UTC) 3993:External links modified 2652:10:34, 5 May 2011 (UTC) 1173:09:14, 9 May 2006 (UTC) 1143:06:11, 9 May 2006 (UTC) 831:21:29, 9 May 2006 (UTC) 790:13:04, 9 May 2006 (UTC) 779:10:43, 9 May 2006 (UTC) 626:09:14, 9 May 2006 (UTC) 472:18:27, 8 May 2006 (UTC) 461:15:12, 8 May 2006 (UTC) 109:WikiProject Mathematics 5398:B-Class vital articles 5096: 5009: 4874: 4690: 4385: 3777: 3496: 3380: 3291: 3111: 3022: 2957: 2446: 2132: 1867: 1589:) 22:59, 29 March 2007 1525: 1420:) 18:00, 29 March 2007 1119: 1027: 759: 694: 615: 548: 384:GNU Scientific Library 198:WikiProject Statistics 5296:Knowledge Ambassadors 5097: 5010: 4875: 4691: 4386: 4171:Binomial distribution 4002:Binomial distribution 3778: 3497: 3354: 3292: 3085: 3023: 2931: 2553:Normal approximations 2447: 2109: 1868: 1526: 1292:comment was added by 1239:Bernstein polynomials 1120: 1028: 760: 674: 616: 528: 345:comment was added by 36:level-4 vital article 5022: 4921: 4765: 4596: 4348: 4221:regular verification 4062:regular verification 3519: 3307: 3038: 2880: 2266:is 0 or a noninteger 2194: 1831: 1778:) and the variance ( 1658:Normal Approximation 1468: 1336:value side turn up? 1243:Binomial coefficient 1038: 942: 644: 492: 309:Confidence Interval? 132:mathematics articles 5222:. Peer reviewers: 4211:After February 2018 4052:After February 2018 4031:parameter below to 2564:· |skewness| ≥ 3.33 2126:= 0.05 (strictly a 1385:"Kitchen's theorem" 1162:Lottery_Mathematics 448:CDF Example Request 221:Statistics articles 5376:Pawel.jamiolkowski 5309:{{IEP assignment}} 5220:on the course page 5092: 5005: 4906:Fisher information 4870: 4722:Jens Adler Nielsen 4686: 4381: 4265:InternetArchiveBot 4216:InternetArchiveBot 4106:InternetArchiveBot 4057:InternetArchiveBot 3773: 3771: 3670: 3668: 3492: 3487: 3424: 3422: 3287: 3282: 3219: 3217: 3143: 3141: 3133: 3018: 2628:). Should be σ = ( 2442: 2437: 2326: 2295: 2232: 1863: 1520: 1519: 1214:Here is the diff: 1181:"nmemonic" section 1115: 1023: 755: 611: 610: 292:2/(4/12)= 24/4= 6. 101:Mathematics portal 45:content assessment 5188:978-3-642-37886-7 5154:978-3-642-37886-7 5090: 5029: 5003: 4961: 4898: 4886:comment added by 4864: 4842: 4810: 4794: 4716: 4704:comment added by 4680: 4607: 4510: 4494:comment added by 4241: 4137: 4082: 3973: 3956:comment added by 3941: 3927:comment added by 2972: 2856:comment added by 2773:comment added by 2676:What do you say? 2642:comment added by 2549: 2535:comment added by 2483:comment added by 2403: 2344: 2305: 2267: 2244: 2200: 2170:comment added by 2033: 2021:comment added by 2006: 1990:comment added by 1758:comment added by 1591: 1577:comment added by 1422: 1408:comment added by 1305: 1141: 1139:(call me collect) 777: 775:(call me collect) 709: 563: 358: 255: 254: 251: 250: 247: 246: 162: 161: 158: 157: 5460: 5324: 5311: 5310: 5293: 5254: 5217: 5216:19 December 2021 5213: 5209: 5193: 5192: 5165: 5159: 5158: 5131: 5101: 5099: 5098: 5093: 5091: 5089: 5066: 5049: 5048: 5036: 5035: 5030: 5027: 5014: 5012: 5011: 5006: 5004: 5002: 5001: 5000: 4978: 4967: 4962: 4960: 4959: 4947: 4933: 4932: 4879: 4877: 4876: 4871: 4869: 4865: 4860: 4859: 4858: 4843: 4835: 4823: 4811: 4803: 4795: 4787: 4695: 4693: 4692: 4687: 4685: 4681: 4673: 4661: 4657: 4618: 4617: 4608: 4600: 4390: 4388: 4387: 4382: 4282:Lead readability 4275: 4266: 4239: 4238: 4217: 4131: 4129:Nicolas Perrault 4116: 4107: 4080: 4079: 4058: 4046: 3972: 3950: 3940: 3921: 3782: 3780: 3779: 3774: 3772: 3768: 3755: 3739: 3738: 3737: 3716: 3639: 3614: 3613: 3612: 3590: 3589: 3588: 3567: 3554: 3538: 3537: 3536: 3501: 3499: 3498: 3493: 3491: 3488: 3484: 3464: 3393: 3379: 3368: 3348: 3296: 3294: 3293: 3288: 3286: 3283: 3279: 3259: 3188: 3187: 3186: 3185: 3174: 3155: 3154: 3153: 3139: 3138: 3134: 3110: 3099: 3079: 3027: 3025: 3024: 3019: 3017: 3016: 2989: 2988: 2979: 2978: 2977: 2964: 2956: 2945: 2865: 2785: 2739:Error in article 2711: 2706: 2701: 2654: 2594: 2590: 2587: 2548: 2529: 2495: 2451: 2449: 2448: 2443: 2441: 2440: 2404: 2401: 2345: 2342: 2306: 2303: 2268: 2265: 2245: 2242: 2201: 2198: 2182: 2080:The article for 2016: 1985: 1887: 1883: 1880: 1872: 1870: 1869: 1864: 1859: 1858: 1840: 1839: 1804: 1800: 1797: 1770: 1590: 1571: 1531: 1528: 1527: 1521: 1517: 1513: 1492: 1488: 1421: 1402: 1287: 1137: 1132: 1124: 1122: 1121: 1116: 1087: 1086: 1059: 1058: 1032: 1030: 1029: 1024: 773: 764: 762: 761: 756: 754: 753: 726: 725: 716: 715: 714: 701: 693: 688: 620: 618: 617: 612: 608: 607: 580: 579: 570: 569: 568: 555: 547: 542: 340: 241:importance scale 223: 222: 219: 216: 213: 192: 185: 184: 179: 171: 164: 134: 133: 130: 127: 124: 103: 98: 97: 87: 80: 79: 74: 66: 59: 42: 33: 32: 25: 24: 16: 5468: 5467: 5463: 5462: 5461: 5459: 5458: 5457: 5388: 5387: 5349: 5330: 5308: 5307: 5305: 5287: 5272:Universemaster1 5260: 5237: 5203: 5198: 5197: 5196: 5189: 5167: 5166: 5162: 5155: 5133: 5132: 5128: 5070: 5040: 5025: 5020: 5019: 4992: 4979: 4968: 4951: 4924: 4919: 4918: 4908: 4850: 4824: 4818: 4763: 4762: 4737: 4668: 4626: 4622: 4609: 4594: 4593: 4590: 4471:from 0 through 4461: 4346: 4345: 4323: 4284: 4269: 4264: 4232: 4225:have permission 4215: 4179:this simple FaQ 4164: 4125: 4110: 4105: 4073: 4066:have permission 4056: 4040: 4010:this simple FaQ 3995: 3958:Andreas Mueller 3951: 3947: 3922: 3917: 3825: 3806: 3770: 3769: 3756: 3723: 3718: 3717: 3640: 3604: 3574: 3569: 3568: 3555: 3528: 3517: 3516: 3486: 3485: 3472: 3466: 3465: 3394: 3350: 3349: 3336: 3325: 3305: 3304: 3281: 3280: 3267: 3261: 3260: 3189: 3157: 3145: 3132: 3131: 3125: 3124: 3113: 3081: 3080: 3067: 3056: 3036: 3035: 3002: 2980: 2959: 2878: 2877: 2871: 2851: 2848: 2842: 2803: 2797: 2791: 2768: 2741: 2709: 2704: 2699: 2693: 2660: 2644:213.197.179.210 2637: 2588: 2585: 2581: 2555: 2530: 2525: 2478: 2475: 2436: 2435: 2398: 2392: 2391: 2339: 2273: 2272: 2239: 2206: 2192: 2191: 2165: 2154: 2113:random variable 2105: 2090:Virgil H. Soule 2078: 2059: 2039: 2012: 1982: 1968: 1919: 1881: 1878: 1874: 1850: 1829: 1828: 1798: 1795: 1791: 1790:distributions. 1753: 1660: 1572: 1503: 1499: 1478: 1474: 1465: 1464: 1403: 1387: 1288:—The preceding 1281: 1259:130.237.179.166 1229: 1183: 1131:BetaRegularized 1130: 1078: 1050: 1036: 1035: 940: 939: 739: 717: 696: 642: 641: 593: 571: 550: 490: 489: 450: 430: 404: 394: 375: 341:—The preceding 311: 260: 220: 217: 214: 211: 210: 177: 131: 128: 125: 122: 121: 99: 92: 72: 43:on Knowledge's 40: 30: 12: 11: 5: 5466: 5464: 5456: 5455: 5450: 5445: 5440: 5435: 5430: 5425: 5420: 5415: 5410: 5405: 5400: 5390: 5389: 5348: 5347:Interpretation 5345: 5329: 5326: 5286: 5283: 5259: 5256: 5212:27 August 2021 5202: 5199: 5195: 5194: 5187: 5160: 5153: 5125: 5124: 5120: 5088: 5085: 5082: 5079: 5076: 5073: 5069: 5064: 5061: 5058: 5055: 5052: 5047: 5043: 5039: 5034: 5017: 4999: 4995: 4991: 4988: 4985: 4982: 4977: 4974: 4971: 4965: 4958: 4954: 4950: 4945: 4942: 4939: 4936: 4931: 4927: 4907: 4900: 4888:176.150.242.62 4868: 4863: 4857: 4853: 4849: 4846: 4841: 4838: 4833: 4830: 4827: 4821: 4817: 4814: 4809: 4806: 4801: 4798: 4793: 4790: 4785: 4782: 4779: 4776: 4773: 4770: 4736: 4733: 4696: 4684: 4679: 4676: 4671: 4667: 4664: 4660: 4656: 4653: 4650: 4647: 4644: 4641: 4638: 4635: 4632: 4629: 4625: 4621: 4616: 4612: 4606: 4603: 4589: 4586: 4585: 4584: 4583: 4582: 4557: 4556: 4555: 4554: 4532: 4531: 4481: 4480: 4460: 4457: 4456: 4455: 4454: 4453: 4452: 4451: 4428: 4427: 4426: 4425: 4415:StephenJohns00 4408: 4407: 4380: 4377: 4374: 4371: 4368: 4365: 4362: 4359: 4356: 4353: 4331:StephenJohns00 4322: 4319: 4283: 4280: 4259: 4258: 4251: 4204: 4203: 4195:Added archive 4193: 4185:Added archive 4163: 4160: 4124: 4121: 4100: 4099: 4092: 4025: 4024: 4016:Added archive 3994: 3991: 3990: 3989: 3946: 3943: 3916: 3913: 3911: 3909: 3908: 3907: 3906: 3905: 3904: 3871: 3870: 3869: 3868: 3861: 3860: 3824: 3821: 3810:71.139.165.140 3805: 3802: 3784: 3783: 3767: 3764: 3761: 3757: 3754: 3751: 3748: 3745: 3742: 3736: 3733: 3730: 3726: 3720: 3719: 3715: 3712: 3709: 3706: 3703: 3700: 3697: 3694: 3691: 3688: 3685: 3682: 3679: 3676: 3673: 3666: 3663: 3660: 3657: 3654: 3651: 3648: 3645: 3641: 3638: 3635: 3632: 3629: 3626: 3623: 3620: 3617: 3611: 3607: 3602: 3599: 3596: 3593: 3587: 3584: 3581: 3577: 3571: 3570: 3566: 3563: 3560: 3556: 3553: 3550: 3547: 3544: 3541: 3535: 3531: 3525: 3524: 3503: 3502: 3490: 3483: 3480: 3477: 3473: 3471: 3468: 3467: 3463: 3460: 3457: 3454: 3451: 3448: 3445: 3442: 3439: 3436: 3433: 3430: 3427: 3420: 3417: 3414: 3411: 3408: 3405: 3402: 3399: 3395: 3392: 3389: 3386: 3383: 3378: 3375: 3372: 3367: 3364: 3361: 3357: 3352: 3351: 3347: 3344: 3341: 3337: 3335: 3332: 3331: 3328: 3324: 3321: 3318: 3315: 3312: 3298: 3297: 3285: 3278: 3275: 3272: 3268: 3266: 3263: 3262: 3258: 3255: 3252: 3249: 3246: 3243: 3240: 3237: 3234: 3231: 3228: 3225: 3222: 3215: 3212: 3209: 3206: 3203: 3200: 3197: 3194: 3190: 3184: 3181: 3178: 3173: 3170: 3167: 3164: 3161: 3152: 3148: 3137: 3130: 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2260: 2257: 2254: 2251: 2248: 2240: 2238: 2235: 2230: 2227: 2224: 2221: 2218: 2215: 2212: 2211: 2209: 2204: 2187: 2172:86.165.211.190 2153: 2150: 2104: 2101: 2077: 2074: 2063:209.94.128.119 2058: 2055: 2038: 2035: 2011: 2008: 1981: 1978: 1967: 1964: 1963: 1962: 1918: 1915: 1914: 1913: 1912: 1911: 1910: 1909: 1908: 1907: 1906: 1905: 1862: 1857: 1853: 1849: 1846: 1843: 1838: 1750: 1749: 1748: 1747: 1701: 1700: 1672:128.195.106.28 1668: 1667: 1659: 1656: 1646: 1645: 1639: 1638: 1624: 1623: 1618: 1617: 1604: 1603: 1567: 1566: 1565: 1564: 1540: 1539: 1538: 1537: 1536: 1534: 1533: 1532: 1516: 1512: 1509: 1506: 1502: 1498: 1495: 1491: 1487: 1484: 1481: 1477: 1473: 1460: 1399: 1386: 1383: 1358:128.135.96.223 1354: 1353: 1328: 1327: 1317: 1316: 1280: 1279:Better Example 1277: 1276: 1275: 1228: 1225: 1203: 1202: 1182: 1179: 1178: 1177: 1176: 1175: 1153:load(distrib); 1146: 1145: 1127: 1126: 1125: 1114: 1111: 1108: 1105: 1102: 1099: 1096: 1093: 1090: 1085: 1081: 1077: 1074: 1071: 1068: 1065: 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249: 248: 245: 244: 237:Top-importance 233: 227: 226: 224: 207:the discussion 193: 181: 180: 178:Top‑importance 172: 160: 159: 156: 155: 144: 138: 137: 135: 118:the discussion 105: 104: 88: 76: 75: 67: 55: 54: 48: 26: 13: 10: 9: 6: 4: 3: 2: 5465: 5454: 5451: 5449: 5446: 5444: 5441: 5439: 5436: 5434: 5431: 5429: 5426: 5424: 5421: 5419: 5416: 5414: 5411: 5409: 5406: 5404: 5401: 5399: 5396: 5395: 5393: 5386: 5385: 5381: 5377: 5371: 5370: 5365: 5364: 5360: 5357: 5356: 5352: 5346: 5344: 5343: 5339: 5335: 5327: 5325: 5323: 5319: 5315: 5303: 5301: 5297: 5292: 5284: 5282: 5281: 5277: 5273: 5269: 5265: 5257: 5255: 5253: 5249: 5245: 5241: 5235: 5233: 5229: 5225: 5221: 5208: 5200: 5190: 5185: 5181: 5177: 5173: 5172: 5164: 5161: 5156: 5151: 5147: 5143: 5139: 5138: 5130: 5127: 5123: 5119: 5118: 5114: 5110: 5106: 5103: 5083: 5080: 5077: 5071: 5067: 5062: 5053: 5045: 5041: 5032: 5015: 4997: 4989: 4986: 4983: 4975: 4972: 4969: 4963: 4956: 4952: 4948: 4943: 4937: 4929: 4925: 4915: 4913: 4905: 4901: 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2776: 2772: 2766: 2762: 2758: 2754: 2750: 2746: 2738: 2736: 2731: 2727: 2723: 2719: 2718: 2717: 2716: 2713: 2712: 2707: 2702: 2691:Accessibility 2690: 2688: 2687: 2683: 2679: 2673: 2670: 2668: 2663: 2657: 2655: 2653: 2649: 2645: 2641: 2635: 2631: 2627: 2623: 2619: 2612: 2608: 2604: 2600: 2599: 2598: 2597: 2592: 2591: 2579: 2575: 2571: 2563: 2560: 2559: 2558: 2552: 2550: 2546: 2542: 2538: 2534: 2522: 2516: 2512: 2508: 2504: 2500: 2499: 2498: 2497: 2496: 2494: 2490: 2486: 2482: 2472: 2470: 2469: 2465: 2461: 2460:Michael Hardy 2455: 2432: 2429: 2426: 2423: 2420: 2414: 2411: 2408: 2395: 2388: 2382: 2379: 2376: 2373: 2370: 2364: 2361: 2355: 2352: 2349: 2336: 2333: 2327: 2319: 2316: 2313: 2296: 2288: 2285: 2282: 2269: 2261: 2255: 2252: 2249: 2233: 2225: 2222: 2219: 2207: 2202: 2190: 2189: 2188: 2185: 2184: 2183: 2181: 2177: 2173: 2169: 2163: 2159: 2151: 2149: 2148: 2144: 2140: 2135: 2131: 2129: 2125: 2121: 2117: 2114: 2108: 2102: 2100: 2099: 2095: 2091: 2087: 2083: 2075: 2073: 2072: 2068: 2064: 2056: 2054: 2053: 2049: 2045: 2036: 2034: 2032: 2028: 2024: 2023:68.50.194.132 2020: 2009: 2007: 2005: 2001: 1997: 1993: 1989: 1979: 1977: 1976: 1973: 1966:Incorrect cdf 1965: 1961: 1958: 1953: 1952: 1951: 1950: 1947: 1941: 1940: 1937: 1933: 1929: 1925: 1916: 1904: 1900: 1896: 1892: 1891: 1890: 1885: 1884: 1855: 1851: 1847: 1844: 1826: 1825: 1824: 1820: 1816: 1812: 1811: 1809: 1808: 1807: 1802: 1801: 1789: 1785: 1781: 1777: 1773: 1772: 1771: 1769: 1765: 1761: 1757: 1746: 1742: 1738: 1734: 1733: 1732: 1731: 1730: 1729: 1725: 1721: 1715: 1714: 1711: 1707: 1704:worth noting 1699: 1696: 1692: 1688: 1683: 1679: 1678: 1677: 1676: 1673: 1665: 1664: 1663: 1657: 1655: 1654: 1651: 1650:Michael Hardy 1644: 1643: 1642: 1637: 1634: 1630: 1626: 1625: 1620: 1619: 1615: 1610: 1609: 1608: 1602: 1599: 1598:Michael Hardy 1594: 1593: 1592: 1588: 1584: 1580: 1576: 1563: 1560: 1559:Michael Hardy 1556: 1555: 1554: 1551: 1550:Michael Hardy 1546: 1541: 1535: 1514: 1510: 1507: 1504: 1500: 1496: 1493: 1489: 1485: 1482: 1479: 1475: 1471: 1463: 1462: 1461: 1458: 1454: 1450: 1446: 1442: 1438: 1434: 1433: 1432: 1431: 1429: 1425: 1424: 1423: 1419: 1415: 1411: 1407: 1397: 1396: 1393: 1392:Michael Hardy 1382: 1381: 1377: 1373: 1368: 1367: 1363: 1359: 1352: 1349: 1345: 1344: 1343: 1342: 1339: 1334: 1326: 1323: 1319: 1318: 1315: 1312: 1308: 1307: 1306: 1303: 1299: 1295: 1291: 1285: 1278: 1274: 1271: 1266: 1265: 1264: 1263: 1260: 1255: 1250: 1248: 1244: 1240: 1235: 1234: 1226: 1224: 1223: 1220: 1216: 1212: 1211: 1208: 1201: 1198: 1194: 1193: 1192: 1191: 1188: 1180: 1174: 1171: 1167: 1163: 1158: 1154: 1150: 1149: 1148: 1147: 1144: 1140: 1136: 1128: 1112: 1106: 1103: 1097: 1094: 1091: 1083: 1079: 1075: 1069: 1066: 1063: 1055: 1051: 1047: 1044: 1041: 1034: 1017: 1014: 1011: 1008: 1005: 999: 996: 993: 990: 984: 981: 978: 969: 966: 963: 957: 954: 951: 938: 937: 935: 934: 919: 916: 912: 910: 907: 903: 899: 898: 897: 896: 895: 894: 893: 892: 891: 890: 889: 888: 887: 886: 871: 867: 863: 862:Zane Dylanger 858: 857: 856: 855: 854: 853: 852: 851: 850: 849: 848: 847: 846: 845: 832: 829: 828:Michael Hardy 825: 821: 817: 813: 812: 811: 810: 809: 808: 807: 806: 805: 804: 803: 802: 791: 788: 784: 783: 782: 781: 780: 776: 772: 768: 750: 747: 744: 736: 733: 730: 722: 718: 706: 703: 690: 685: 682: 679: 675: 671: 665: 662: 659: 656: 653: 647: 640: 639: 637: 636: 635: 634: 633: 632: 627: 624: 604: 601: 598: 590: 587: 584: 576: 572: 560: 557: 544: 539: 536: 533: 529: 525: 519: 516: 513: 510: 507: 501: 498: 495: 488: 487: 486: 485: 479: 478: 477: 476: 473: 470: 469:Michael Hardy 465: 464: 463: 462: 459: 454: 447: 443: 440: 439:Richard Clegg 436: 435: 434: 427: 421: 420: 417: 414: 413:Michael Hardy 410: 409: 408: 402:HIV positive? 401: 397: 393: 389: 385: 381: 380: 379: 372: 364: 363: 362: 361: 356: 352: 348: 347:64.122.234.42 344: 333: 332: 329: 326: 321: 320: 319: 315: 308: 306: 305: 298: 291: 288: 287: 286: 282: 280: 273: 270: 266: 263: 257: 242: 238: 232: 229: 228: 225: 208: 204: 200: 199: 194: 191: 187: 186: 182: 176: 173: 170: 166: 153: 149: 143: 140: 139: 136: 119: 115: 111: 110: 102: 96: 91: 89: 86: 82: 81: 77: 71: 68: 65: 61: 56: 52: 46: 38: 37: 27: 23: 18: 17: 5372: 5368: 5366: 5362: 5361: 5358: 5354: 5353: 5350: 5331: 5304: 5298:through the 5288: 5267: 5263: 5261: 5236: 5204: 5170: 5163: 5136: 5129: 5121: 5107: 5104: 5016: 4916: 4911: 4909: 4903: 4882:— Preceding 4758: 4756: 4745:69.119.31.14 4743: 4738: 4718: 4700:— Preceding 4591: 4564:alias method 4516: 4490:— Preceding 4484: 4482: 4472: 4468: 4462: 4327:divided by n 4326: 4324: 4300: 4285: 4263: 4260: 4235:source check 4214: 4208: 4205: 4168: 4165: 4144: 4142: 4126: 4104: 4101: 4076:source check 4055: 4049: 4036: 4032: 4028: 4026: 3999: 3996: 3952:— Preceding 3948: 3923:— Preceding 3918: 3910: 3872: 3832: 3826: 3807: 3789: 3785: 3511: 3507: 3504: 3299: 3030: 2872: 2858:72.43.218.26 2852:— Preceding 2849: 2843: 2837: 2833: 2806:Docsteve.518 2804: 2798: 2792: 2769:— Preceding 2764: 2760: 2756: 2752: 2748: 2744: 2742: 2734: 2697: 2694: 2674: 2671: 2664: 2661: 2633: 2629: 2625: 2621: 2617: 2615: 2583: 2577: 2573: 2569: 2567: 2561: 2556: 2526: 2502: 2476: 2458: 2161: 2157: 2155: 2136: 2133: 2123: 2119: 2115: 2110: 2106: 2085: 2079: 2060: 2040: 2013: 2010:bad language 1983: 1969: 1942: 1927: 1920: 1876: 1793: 1783: 1779: 1775: 1760:24.29.95.138 1751: 1720:129.13.186.1 1716: 1705: 1702: 1690: 1686: 1681: 1669: 1661: 1647: 1640: 1613: 1605: 1573:— Preceding 1568: 1544: 1459:. Show that 1456: 1452: 1448: 1444: 1440: 1436: 1427: 1404:— Preceding 1398: 1388: 1369: 1355: 1332: 1329: 1294:18.216.0.100 1286: 1282: 1253: 1251: 1247:distribution 1246: 1236: 1232: 1230: 1213: 1204: 1184: 1165: 1156: 1155:followed by 1152: 901: 823: 819: 815: 455: 451: 431: 405: 376: 316: 312: 300: 295: 283: 275: 271: 267: 264: 261: 236: 196: 148:Mid-priority 147: 107: 73:Mid‑priority 51:WikiProjects 34: 4143:Well... it 4043:Sourcecheck 3300:or indeed: 2799:and I get 2667:development 2638:—Preceding 2531:—Preceding 2479:—Preceding 2473:Paramater n 2166:—Preceding 2103:Bad example 2057:Bad Figures 2017:—Preceding 1986:—Preceding 1980:derivations 1936:Gerald Tros 1932:Gerald Tros 1924:Gerald Tros 1754:—Preceding 1689:-test, the 1579:84.66.3.105 1410:84.66.3.105 1338:Gerald Tros 1322:Gerald Tros 1170:New Thought 915:New Thought 906:New Thought 787:New Thought 623:New Thought 458:New Thought 373:Simulation? 304:Dick Beldin 279:Dick Beldin 123:Mathematics 114:mathematics 70:Mathematics 5392:Categories 5334:OveGjerlow 5228:C.Hua Wang 5122:References 4272:Report bug 4113:Report bug 3929:Kjetil1001 3875:Tal Galili 3834:Tal Galili 2722:Alzarian16 2603:12.7.202.2 2160:= 1.0 and 2122:= 100 and 1930:ready :-) 1447:, and let 1333:unfamiliar 1157:functions; 212:Statistics 203:statistics 175:Statistics 5232:Jiang1725 5224:Ziyanggod 4496:TheKing44 4255:this tool 4248:this tool 4123:Spoilers! 4096:this tool 4089:this tool 3791:Lfahlberg 2580:omitted? 2501:The case 2107:Removed: 1992:Student29 1957:Sander123 1710:4.79.81.6 1633:Sander123 1348:Sander123 1219:Sander123 1135:MarkSweep 771:MarkSweep 388:MarkSweep 39:is rated 5314:PrimeBOT 5268:binomial 5264:Binomial 5244:PrimeBOT 4912:expected 4904:expected 4884:unsigned 4702:unsigned 4504:contribs 4492:unsigned 4261:Cheers.— 4102:Cheers.— 3966:contribs 3954:unsigned 3937:contribs 3925:unsigned 3873:Cheers, 3804:Graph, n 2854:unsigned 2821:Melcombe 2771:unsigned 2678:Ofermano 2662:Hi all, 2640:unsigned 2601:Fixed. - 2545:contribs 2533:unsigned 2507:Melcombe 2481:unsigned 2168:unsigned 2139:Madkaugh 2076:Sampling 2044:Telliott 2037:Variance 2019:unsigned 2000:contribs 1988:unsigned 1756:unsigned 1737:Melcombe 1587:contribs 1575:unsigned 1418:contribs 1406:unsigned 1302:contribs 1290:unsigned 1197:Rjmorris 343:unsigned 4740:=k: --> 4588:Entropy 4175:my edit 4029:checked 4006:my edit 3977:fixed. 3894:Blahb31 3850:Blahb31 1166:exactly 239:on the 150:on the 41:B-class 5109:Ddreif 4761:bound 4576:videos 4572:carbon 4548:videos 4544:carbon 4445:videos 4441:carbon 4401:videos 4397:carbon 4308:Zaheen 4289:Zaheen 4149:Bosons 4037:failed 1946:Wrayal 1928:almost 1788:normal 1629:wp:nor 1622:paper. 1372:Bosons 1254:chosen 1107:17.647 339:=95%. 47:scale. 5320:) on 4759:lower 4521:McKay 3979:McKay 2759:(1 − 2747:(1 − 2632:(1 − 2620:(1 − 2589:pasha 2572:when 1882:pasha 1799:pasha 1695:McKay 1614:prove 1483:: --> 1426:From 1311:McKay 1270:McKay 1207:Tabby 1187:McKay 325:McKay 28:This 5380:talk 5338:talk 5318:talk 5276:talk 5248:talk 5214:and 5184:ISBN 5150:ISBN 5113:talk 4892:talk 4749:talk 4726:talk 4710:talk 4525:talk 4500:talk 4419:talk 4335:talk 4312:talk 4301:The 4293:talk 4153:talk 4134:talk 4033:true 3983:talk 3962:talk 3933:talk 3898:talk 3879:talk 3867:fit? 3854:talk 3838:talk 3814:talk 3795:talk 3653:< 3562:< 3413:< 3343:< 3208:< 3074:< 2862:talk 2825:talk 2810:talk 2779:talk 2767:)) 2726:talk 2682:talk 2648:talk 2607:talk 2541:talk 2537:ATBS 2511:talk 2489:talk 2464:talk 2304:and 2199:mode 2176:talk 2143:talk 2094:talk 2086:with 2067:talk 2048:talk 2027:talk 1996:talk 1899:talk 1819:talk 1764:talk 1741:talk 1724:talk 1583:talk 1508:< 1455:and 1443:and 1435:Let 1414:talk 1376:talk 1362:talk 1298:talk 1084:0.05 1056:0.95 1018:0.05 866:talk 386:. -- 351:talk 5312:by 5302:. 5242:by 5176:doi 5142:doi 4880:. 4813:exp 4611:log 4229:RfC 4199:to 4189:to 4070:RfC 4047:). 4035:or 4020:to 2710:466 2402:if 2343:if 2243:if 1972:ПБХ 1895:PAR 1815:PAR 1782:(1− 1706:why 1545:new 1304:) . 1249:. 1098:471 1064:471 1012:500 231:Top 142:Mid 5394:: 5382:) 5340:) 5278:) 5270:. 5250:) 5234:. 5230:, 5226:, 5182:. 5148:. 5115:) 5081:− 4987:− 4973:− 4894:) 4845:− 4829:16 4826:− 4816:⁡ 4808:15 4800:≥ 4751:) 4728:) 4712:) 4649:− 4631:π 4620:⁡ 4578:) 4574:• 4550:) 4546:• 4527:) 4506:) 4502:• 4447:) 4443:• 4421:) 4403:) 4399:• 4367:… 4337:) 4314:) 4295:) 4242:. 4237:}} 4233:{{ 4155:) 4145:is 4083:. 4078:}} 4074:{{ 4045:}} 4041:{{ 3985:) 3968:) 3964:• 3939:) 3935:• 3900:) 3881:) 3856:) 3840:) 3816:) 3797:) 3763:≥ 3711:− 3647:≤ 3479:≥ 3429:∈ 3407:≤ 3401:− 3374:− 3356:∑ 3274:≥ 3224:∈ 3202:≤ 3196:− 3180:− 3166:− 3105:− 3087:∑ 3011:− 2997:− 2954:⌋ 2948:⌊ 2933:∑ 2920:≤ 2911:Pr 2864:) 2827:) 2812:) 2781:) 2763:)/ 2751:) 2728:) 2696:-- 2684:) 2650:) 2630:np 2624:)/ 2609:) 2593:» 2586:st 2582:… 2547:) 2543:• 2513:) 2491:) 2466:) 2433:1. 2377:… 2365:∈ 2334:− 2331:⌋ 2308:⌊ 2300:⌋ 2277:⌊ 2237:⌋ 2214:⌊ 2178:) 2145:) 2130:). 2096:) 2069:) 2050:) 2029:) 2002:) 1998:• 1901:) 1886:» 1879:st 1875:… 1852:σ 1845:μ 1821:) 1803:» 1796:st 1792:… 1780:np 1776:np 1766:) 1743:) 1726:) 1585:• 1416:• 1378:) 1364:) 1300:• 1110:% 1104:≈ 1092:30 1070:30 1048:− 1006:29 997:− 985:29 982:≤ 973:Pr 970:− 958:30 955:≥ 946:Pr 868:) 826:. 822:≤ 769:-- 748:− 734:− 676:∑ 621:-- 602:− 588:− 530:∑ 353:) 5378:( 5336:( 5316:( 5274:( 5246:( 5191:. 5178:: 5157:. 5144:: 5111:( 5087:) 5084:p 5078:1 5075:( 5072:p 5068:n 5063:= 5060:] 5057:) 5054:p 5051:( 5046:n 5042:g 5038:[ 5033:X 5028:E 4998:2 4994:) 4990:p 4984:1 4981:( 4976:x 4970:n 4964:+ 4957:2 4953:p 4949:x 4944:= 4941:) 4938:p 4935:( 4930:n 4926:g 4890:( 4867:) 4862:n 4856:2 4852:) 4848:k 4840:2 4837:n 4832:( 4820:( 4805:1 4797:) 4792:2 4789:1 4784:, 4781:n 4778:; 4775:k 4772:( 4769:F 4747:( 4724:( 4708:( 4683:) 4678:n 4675:1 4670:( 4666:O 4663:+ 4659:) 4655:) 4652:p 4646:1 4643:( 4640:p 4637:n 4634:e 4628:2 4624:( 4615:2 4605:2 4602:1 4570:( 4542:( 4538:– 4523:( 4517:n 4498:( 4485:n 4473:n 4469:k 4439:( 4417:( 4395:( 4379:. 4376:} 4373:n 4370:, 4364:, 4361:1 4358:, 4355:0 4352:{ 4333:( 4310:( 4291:( 4274:) 4270:( 4257:. 4250:. 4151:( 4136:) 4132:( 4115:) 4111:( 4098:. 4091:. 3981:( 3960:( 3931:( 3896:( 3877:( 3852:( 3836:( 3812:( 3793:( 3766:n 3760:x 3753:1 3750:= 3747:) 3744:x 3741:( 3735:1 3732:+ 3729:n 3725:F 3714:1 3708:n 3705:, 3702:. 3699:. 3696:. 3693:, 3690:2 3687:, 3684:1 3681:, 3678:0 3675:= 3672:k 3665:, 3662:1 3659:+ 3656:k 3650:x 3644:k 3637:) 3634:k 3631:( 3628:f 3625:+ 3622:) 3619:x 3616:( 3610:k 3606:F 3601:= 3598:) 3595:x 3592:( 3586:1 3583:+ 3580:k 3576:F 3565:0 3559:x 3552:0 3549:= 3546:) 3543:x 3540:( 3534:0 3530:F 3482:n 3476:x 3470:1 3462:} 3459:n 3456:, 3453:. 3450:. 3447:. 3444:, 3441:2 3438:, 3435:1 3432:{ 3426:k 3419:, 3416:k 3410:x 3404:1 3398:k 3391:) 3388:j 3385:( 3382:f 3377:1 3371:k 3366:0 3363:= 3360:j 3346:0 3340:x 3334:0 3327:{ 3323:= 3320:) 3317:x 3314:( 3311:F 3277:n 3271:x 3265:1 3257:} 3254:n 3251:, 3248:. 3245:. 3242:. 3239:, 3236:2 3233:, 3230:1 3227:{ 3221:k 3214:, 3211:k 3205:x 3199:1 3193:k 3183:k 3177:n 3172:) 3169:p 3163:1 3160:( 3151:k 3147:p 3136:) 3129:k 3122:n 3115:( 3108:1 3102:k 3097:0 3094:= 3091:j 3077:0 3071:x 3065:0 3058:{ 3054:= 3051:) 3048:x 3045:( 3042:F 3014:i 3008:n 3004:) 3000:p 2994:1 2991:( 2986:i 2982:p 2975:) 2970:i 2967:n 2962:( 2951:k 2943:0 2940:= 2937:i 2929:= 2926:) 2923:k 2917:X 2914:( 2908:= 2905:) 2902:p 2899:, 2896:n 2893:; 2890:k 2887:( 2884:F 2860:( 2823:( 2808:( 2777:( 2765:n 2761:p 2757:p 2753:n 2749:p 2745:p 2724:( 2705:N 2700:J 2680:( 2646:( 2634:p 2626:n 2622:p 2618:p 2605:( 2578:n 2574:p 2570:n 2562:n 2539:( 2509:( 2503:n 2487:( 2462:( 2430:+ 2427:n 2424:= 2421:p 2418:) 2415:1 2412:+ 2409:n 2406:( 2396:n 2389:, 2386:} 2383:n 2380:, 2374:, 2371:1 2368:{ 2362:p 2359:) 2356:1 2353:+ 2350:n 2347:( 2337:1 2328:p 2323:) 2320:1 2317:+ 2314:n 2311:( 2297:p 2292:) 2289:1 2286:+ 2283:n 2280:( 2270:, 2262:p 2259:) 2256:1 2253:+ 2250:n 2247:( 2234:p 2229:) 2226:1 2223:+ 2220:n 2217:( 2208:{ 2203:= 2174:( 2162:n 2158:p 2141:( 2124:p 2120:n 2116:X 2092:( 2065:( 2046:( 2025:( 1994:( 1897:( 1861:) 1856:2 1848:, 1842:( 1837:N 1817:( 1784:p 1762:( 1739:( 1722:( 1691:F 1687:t 1682:n 1581:( 1515:} 1511:r 1505:Y 1501:{ 1497:P 1494:= 1490:} 1486:n 1480:X 1476:{ 1472:P 1457:p 1453:n 1449:Y 1445:p 1441:r 1437:X 1412:( 1374:( 1360:( 1296:( 1113:. 1101:) 1095:, 1089:( 1080:I 1076:= 1073:) 1067:, 1061:( 1052:I 1045:1 1042:= 1021:) 1015:, 1009:; 1003:( 1000:F 994:1 991:= 988:] 979:X 976:[ 967:1 964:= 961:] 952:X 949:[ 902:k 864:( 824:n 820:k 816:k 751:j 745:n 741:) 737:p 731:1 728:( 723:j 719:p 712:) 707:j 704:n 699:( 691:k 686:0 683:= 680:j 672:= 669:) 666:p 663:, 660:n 657:; 654:k 651:( 648:F 605:k 599:n 595:) 591:p 585:1 582:( 577:k 573:p 566:) 561:k 558:n 553:( 545:n 540:1 537:= 534:k 526:= 523:) 520:p 517:, 514:n 511:; 508:k 505:( 502:f 499:d 496:c 392:✍ 357:. 349:( 243:. 154:. 53::

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