Talk:Probability density function

2526:

then it can/should certainly be included. Just because you don't like the term is no reason for deleting it. Reverting the deletion was a quick answer to an ill-considered edit, in exactly the same spirit as the deletion. Of course discussion of the term could be moved later on and given proper context if necessary. On following the citation given, I see that it does not actually seem to lead to anything directly related to equivalence to ""probability density function". As to established usage, the Oxford Dicionary of Statistical Terms says: "It is customary , but not the universal practice, to use 'probability distribution' to denote the probability mass or probability density of either discontinuous or continuous variable and some such expression as 'cumulative probability distribution' to denote the probability of values up to and including the argument x."

2711:

result value is between zero and one, depending of the integration limits of the random variable).By the shape of f(x) we see values of the random variable wich have more density of probability than others , but the probability to any point (any value of the random variable) is always zero in the continuous pdf. Proof is the integration with limits the same point. Probability is the integral of the pdf. Thinking,as example, in the Normal pdf, N(0,1) of a random variable in meters, at value zero meters have 0.4/meter of density and at value one meter have 0.243/meter of density. This is 1.65 (=0.4/0.243) more density at the value zero meters than at the value one meter of that random variable, but the Probability at zero meter is equal to the Probability at one meter, equal to Zero (dimensionless). Rferreirapt ,as I answered above.

1429:

units(wich integrated result value is between zero and one, depending of the integration limits of the random variable).By the shape of f(x) we see values of the random variable wich have more density of probability than others , but the probability to any point (any value of the random variable) is always zero in the continuous pdf. Proof is the integration with limits the same point. Probability is the integral of the pdf. Thinking in the Normal pdf, N(0,1) of a random variable in meters, at value zero meters have 0.4/meter of density and at value one meter have 0.243/meter of density. This is o.4/0.243=1.65 more density at the value zero meters than at the value one meter of that random variable, but the Probability at zero meter is equal to the Probability at one meter, equal to Zero (dimensionless). Rferreira1204 .

1137:) is in seconds. Multiply them and get meters. That's not just syntactical delimiting. Moreover, the "proper definitions" were obviously not what Leibniz had in mind when he introduced this notation in the 17th century. The intuitive explanation given is in line with the way Leibniz did it. And it is useful. Was Leibniz "making up an interpretation distinct from the actual definition", when in fact the "actual definition" came two centuries later in the 1800s? Suprahili, have you ever heard that the world existed before the 21st century? 85: 64: 184: 174: 153: 31: 3305:, which is similar in topic, uses the term (and defines) "generalized probability density function". No doubt there are others ... the notion has been around since the 70's at least, often as a way of treating characteristic functions via a single simple formula that looks like a simple Fourier transform. Also, in the case of the differential equations for the pdf of a stochastic diffusion equation, the intial condition can be conveniently represented as a delta function. 1391:, then it can take many values. But, most of the time those values are not equally likely, some of them occur more often than others. So, if a value of this variable is more likely, the density of that variable is higher at that value. If certain value does not occur at all, the density at that value is zero. This explanation is not at all rigurious, but it might drive the point home. 3229:". I mean, it is a correct application of the notion of generalized function to the theory of distributions; but the problem is, whether this possibility was mentioned in the literature (outside Knowledge). I guess, such formal manipulations with delta-functions were made by physicists; after all, the delta-function was invented by Dirac! The question is, where to look for a source. 22: 3278:"generalized function" indicates clearly that it is meant to be also a kind of function, in some sense. The probability measure really is the derivative of the cumulative distribution function, provided that (a) derivative is understood according to the theory of generalized functions, and (b) measures are treated as special case of generalized functions. 3147:

could be "points of the sample space" or "possible values of random variables". And in the following section, "any domain D in the n-dimensional space" could be rather "any measurable subset of the n-dimensional space". Anyway, the text needs to be rewritten for clarity. And maybe all that is just not enough important for this article. --

2599:"The definition of a probability density function at the start of this page makes it possible to describe the variable associated with a continuous distribution using a set of binary discrete variables associated with the intervals (for example, a variable being worth 1 if X is in , and 0 if not)." — Does anyone understand it? I do not. 3828:

word; or has it a different meaning? Very vague. It should be about (at best) infinitesimal intervals, not values. And then "metric dependence" manifests itself: we should consider two infinitesimal intervals of the same length. In other words: "intervals of equal length" makes sense, while "points of equal size" does not.

3553: 1097:, not some kind of second variable. I recognize many people do find your view intutive, but I feel it's kind of like making up an interpretation of the symbols distinct from their actual definition. Thus there should at least be a link explaining this viewpoint (can this stuff be made rigorous with differentials?). 1531:(AC) if there is a positive bounded measure which is absolutely continuous w.r.t. to the probability measure (and thus by Radon-Nikodym Theorem ensures us a density exists). Another reference is the beginning of chapter 10 in Resnick's "A Probability Path" which is a bit more readable then Billingsley. 883:. If f is continuous at x then it will hold by the Fundamental Theorem of Calculus, but even if F is differentiable it won't necessarily be true since the pdf is only unique almost everywhere. Not defining a univariate distribution seems okay though, since there is a link to the page in the section. -- 2121:

Well put. This article needs a complete rewrite; what's defined as a density here is just a special case for univariate random variables. And densities can't be "informally" thought of as a smoothed histogram. Etcetera. And while we're at it, cut out all the peripheral stuff, like transformations and

1530:

I believe you are confusing absolute continuity of a function with absolute continuity of a measure. The article is in reference to the latter. See a probability and measure text such as Billingsley for a detailed treatment of the subject. Essentially a distribution is called absolutely continuous

1378:

Given that it is a common mistake to interpret the y-axis of the probability density function as representing probability (as is often done with the normal curve), it would be helpful to have a common-sense description of what probability *density* is. It's clearly related to actual probability, but

3183:

Although the section "Link between discrete and continuous distributions" is useful and intuitive, I have not been able to find any sources that support its claims. I spent the day searching online and checking by University's library for any references to generalized PDFs, and I have found none. Of

2710:

Density means "one divide by something", this is the inverse of something. As dx have the physical units of the considered random variable, then the pdf, f(x) ,have units inverse of the random variable physical units. As they multiply inside the integral we obtain dimensionless units(wich integrated

2194:

I've replaced most of the html or wiki encodings of equations with explicit calls to LaTeX math mode. This looks better if the preferences are set to "always png", otherwise the font changes between some of the expressions. I didn't fix all of the variable or functions that stood alone, but may do

3811:

I reverted the change to the opening sentence that didn't like the term "relative likelihood". It was fine as it was -- to give an intuitive sense for what a PDF is, not as a technical definition. Moreover, the way it was changed, saying the density is a function that describes the local density,

1399:

A simple example would help: Failure probablilty vs. failure rate. Any device fails after some time (failure probability==1, which is the integral from 0 to infinity), but the failure rate is high in the beginning (infant mortalility) and late (as the device wears out), but low in the middle during

1358:

And why oh why say "However special care should be taken around this term, since it is not standard among probabilists and statisticians and in other sources “probability distribution function” may be used when the probability distribution is defined as a function over general sets of values, or it

369:

Make the structure like in every usual math book: first paragraph stays as is, second paragraph is the formal, general, unambiguous definition and then we specialize to the reals and give some examples. Why should one 'generalize' the definition to the 'measure theoretic definition of probability'?

2776:

Important to note that the values of the pdf are not the probability. Probability is the Integral of the pdf. This is why ,in continuous pdf, the pontual probability is zero to any point (integral from a point to a point is zero), but pdf have values in that points. The pdf values are also used to

2525:

That's easy. You gave the justification for deleting it that it is "deprecated" ...which is no reason for deleting something from an encyclopedia. After all the use of the term was given a citation as part of what you deleted. If an alternative term for "probability density function" has been used

1765:

By definition a Random Variable has real values as its range. It maps from the sample space of the probaility space to the reals. However, random elements of the metric space can map to a set besides the reals. (Also, by definition an n-dimentional random vector maps to n-dimentional R-space).

3827:

One problem with this formulation is that "the probability for this random variable to take on a given value" is just zero (since we mean continuous distributions), and "likelihood for this random variable to take on a given value" is an attempt to legalize the illegal notion by using a different

3146:

Yes... "the domain of a family of densities" is the sample space; these densities describe how probability is distributed over the sample space. "The domain is the actual random variable" probably should be "the sample space is the domain of the actual random variable". "Variables in the domain"

3122:

is very unclear, I think due to various meanings of the word "domain". There's "the domain of a family of densities", then "the domain is the actual random variable...", then "variables in the domain"--and in the following section, "any domain D in the n-dimensional space". I think the last of

1443:

None of these answers help me to appreciate the literal understanding. There is a Kahn academy video that produces a very nice example using the pdf for Rainfall and helps explain why the y axis doesn't represent a probabilty for a single point, but for a range, and why unlike a cdf, the y axis

2856:

I'm uncomfortable with the following phrase in the opening sentence: "a function that describes the relative chance for this random variable to occur at a given point in the observation space." First, the passage only refers to "a given point", but a relative chance has to relate two different

2414:

I also find it unfortunate. This is how scientists and mathematicians actually think of PDFs. If the infinitesimal nature was bothersome to you, a finite increment could have been discussed. The word "Loosely" covers the technicality of the probability going to zero and should have covered that

1428:

OTHER Answer: Density means "one divide by something", this is the inverse of something. As dx have the physical units of the considered random variable, then the pdf f(x) ,have units inverse of the random variable physical units. As they multiply inside the integral we obtain dimensionless

3277:

Surely, a generalized function is not a function, thus, not a PDF if we insist that PDF means "probability density function" (rather than "probability density generalized function" - PDGF?). On the other hand, for a physicist or engineer this rigor is of little interest. And the very term

1409:

The only description that made any sense to me was the paragraph beginning "In the field of statistical physics". I gather that somehow while the y-axis values do not represent probabilities of corresponding x-axis values, the y-axis values do represent probabilities of the interval from

1520:

The article begins by saying that only when the Distribution Function is Absolutely Continuous, the random variable will have a Probability Density Function, but then it leaps into the PDF of Discrete Distributions, using the Dirac Delta "Function"! I don't think this is consistent :-)

373:'Note that it is not possible to define a density with reference to an arbitrary measure (e.g. one can't choose the counting measure as a reference for a continuous random variable).' is deleted and something like the following is added: 'Note that it is possible for a random variable 1744:

Is it true that all continuous random variables have to take on real values? What about the a random variable that represents a colour. Can it not have a probability density function over R3? Perhaps we could alter the formal definition to talk about ranges instead of intervals?

1410:

correponding x-axis value to that value plus an infinitely small amount. While this statement is easy to understand in the reading of it, I'm still puzzled about how an infinitely small amount can make any difference if we limit ourselves to the real number system.

1382:

Answer: If "probability" is equivalent to "distance travelled" then "probability density" is equivalent to "speed". So the "probability density function of input variable x " is equivalent to "speed function of input variable t" where t stands for time. -ohanian

2354:

I've deleted the sentence above because I think it is confusing. f(x) dx would go to zero as dx went to an infinitely small number. The probability of the value being in an infinetly small interval is zero (if you are using a PDF with finite values).

3346: 2646:

Maybe you can turn it into that, but for now, rereading the paragraph quoted above I see something very different. Your version does not explain why "the variable associated with a continuous distribution" and why the indicators of intervals.

702:

Well, for a mathematician, you are right (except for one mistake, see below). The problem is that most of the users of this notion are far not mathematicians, have no idea of the measure theory, and are not interested to ever learn such

3247:

It may be a probability measure, but this section words this as though it is a PDF that gives "The density of probability associated with this variable ". Saying it is a probability measure is different from saying that it is a PDF.

1718: 3772:

Agree. Even if the pedagogy and style were perfect (and I do have complaints), it's way too long. Nobody will read it, nor will they read anything else in the article. Wikiversity makes sense to me, simple wikipedia not so much.

2796:

Mathematical theories are always presented in the unitless form; that is, units are assumed to be chosen once and for all; the dimension analysis is left to physicists. The general theory of probability density is also like

1971: 3075:

It used to say “relative likelihood”, but then somebody got offended at the word “likelihood” and changed it to “relative chance”. Anyways, there is no need to specify a second point since the statement is valid for

2174:

Would it be useful to explain in simple terms the use of this function, and contrast that with cumulative distribution functions? I gather that one is the integral of the other but beyond that I am having trouble.

793: 881: 1352: 673:

Also, the rest is not really 'clean'. Everywhere one can read 'it is possible to define density for ...' *NO!* We *have* already defined it. It should read 'In this case, the density is given by ...'.

2692:"Probability is not dimensionless: it is outcomes per trial" — no, sorry, I disagree; "number of outcomes" is dimensionless, and "number of trials" is dimensionless; if in doubt ask a physicist or see 2793:

That is a problem, since your text is a not well-done essay and, more important, is your original research (unless you find a reference); it will not survive here, even though it is basically correct.

2435:

Are "probability distribution function" and "probability density function" synonymous? To me a "probability distribution function" is the distribution function, not the probability density function..

1264: 2952:

I would suggest two things: (1) Move this statement out of the intro, since it is likely, without further explanation that would be too detailed for the intro, to give the wrong impression that f(x

135: 3658: 2004: 2777:

ratios( with the standard normal, Φ(0)=0.4 and Φ(1)=0.24 so the density of probability in zero is 1.66 greater than in one, and different conclusions to the same points of others continuos pdf).

281:

Well, what keeps you from choosing the measure like this? What keeps you from defining it? The question is whether there exists an object that satisfies the definition or not. Let us continue:

3184:

the dozens of books on generalized functions (aka, distributions), not one mentioned their applicability to PDFs. Similarly, none of the books on probability made mention of generalized PDFs.

467: 2857:

points' chances to each other; so I think the passage should be reworded to reflect that. Second and more important, in what sense does the density describe the "relative chance"? Is it f(x

2144:

Because when statisticians retire they spend their time refuting global warming instead of contributing to Knowledge? (I'm a retired computer science logician, and spent a long time writing

240: 3687:-1)-dimensional volume on the hypersurface which is hinted at in the (rather unclear) explanatory phrase after the formula. When using the parametrization, indeed, Jacobians are relevant. 407: 3059:). This limit is equal to the ratio of the densities, under natural assumptions. (Namely, that both points are points of continuity of the density, and the denominator does not vanish.) 521: 4076: 1359:

may refer to the cumulative distribution function, or it may be a probability mass function rather than the density." That is an awful sentence. And a probabilist is a statistician.

3161:

Thanks. I've made some small changes based on your comments. Probably the paragraph should be rewritten entirely, but I'll leave that for someone with more expertise in the subject.

2127:

While I'm ranting... why are nearly all stats articles on wikipedia such badly written junk? Compared to math articles, for example, which tend to be short, precise and to the point.

3558:

looks wrong to me: the denominator should look like a Jacobian, not the square root of a sum. I am not too sure about the correct formula though, so help with that would be welcome.

951: 277:

Note that it is not possible to define a density with reference to an arbitrary measure (e.g. one can't choose the counting measure as a reference for a continuous random variable).

1192:

In simple english. The probability density function is any function f(x) that describes the probability density in terms of the input variable x. With two further conditions that

3548:{\displaystyle \int \limits _{y=g(x_{1},\cdots ,x_{n})}{\frac {f(x_{1},\cdots ,x_{n})}{\sqrt {\sum _{j=1}^{n}{\frac {\partial g}{\partial x_{j}}}(x_{1},\cdots ,x_{n})^{2}}}}\;dV} 3297:

There are various publications that use generalized functions in something that looks like and is treated like a probability density function, as a convenience tool: for example

1818: 1002: 3581:

variables. The square root of the sum is the norm of the gradient vector. It is in the denominator since it is inversely proportional to the thickness of an infinitesimal layer

2079: 3187:

Specifically, if you allow f to be a generalized function, like this section suggests, you run into problems. How can you enforce that the PDF integrates to one? The statement

3080:

other point. As for your second remark, note that the lead sentence is intended to serve as an informal introduction. Of course, the technical meaning of the density is that

1843: 2544:

And here I thought that not liking something is a sufficient reason to delete it :) As for the term “probability distribution” in Oxford Dictionary — guess what — we have a

2104: 2050: 2028: 1894: 643: 328: 3131:--specifically the function sending the parameter values to the corresponding distribution--but this doesn't seem consistent with what follows. Can anyone clarify this? 3057: 3029: 3001: 2947: 2919: 2891: 2289: 2248: 1448:

In particular his reminder that the "area" of the range is the probability. Reference to the discrete integral usually helps people understand what this concept means.

591: 4066: 621: 358: 2800:

Thus I think that your text is helpful but regretfully not suitable for Knowledge; try something else, like Wikibook (or find a reliable source for your statements).

2326: 1872: 487: 427: 3681: 2349: 1075: 4081: 3994: 3990: 3976: 1095: 541: 3846:"; but the latter is about a single possible value of two random variables (distributed differently), not about two possible values of a single random variable. 2949:), or something like that. That interpretation is unfamiliar to me (albeit intuitive as a counterpart to the interpretation of discrete probability functions). 1445: 3298: 1625: 35: 4106: 4091: 2614:

I think the general idea that section tries to convey is that a discrete r.v. can be represented by the"density" (and btw. it’s improper to call it simply

230: 125: 3962: 2081:. Any measure can also be used as reference, it is for instance perfectly possible to define a PDF with respect to area but spanned by polar coordinates. 2482:” piece. First of all it's clearly not often. Second, the majority of textbooks which do employ that term use it in the “cdf” meaning. In fact, I cannot 4061: 3092:, but informally it is perfectly reasonable to say that higher values of the density function are where the random variable is more likely to occur. 4096: 4071: 2504:

Stating in the very first sentence that “probability density function” is the same as “probability distribution function” is at least misleading.

1269:

The actual probability can then be calculated by taking the integral of the function f(x) by the integration interval of the input variable x .

1902: 2577: 289:

Well, without referencing to the fact that we agreed to choose the Lebesgue measure on the reals, this sentence is just wrong. As noted above:

206: 4101: 2436: 2364: 101: 4086: 3255: 3203: 2398:

notation in calculus, except that the way calculus has been taught to freshmen in recent years has been influenced by undue squeamishness.

1430: 1415: 2490:

probability distribution function as density, although some of them seem to use the term in the “density” sense without ever defining it.

3874:"...is a function that quantifies how likely it is for this random variable be very close to a given value" or something like that?? -- 2483: 3306: 1584:

We know fX(x) for X, and fY(Y) for Y, X & Y are independent, Z=X+Y, then what is the density function for Z? And Z2=kX+Y. Thanks.

3972:

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

3812:

is unhelpful. There's probably some room for improvement over the way it is now, but it should still convey the same overall gist.

2783:

And the units of E? Inside of that integral are x.f(x).dx .This is Unit.(1/Unit).Unit, and the conclusion E have the same unit as X.

2718: 2451: 732: 197: 158: 823: 1278: 92: 69: 4056: 2145: 3196:

I have flagged this section as disputed, and unless someone has supporting evidence for it, I will remove it in a few weeks.

1446:

https://www.khanacademy.org/math/probability/random-variables-topic/random_variables_prob_dist/v/probability-density-functions

3124: 2968:); and (2) put it later in the article, with a careful statement of what is meant and with a citation. Comments, anyone? 2622:. It is clear the entire section has to be rewritten to make it comprehensible to a person who does not know that already. 1206: 690: 661: 4037: 3946: 3584: 3119: 44: 3963:

https://web.archive.org/web/20110807023948/http://planetmath.org/?method=png&from=objects&id=2884&op=getobj

1979: 1493:

links to Probability Amplitude, which is about quantum mechanics. I think that should be a disambiguation page. --anon

2474:

Why would somebody put incorrect information into the lead and then insist to keep it there? I'm talking about the “…

904:

Intuitively, if a probability distribution has density f(x), then the infinitesimal interval has probability f(x) dx.

2873:)? No, since the right side of this is zero over zero. So the intended meaning must be something about the limit as 432: 3744: 3733: 1272:

For example: the variable x being within the interval 4.3 < x < 7.8 would have the actual probability of

3860:

On the other hand I agree that saying "the density is a function that describes the local density" is unhelpful.

2394:

is also. When BOTH approach zero, their ratio may approach a finite nonzero value. This is standard use of the

1569: 285:

Not every probability distribution has a density function: the distributions of discrete random variables do not.

3993:

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

3966: 376: 2545: 2497:” was used by Maxwell to describe the probability density function of gas particles multiplied by the physical 2440: 1434: 1419: 3259: 3207: 492: 1785:

I do not think the "formal" definition of a PDF given in the beginning of the article is the widely accepted

4028: 3954: 3817: 3218:

This generalized function is a probability measure, - a special case of a generalized function. (In fact, a

2403: 2360: 2157: 1142: 1037: 810: 2737:" and here), and (b) so what? do you propose some change to the article? or just use this page as a forum? 2722: 3711: 3310: 1550:

if we know the joint density function f(x,y), how to get the marginal density function fY(x) & fX(y)?

911: 1771: 1536: 1411: 4012:

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

4000: 3740: 3729: 3302: 2494: 2420: 50: 3953:. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit 888: 806: 686: 657: 183: 3915: 1795: 960: 84: 63: 3879: 3778: 3707: 3251: 3199: 3128: 2811: 2714: 2693: 2689:

The edit by Rferreirapt is not well-done, but it should be improved rather than deleted, I think so.

2619: 2111: 2055: 1565: 1506: 1392: 1364: 678: 649: 1767: 1532: 1098: 21: 2973: 2200: 1502: 1498: 1490: 1102: 645:. So it is important to be precise about the measure in question when one talks about densities.' 3314: 3211: 2733:

OK, I agree; but I have two questions to you: (a) why do you insist on stating it twice (both in "

1823: 682: 653: 205:

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

100:

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

3813: 3222:

generalized function is well-known to be a locally finite measure.) The integral is well-defined.

3166: 3136: 2764:

The unit for the random variable X , are that of the entity measured, meters, seconds,liters,etc.

2531: 2459: 2399: 2356: 2153: 2087: 2033: 2011: 1877: 1730: 1720: 1609: 1599: 1561: 1453: 1138: 1033: 626: 189: 3997:

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

706:

And here is the mistake: the density with respect to the pushforward measure is just 1, anyway.

173: 152: 4013: 3736:) 18:08, 16 November 2016 (UTC) I suggest that the material be moved to "Simple Knowledge" at: 2195:

so. One that I wasn't sure how to fix was "ƒf" in Further details. How should that render? --

1756: 292: 3096: 3042: 3014: 2986: 2932: 2904: 2876: 2626: 2552: 2508: 2180: 2257: 2216: 546: 3923: 3905: 3865: 3851: 3843: 3833: 3759: 3692: 3325: 3283: 3234: 3152: 3064: 2819: 2742: 2701: 2672: 2652: 2604: 2585: 2416: 2134: 2122:

expectations of random variables, which belong on their own pages. Anyone with a thick skin?

1746: 711: 4020: 1560:

I think you integrate in x to get fX and integrate in y to get Fy. But I am not sure. Ask

596: 333: 4042: 3927: 3909: 3883: 3875: 3869: 3855: 3837: 3821: 3796: 3792: 3782: 3774: 3763: 3748: 3715: 3696: 3567: 3563: 3329: 3287: 3263: 3238: 3170: 3156: 3140: 3107: 3068: 2977: 2823: 2770:

The pdf is a density , with inverse units of the X units: 1/meter ,1/second ,1/liter, etc.

2746: 2726: 2705: 2676: 2656: 2637: 2608: 2589: 2563: 2535: 2519: 2463: 2444: 2424: 2407: 2204: 2184: 2161: 2138: 2115: 2107: 1775: 1766:

These are just naming conventions, but they are pretty universally used throughout texts.

1759: 1749: 1733: 1723: 1612: 1602: 1588: 1573: 1554: 1540: 1525: 1509: 1457: 1438: 1423: 1404: 1388: 1368: 1146: 1106: 1041: 892: 814: 715: 694: 665: 3127:. I thought on a first reading that should be "the domain of a family of densities" the 2302: 1848: 1755:

That is covered in the section "Probability function associated to multiple variables". -

472: 412: 2578:

Knowledge talk:WikiProject Mathematics#Codomain of a random variable: observation space?

1713:{\displaystyle \sigma ={\sqrt {\int _{-\infty }^{\infty }\,(x-{\bar {x}})^{2}f(x)\,dx}}} 1156:

It would be nice to have a picture of a PDF, of, say, the normal distribution. -iwakura

3979:, "External links modified" talk page sections are no longer generated or monitored by 3897: 3893: 3663: 2969: 2331: 2196: 1118: 1057: 370:

What other definition of 'probability' is there apart from the 'measure theoretic' one?

4019:

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

2780:

If we deal with Bidimensional pdf, f(X,Y), the unit of that density is 1/unitX.unitY .

1080: 1004:

is intuitively thought of as the sum of infinitely many infinitely small quantities ƒ(

526: 4050: 3914:

I like a text in “The elements of continuum biomechanics” by Marcelo Epstein, quoted

3162: 3132: 2527: 2455: 2149: 1449: 1401: 1379:

does it have a "real-world" correlate? How should "density" be interpreted? --anon

3226: 2176: 1585: 1551: 1522: 884: 2767:

This implies the unit for a variation in X ,the dx ,are the same as the unit of X.

3986: 3919: 3901: 3861: 3847: 3829: 3755: 3688: 3321: 3279: 3230: 3148: 3060: 2815: 2738: 2697: 2668: 2648: 2600: 2581: 2454:... this seems correct in saying that different sources use different meanings. 2291:

as the probability that a random variable whose probability density function is

2130: 1595: 707: 202: 3985:. No special action is required regarding these talk page notices, other than 3788: 3559: 2804: 1360: 1032:. That applies to integrals generally, not just those in probability theory. 179: 97: 3737: 3967:

http://planetmath.org/?method=png&from=objects&id=2884&op=getobj

1966:{\displaystyle P(A)=\int \limits _{x\in \mathbf {A} }{f(x)d\mathbf {X} }\,} 1622:

It would be helpful to add the standard deviation formula for completeness

1845:(usually but not necessarily the Lebesgue measure), one can define a PDF 330:*is* the density function for a discrete random variable with respect to 2841: 1474: 3889: 3721:

Recent long (and possibly tedious) addition on the derivation of a pdf.

2498: 1012:, each equal to the area below the graph of ƒ above the interval from 3728:

is very long and, I suggest, tedious. I propose that it be reverted.

1729:

I've added this at the bottom (the variance, not the SD, but close).

2370:

I find that deletion unfortunate. The probability of being between

2084:

I think it's wrong and misleading to present the case of a PDF over

3900:. On the level of lead, they could (or should?) look very similar. 2106:

with respect to the Lebesgue measure as the "definition" of a PDF.

3888:

I wonder, why this density is described so differently from, say,

3706:

function come from? Some book or a source you took this from? --

3573:

Jacobian? No, you cannot make Jacobian out of a single function

3336:

Dependent variables and change of variables : Multiple variables

2212: 2761:

As seen below inside the integral are the pdf multiplied by dx.

2493:

The confusion seems to originate from physics, where the term “

788:{\displaystyle f(x)={\frac {\mathrm {d} }{\mathrm {d} x}}F(x).} 3702:

Could you please add where the formula for the non-continuous

876:{\displaystyle f(x)={\frac {\mathrm {d} }{\mathrm {d} x}}F(x)} 795:

should hold, i.e. why F is differentiable in the first place.

729:

First is, that I don't see (because it is not explained) why :

15: 3660:

The formula is correct as far as the reader understands that

1199:

The total area under the graph is 1. Refer to equation below.

623:

but no density with respect to the usual Lebesgue measure on

1347:{\displaystyle p(4.3<x<7.8)=\int _{4.3}^{7.8}f(x)\,dx} 451: 441: 394: 3225:

On the other hand, maybe indeed this section is so-called "

3957:

for additional information. I made the following changes:

3320:

Thank you! Now we see that this is not "OR by synthesis".

1077:

is just some syntactical delimiter capturing the variable

3918:(pages 50-51). Regretfully, it is too long for a lead... 2618:, without the quotation marks) that is a weighted sum of 1196:

f(x) is greater than or equal to zero for all values of x

1175:

Here, f(x) is an arbitrary probability density function.

269:

the density of X with respect to a reference measure ...

798:

Second, the title of the section is not explained. What

3950: 3726: 1400:

its useful lifetime, forming the famous bathtub curve.

1259:{\displaystyle \int _{-\infty }^{\infty }\,f(x)\,dx=1} 1117:

is not just a syntactical delimeter. It does in fact

1054:

if one is used to the proper definition, because then

3666: 3587: 3349: 3179:

Generalized probability density functions don't exist

3045: 3017: 2989: 2935: 2907: 2879: 2334: 2305: 2260: 2219: 2090: 2058: 2036: 2014: 1982: 1905: 1880: 1851: 1826: 1798: 1628: 1281: 1209: 1083: 1060: 963: 914: 908:

Arrgh, now you tell me this has something to do with

826: 735: 629: 599: 549: 529: 495: 475: 435: 415: 379: 336: 295: 1168:

Indefinite Integral (f(x)^((1/r)+1)) dx where r: -->

201:, a collaborative effort to improve the coverage of 96:, a collaborative effort to improve the coverage of 3989:using the archive tool instructions below. Editors 3653:{\displaystyle y<g(x_{1},\dots ,x_{n})<y+dy.} 2734: 3675: 3652: 3547: 3120:Probability_density_function#Families_of_densities 3051: 3023: 2995: 2941: 2913: 2885: 2595:Link between discrete and continuous distributions 2548:for it, quite distinct from both the pdf and cdf. 2343: 2320: 2283: 2242: 2098: 2073: 2044: 2022: 1999:{\displaystyle \mathbf {A} \subseteq \mathbf {X} } 1998: 1965: 1888: 1866: 1837: 1812: 1712: 1489:I don't have time to correct it now, but the page 1346: 1258: 1089: 1069: 996: 945: 875: 787: 637: 615: 585: 535: 515: 489:on the same space. Example: For a random variable 481: 461: 421: 401: 352: 322: 2810:Maybe your thoughts could appear as examples to " 2572:Codomain of a random variable: observation space? 1184:the article should touch this topic. - rodrigob. 565: 4077:Knowledge level-4 vital articles in Mathematics 3193:has no meaning if f is a generalized function. 2983:Yes, one may say that it is about the limit as 2773:This explains why Probability is dimensionless. 1028:runs through the set of all numbers in the set 469:while having no density for some other measure 462:{\displaystyle ({\mathcal {X}},{\mathcal {A}})} 3975:This message was posted before February 2018. 409:to have a density with respect to one measure 8: 2842:http://en.wikipedia.org/Dimensional_analysis 1475:http://en.wikipedia.org/Dimensional_analysis 820:Yeah, it shouldn't be true in general that 402:{\displaystyle X:\Omega \to {\mathcal {X}}} 3945:I have just modified one external link on 2148:instead of (ok, as well as) ranting about 1172:in terms of Indefinite Integral (f(x)) dx 147: 58: 3842:Also, "relative likelihood" sounds like " 3665: 3623: 3604: 3586: 3528: 3518: 3499: 3483: 3465: 3459: 3448: 3433: 3414: 3401: 3390: 3371: 3354: 3348: 3044: 3016: 2988: 2934: 2906: 2878: 2807:" does not mention (physical) dimensions. 2751:Maybe you want to restore your old text: 2333: 2304: 2259: 2218: 2092: 2091: 2089: 2065: 2061: 2060: 2057: 2038: 2037: 2035: 2015: 2013: 1991: 1983: 1981: 1955: 1939: 1932: 1925: 1904: 1881: 1879: 1850: 1830: 1825: 1805: 1797: 1682: 1667: 1666: 1649: 1641: 1635: 1627: 1318: 1313: 1280: 1222: 1214: 1208: 1082: 1059: 1050:I completely agree that this view is not 968: 962: 919: 913: 850: 844: 842: 825: 759: 753: 751: 734: 631: 630: 628: 604: 598: 548: 528: 516:{\displaystyle X:\Omega \to \mathbb {R} } 509: 508: 494: 474: 450: 449: 440: 439: 434: 414: 393: 392: 378: 341: 335: 294: 3537: 593:with respect to its pushforward measure 4067:Knowledge vital articles in Mathematics 2834: 2273: 2232: 1961: 1700: 1655: 1467: 1336: 1242: 1228: 986: 149: 60: 19: 2486:any published source which would have 726:I got two problems with this section. 4082:B-Class vital articles in Mathematics 2170:Uses of PDF vs. distribution function 946:{\displaystyle \int _{\Omega }f(x)dx} 7: 1608:I've added a short section on this. 523:with a finite image it is true that 195:This article is within the scope of 90:This article is within the scope of 3807:Reverted change to opening sentence 1121:, but consider the situation where 722:Contiunous univariate distributions 49:It is of interest to the following 4107:High-priority mathematics articles 4092:Top-importance Statistics articles 3476: 3468: 3123:these is a domain in the sense of 3114:Families of densities and domains? 2351:is an infinitely small increment. 1792:Given elements in an abstract set 1650: 1645: 1223: 1218: 920: 851: 845: 760: 754: 502: 386: 262:This article is really confusing: 14: 3949:. Please take a moment to review 3299:this American Statistican article 2814:"? But this is also problematic. 2480:probability distribution function 2470:Probability distribution function 2452:Probability distribution function 1813:{\displaystyle x\in \mathbf {X} } 1165:Can someone help me in computing 997:{\displaystyle \int _{A}f(x)\,dx} 215:Knowledge:WikiProject Mathematics 4062:Knowledge level-4 vital articles 3683:is an infinitesimal element of ( 2074:{\displaystyle \mathbb {R} ^{n}} 2030:is not necessarily a subset of 2016: 1992: 1984: 1956: 1933: 1882: 1831: 1806: 1505:, which I made into a disambig. 1444:doesn't have to b 1 at the top. 218:Template:WikiProject Mathematics 182: 172: 151: 110:Knowledge:WikiProject Statistics 83: 62: 29: 20: 4097:WikiProject Statistics articles 3227:"original research by synthesis 2735:#what is a probability density? 2190:Math mode versus html encodings 2146:Introduction to Boolean algebra 1387:One more answer: If you have a 235:This article has been rated as 130:This article has been rated as 113:Template:WikiProject Statistics 4072:B-Class level-4 vital articles 3629: 3597: 3525: 3492: 3439: 3407: 3396: 3364: 3125:Domain (mathematical analysis) 2706:18:54, 23 September 2010 (UTC) 2677:18:55, 23 September 2010 (UTC) 2657:17:27, 20 September 2010 (UTC) 2638:16:28, 20 September 2010 (UTC) 2609:07:43, 20 September 2010 (UTC) 2270: 2264: 2229: 2223: 1949: 1943: 1915: 1909: 1861: 1855: 1697: 1691: 1679: 1672: 1657: 1458:08:25, 14 September 2013 (UTC) 1374:what is a probability density? 1333: 1327: 1303: 1285: 1239: 1233: 1129:) is in meters per second and 983: 977: 934: 928: 870: 864: 836: 830: 779: 773: 745: 739: 580: 568: 559: 553: 505: 456: 436: 389: 317: 305: 299: 1: 3797:02:40, 17 November 2016 (UTC) 3783:22:48, 16 November 2016 (UTC) 3764:18:52, 16 November 2016 (UTC) 3749:18:17, 16 November 2016 (UTC) 3315:23:13, 28 February 2013 (UTC) 3264:21:36, 28 February 2013 (UTC) 3239:20:58, 28 February 2013 (UTC) 3212:20:11, 28 February 2013 (UTC) 3171:05:19, 10 November 2012 (UTC) 3108:18:08, 14 November 2010 (UTC) 3069:16:12, 14 November 2010 (UTC) 2978:15:38, 14 November 2010 (UTC) 2564:20:05, 27 November 2009 (UTC) 2536:10:57, 27 November 2009 (UTC) 2520:20:27, 26 November 2009 (UTC) 2365:18:35, 18 February 2009 (UTC) 2205:21:40, 28 December 2008 (UTC) 2139:03:57, 1 September 2008 (UTC) 2116:23:54, 14 February 2008 (UTC) 1838:{\displaystyle d\mathbf {X} } 1750:21:34, 27 February 2007 (UTC) 1414:05:38, 12 October 2007 (UTC) 209:and see a list of open tasks. 104:and see a list of open tasks. 4102:B-Class mathematics articles 3947:Probability density function 3928:20:49, 3 December 2016 (UTC) 3910:20:43, 3 December 2016 (UTC) 3884:19:28, 3 December 2016 (UTC) 3870:07:04, 2 December 2016 (UTC) 3856:07:00, 2 December 2016 (UTC) 3838:06:14, 2 December 2016 (UTC) 3822:00:31, 2 December 2016 (UTC) 3697:17:06, 16 October 2013 (UTC) 3568:15:46, 16 October 2013 (UTC) 3157:14:09, 9 November 2012 (UTC) 3141:12:08, 9 November 2012 (UTC) 2685:Probability is dimensionless 2464:09:06, 21 October 2009 (UTC) 2445:08:52, 21 October 2009 (UTC) 2162:05:01, 13 January 2011 (UTC) 2099:{\displaystyle \mathbb {R} } 2045:{\displaystyle \mathbb {R} } 2023:{\displaystyle \mathbf {X} } 1889:{\displaystyle \mathbf {X} } 1734:19:44, 5 December 2006 (UTC) 1724:19:02, 5 December 2006 (UTC) 1613:19:59, 5 December 2006 (UTC) 1603:19:44, 5 December 2006 (UTC) 1589:18:00, 1 December 2006 (UTC) 1574:16:17, 1 December 2006 (UTC) 1555:01:37, 1 December 2006 (UTC) 1526:18:14, 9 November 2006 (UTC) 802:a univariate distribution? 638:{\displaystyle \mathbb {R} } 4087:B-Class Statistics articles 3716:16:02, 4 January 2016 (UTC) 2803:You see, also the article " 2667:Anyway, I have deleted it. 1976:for all measurable subsets 1510:18:04, 18 August 2005 (UTC) 4123: 4006:(last update: 5 June 2024) 3942:Hello fellow Wikipedians, 2590:16:54, 27 March 2010 (UTC) 2425:23:03, 22 April 2011 (UTC) 2254:Loosely, one may think of 2213:Loosely, one may think of 2185:03:29, 13 April 2008 (UTC) 1776:00:49, 18 March 2013 (UTC) 1760:05:47, 17 April 2007 (UTC) 1541:00:44, 18 March 2013 (UTC) 1405:20:44, 20 March 2006 (UTC) 1369:02:25, 26 April 2010 (UTC) 1182:what is a multimodal pdf ? 323:{\displaystyle a\mapsto P} 4043:22:56, 26 July 2017 (UTC) 3330:06:53, 1 March 2013 (UTC) 3288:06:49, 1 March 2013 (UTC) 3052:{\displaystyle \epsilon } 3024:{\displaystyle \epsilon } 2996:{\displaystyle \epsilon } 2942:{\displaystyle \epsilon } 2914:{\displaystyle \epsilon } 2886:{\displaystyle \epsilon } 2382:is infinitely small, and 1546:marginal density function 1042:00:50, 12 June 2008 (UTC) 893:18:52, 29 June 2010 (UTC) 815:17:01, 2 March 2010 (UTC) 234: 167: 129: 78: 57: 2824:20:07, 30 May 2013 (UTC) 2786:By last, the units of E? 2747:19:49, 30 May 2013 (UTC) 2727:18:07, 30 May 2013 (UTC) 2408:00:43, 20 May 2009 (UTC) 2295:is in the interval from 2284:{\displaystyle f(x)\,dx} 2243:{\displaystyle f(x)\,dx} 1820:equipped with a measure 1439:15:53, 30 May 2013 (UTC) 1424:15:53, 30 May 2013 (UTC) 1147:00:48, 20 May 2009 (UTC) 1107:22:14, 19 May 2009 (UTC) 957:Certainly. An integral 716:07:12, 6 June 2015 (UTC) 695:09:40, 5 June 2015 (UTC) 666:09:04, 5 June 2015 (UTC) 586:{\displaystyle f(x)=\Pr} 241:project's priority scale 3938:External links modified 1395:15:03, 2 Jun 2005 (UTC) 198:WikiProject Mathematics 4057:B-Class vital articles 3677: 3654: 3549: 3464: 3053: 3025: 2997: 2943: 2915: 2887: 2852:"Relative probability" 2345: 2322: 2285: 2244: 2100: 2075: 2046: 2024: 2000: 1967: 1890: 1868: 1839: 1814: 1714: 1598:of the two densities. 1348: 1260: 1091: 1071: 998: 947: 877: 789: 639: 617: 616:{\displaystyle X_{*}P} 587: 537: 517: 483: 463: 423: 403: 354: 353:{\displaystyle X_{*}P} 324: 93:WikiProject Statistics 3678: 3655: 3550: 3444: 3178: 3054: 3026: 2998: 2944: 2916: 2888: 2620:Dirac delta functions 2495:distribution function 2346: 2323: 2286: 2245: 2101: 2076: 2047: 2025: 2001: 1968: 1891: 1869: 1840: 1815: 1715: 1349: 1261: 1092: 1072: 999: 948: 878: 790: 640: 618: 588: 538: 518: 484: 464: 424: 404: 355: 325: 36:level-4 vital article 3987:regular verification 3664: 3585: 3347: 3129:domain of a function 3043: 3015: 2987: 2933: 2905: 2877: 2812:Dimensional analysis 2694:Dimensional analysis 2501:of those particles. 2332: 2321:{\displaystyle x+dx} 2303: 2258: 2217: 2088: 2056: 2034: 2012: 1980: 1903: 1878: 1867:{\displaystyle f(x)} 1849: 1824: 1796: 1626: 1279: 1207: 1081: 1058: 961: 912: 824: 733: 627: 597: 547: 527: 493: 482:{\displaystyle \nu } 473: 433: 422:{\displaystyle \mu } 413: 377: 334: 293: 221:mathematics articles 3977:After February 2018 3754:Or to Wikiversity? 3003:goes to zero of P(x 2893:goes to zero of P(x 2478:referred to as the 1654: 1503:Probability density 1499:Probability Density 1491:Probability Density 1485:Probability Density 1323: 1227: 116:Statistics articles 4031:InternetArchiveBot 3982:InternetArchiveBot 3676:{\displaystyle dV} 3673: 3650: 3545: 3538: 3400: 3049: 3021: 2993: 2939: 2911: 2883: 2344:{\displaystyle dx} 2341: 2318: 2281: 2274: 2240: 2233: 2096: 2071: 2042: 2020: 1996: 1963: 1962: 1938: 1886: 1864: 1835: 1810: 1710: 1701: 1656: 1637: 1618:Standard deviation 1564:, he should know. 1516:Self Inconsistency 1344: 1337: 1309: 1256: 1243: 1229: 1210: 1087: 1070:{\displaystyle dx} 1067: 994: 987: 943: 873: 785: 635: 613: 583: 533: 513: 479: 459: 419: 399: 350: 320: 190:Mathematics portal 45:content assessment 4007: 3725:This recent edit 3535: 3534: 3490: 3350: 3254:comment added by 3202:comment added by 2756:Units for the pdf 2717:comment added by 2141: 1921: 1708: 1675: 1090:{\displaystyle x} 859: 805:Please fix this. 768: 698: 681:comment added by 669: 652:comment added by 536:{\displaystyle X} 255: 254: 251: 250: 247: 246: 146: 145: 142: 141: 4114: 4041: 4032: 4005: 4004: 3983: 3844:likelihood ratio 3741:Isambard Kingdom 3730:Isambard Kingdom 3682: 3680: 3679: 3674: 3659: 3657: 3656: 3651: 3628: 3627: 3609: 3608: 3554: 3552: 3551: 3546: 3536: 3533: 3532: 3523: 3522: 3504: 3503: 3491: 3489: 3488: 3487: 3474: 3466: 3463: 3458: 3443: 3442: 3438: 3437: 3419: 3418: 3402: 3399: 3395: 3394: 3376: 3375: 3303:this other paper 3266: 3214: 3106: 3102: 3099: 3058: 3056: 3055: 3050: 3030: 3028: 3027: 3022: 3002: 3000: 2999: 2994: 2948: 2946: 2945: 2940: 2920: 2918: 2917: 2912: 2892: 2890: 2889: 2884: 2844: 2839: 2729: 2636: 2632: 2629: 2562: 2558: 2555: 2546:separate article 2518: 2514: 2511: 2350: 2348: 2347: 2342: 2327: 2325: 2324: 2319: 2290: 2288: 2287: 2282: 2249: 2247: 2246: 2241: 2128: 2105: 2103: 2102: 2097: 2095: 2080: 2078: 2077: 2072: 2070: 2069: 2064: 2051: 2049: 2048: 2043: 2041: 2029: 2027: 2026: 2021: 2019: 2005: 2003: 2002: 1997: 1995: 1987: 1972: 1970: 1969: 1964: 1960: 1959: 1937: 1936: 1895: 1893: 1892: 1887: 1885: 1873: 1871: 1870: 1865: 1844: 1842: 1841: 1836: 1834: 1819: 1817: 1816: 1811: 1809: 1719: 1717: 1716: 1711: 1709: 1687: 1686: 1677: 1676: 1668: 1653: 1648: 1636: 1477: 1472: 1353: 1351: 1350: 1345: 1322: 1317: 1265: 1263: 1262: 1257: 1226: 1221: 1096: 1094: 1093: 1088: 1076: 1074: 1073: 1068: 1003: 1001: 1000: 995: 973: 972: 952: 950: 949: 944: 924: 923: 882: 880: 879: 874: 860: 858: 854: 848: 843: 794: 792: 791: 786: 769: 767: 763: 757: 752: 697: 675: 668: 646: 644: 642: 641: 636: 634: 622: 620: 619: 614: 609: 608: 592: 590: 589: 584: 543:has the density 542: 540: 539: 534: 522: 520: 519: 514: 512: 488: 486: 485: 480: 468: 466: 465: 460: 455: 454: 445: 444: 428: 426: 425: 420: 408: 406: 405: 400: 398: 397: 364:Suggested edits: 359: 357: 356: 351: 346: 345: 329: 327: 326: 321: 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: 4122: 4121: 4117: 4116: 4115: 4113: 4112: 4111: 4047: 4046: 4035: 4030: 3998: 3991:have permission 3981: 3955:this simple FaQ 3940: 3920:Boris Tsirelson 3902:Boris Tsirelson 3862:Boris Tsirelson 3848:Boris Tsirelson 3830:Boris Tsirelson 3809: 3756:Boris Tsirelson 3723: 3689:Boris Tsirelson 3662: 3661: 3619: 3600: 3583: 3582: 3524: 3514: 3495: 3479: 3475: 3467: 3429: 3410: 3403: 3386: 3367: 3345: 3344: 3338: 3322:Boris Tsirelson 3301:. Additionally 3280:Boris Tsirelson 3249: 3231:Boris Tsirelson 3197: 3190:\int_X f(x) dx 3181: 3149:Boris Tsirelson 3116: 3100: 3097: 3093: 3061:Boris Tsirelson 3041: 3040: 3038: 3034: 3013: 3012: 3010: 3006: 2985: 2984: 2967: 2963: 2959: 2955: 2931: 2930: 2928: 2924: 2903: 2902: 2900: 2896: 2875: 2874: 2872: 2868: 2864: 2860: 2854: 2849: 2848: 2847: 2840: 2836: 2816:Boris Tsirelson 2739:Boris Tsirelson 2712: 2698:Boris Tsirelson 2687: 2669:Boris Tsirelson 2649:Boris Tsirelson 2630: 2627: 2623: 2601:Boris Tsirelson 2597: 2582:Boris Tsirelson 2574: 2556: 2553: 2549: 2512: 2509: 2505: 2472: 2437:131.175.127.242 2433: 2330: 2329: 2301: 2300: 2256: 2255: 2252: 2215: 2214: 2192: 2172: 2086: 2085: 2059: 2054: 2053: 2032: 2031: 2010: 2009: 1978: 1977: 1901: 1900: 1876: 1875: 1847: 1846: 1822: 1821: 1794: 1793: 1783: 1742: 1678: 1624: 1623: 1620: 1582: 1566:Oleg Alexandrov 1548: 1518: 1507:Oleg Alexandrov 1487: 1482: 1481: 1480: 1473: 1469: 1393:Oleg Alexandrov 1389:random variable 1376: 1277: 1276: 1205: 1204: 1190: 1154: 1119:bind a variable 1079: 1078: 1056: 1055: 964: 959: 958: 915: 910: 909: 901: 849: 822: 821: 758: 731: 730: 724: 708:Boris Tsirelson 676: 647: 625: 624: 600: 595: 594: 545: 544: 525: 524: 491: 490: 471: 470: 431: 430: 411: 410: 375: 374: 337: 332: 331: 291: 290: 260: 220: 217: 214: 211: 210: 188: 181: 161: 115: 112: 109: 106: 105: 72: 43:on Knowledge's 40: 30: 12: 11: 5: 4120: 4118: 4110: 4109: 4104: 4099: 4094: 4089: 4084: 4079: 4074: 4069: 4064: 4059: 4049: 4048: 4025: 4024: 4017: 3970: 3969: 3961:Added archive 3939: 3936: 3935: 3934: 3933: 3932: 3931: 3930: 3912: 3898:Energy density 3894:Charge density 3858: 3840: 3808: 3805: 3804: 3803: 3802: 3801: 3800: 3799: 3767: 3766: 3722: 3719: 3700: 3699: 3672: 3669: 3649: 3646: 3643: 3640: 3637: 3634: 3631: 3626: 3622: 3618: 3615: 3612: 3607: 3603: 3599: 3596: 3593: 3590: 3556: 3555: 3544: 3541: 3531: 3527: 3521: 3517: 3513: 3510: 3507: 3502: 3498: 3494: 3486: 3482: 3478: 3473: 3470: 3462: 3457: 3454: 3451: 3447: 3441: 3436: 3432: 3428: 3425: 3422: 3417: 3413: 3409: 3406: 3398: 3393: 3389: 3385: 3382: 3379: 3374: 3370: 3366: 3363: 3360: 3357: 3353: 3337: 3334: 3333: 3332: 3295: 3294: 3293: 3292: 3291: 3290: 3270: 3269: 3268: 3267: 3256:174.63.120.183 3242: 3241: 3223: 3204:174.63.120.183 3180: 3177: 3176: 3175: 3174: 3173: 3115: 3112: 3111: 3110: 3072: 3071: 3048: 3036: 3032: 3020: 3008: 3004: 2992: 2965: 2961: 2957: 2953: 2938: 2926: 2922: 2910: 2898: 2894: 2882: 2870: 2866: 2862: 2858: 2853: 2850: 2846: 2845: 2833: 2832: 2828: 2827: 2826: 2808: 2801: 2798: 2794: 2791: 2790: 2789: 2787: 2784: 2781: 2778: 2774: 2771: 2768: 2765: 2762: 2759: 2758: 2757: 2749: 2686: 2683: 2682: 2681: 2680: 2679: 2662: 2661: 2660: 2659: 2641: 2640: 2596: 2593: 2573: 2570: 2569: 2568: 2567: 2566: 2539: 2538: 2471: 2468: 2467: 2466: 2432: 2429: 2428: 2427: 2411: 2410: 2340: 2337: 2317: 2314: 2311: 2308: 2280: 2277: 2272: 2269: 2266: 2263: 2251: 2239: 2236: 2231: 2228: 2225: 2222: 2211: 2209: 2191: 2188: 2171: 2168: 2167: 2166: 2165: 2164: 2124: 2123: 2094: 2068: 2063: 2040: 2018: 1994: 1990: 1986: 1974: 1973: 1958: 1954: 1951: 1948: 1945: 1942: 1935: 1931: 1928: 1924: 1920: 1917: 1914: 1911: 1908: 1884: 1863: 1860: 1857: 1854: 1833: 1829: 1808: 1804: 1801: 1782: 1779: 1763: 1762: 1741: 1738: 1737: 1736: 1707: 1704: 1699: 1696: 1693: 1690: 1685: 1681: 1674: 1671: 1665: 1662: 1659: 1652: 1647: 1644: 1640: 1634: 1631: 1619: 1616: 1606: 1605: 1581: 1578: 1577: 1576: 1547: 1544: 1517: 1514: 1513: 1512: 1486: 1483: 1479: 1478: 1466: 1465: 1461: 1431:197.242.10.103 1416:197.242.10.103 1412:207.189.230.42 1397: 1396: 1375: 1372: 1356: 1355: 1343: 1340: 1335: 1332: 1329: 1326: 1321: 1316: 1312: 1308: 1305: 1302: 1299: 1296: 1293: 1290: 1287: 1284: 1267: 1266: 1255: 1252: 1249: 1246: 1241: 1238: 1235: 1232: 1225: 1220: 1217: 1213: 1201: 1200: 1197: 1189: 1188:Simple English 1186: 1159: 1153: 1150: 1112: 1111: 1110: 1109: 1086: 1066: 1063: 1045: 1044: 993: 990: 985: 982: 979: 976: 971: 967: 942: 939: 936: 933: 930: 927: 922: 918: 900: 897: 896: 895: 872: 869: 866: 863: 857: 853: 847: 841: 838: 835: 832: 829: 784: 781: 778: 775: 772: 766: 762: 756: 750: 747: 744: 741: 738: 723: 720: 719: 718: 704: 671: 670: 633: 612: 607: 603: 582: 579: 576: 573: 570: 567: 564: 561: 558: 555: 552: 532: 511: 507: 504: 501: 498: 478: 458: 453: 448: 443: 438: 418: 396: 391: 388: 385: 382: 371: 349: 344: 340: 319: 316: 313: 310: 307: 304: 301: 298: 265:First it sais 259: 256: 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: 4119: 4108: 4105: 4103: 4100: 4098: 4095: 4093: 4090: 4088: 4085: 4083: 4080: 4078: 4075: 4073: 4070: 4068: 4065: 4063: 4060: 4058: 4055: 4054: 4052: 4045: 4044: 4039: 4034: 4033: 4022: 4018: 4015: 4011: 4010: 4009: 4002: 3996: 3992: 3988: 3984: 3978: 3973: 3968: 3964: 3960: 3959: 3958: 3956: 3952: 3948: 3943: 3937: 3929: 3925: 3921: 3917: 3913: 3911: 3907: 3903: 3899: 3895: 3891: 3887: 3886: 3885: 3881: 3877: 3873: 3872: 3871: 3867: 3863: 3859: 3857: 3853: 3849: 3845: 3841: 3839: 3835: 3831: 3826: 3825: 3824: 3823: 3819: 3815: 3814:Deacon Vorbis 3806: 3798: 3794: 3790: 3786: 3785: 3784: 3780: 3776: 3771: 3770: 3769: 3768: 3765: 3761: 3757: 3753: 3752: 3751: 3750: 3746: 3742: 3738: 3735: 3731: 3727: 3720: 3718: 3717: 3713: 3709: 3705: 3698: 3694: 3690: 3686: 3670: 3667: 3647: 3644: 3641: 3638: 3635: 3632: 3624: 3620: 3616: 3613: 3610: 3605: 3601: 3594: 3591: 3588: 3580: 3576: 3572: 3571: 3570: 3569: 3565: 3561: 3542: 3539: 3529: 3519: 3515: 3511: 3508: 3505: 3500: 3496: 3484: 3480: 3471: 3460: 3455: 3452: 3449: 3445: 3434: 3430: 3426: 3423: 3420: 3415: 3411: 3404: 3391: 3387: 3383: 3380: 3377: 3372: 3368: 3361: 3358: 3355: 3351: 3343: 3342: 3341: 3335: 3331: 3327: 3323: 3319: 3318: 3317: 3316: 3312: 3308: 3304: 3300: 3289: 3285: 3281: 3276: 3275: 3274: 3273: 3272: 3271: 3265: 3261: 3257: 3253: 3246: 3245: 3244: 3243: 3240: 3236: 3232: 3228: 3224: 3221: 3217: 3216: 3215: 3213: 3209: 3205: 3201: 3194: 3191: 3188: 3185: 3172: 3168: 3164: 3160: 3159: 3158: 3154: 3150: 3145: 3144: 3143: 3142: 3138: 3134: 3130: 3126: 3121: 3113: 3109: 3104: 3103: 3091: 3087: 3083: 3079: 3074: 3073: 3070: 3066: 3062: 3046: 3035:< x < x 3018: 3007:< x < x 2990: 2982: 2981: 2980: 2979: 2975: 2971: 2950: 2936: 2925:< x < x 2908: 2897:< x < x 2880: 2851: 2843: 2838: 2835: 2831: 2825: 2821: 2817: 2813: 2809: 2806: 2802: 2799: 2795: 2792: 2788: 2785: 2782: 2779: 2775: 2772: 2769: 2766: 2763: 2760: 2755: 2754: 2753: 2752: 2750: 2748: 2744: 2740: 2736: 2732: 2731: 2730: 2728: 2724: 2720: 2716: 2708: 2707: 2703: 2699: 2695: 2690: 2684: 2678: 2674: 2670: 2666: 2665: 2664: 2663: 2658: 2654: 2650: 2645: 2644: 2643: 2642: 2639: 2634: 2633: 2621: 2617: 2613: 2612: 2611: 2610: 2606: 2602: 2594: 2592: 2591: 2587: 2583: 2579: 2571: 2565: 2560: 2559: 2547: 2543: 2542: 2541: 2540: 2537: 2533: 2529: 2524: 2523: 2522: 2521: 2516: 2515: 2502: 2500: 2496: 2491: 2489: 2485: 2481: 2477: 2469: 2465: 2461: 2457: 2453: 2449: 2448: 2447: 2446: 2442: 2438: 2430: 2426: 2422: 2418: 2413: 2412: 2409: 2405: 2401: 2400:Michael Hardy 2397: 2393: 2389: 2385: 2381: 2377: 2373: 2369: 2368: 2367: 2366: 2362: 2358: 2357:Richard Giuly 2352: 2338: 2335: 2315: 2312: 2309: 2306: 2298: 2294: 2278: 2275: 2267: 2261: 2237: 2234: 2226: 2220: 2210: 2207: 2206: 2202: 2198: 2189: 2187: 2186: 2182: 2178: 2169: 2163: 2159: 2155: 2154:Vaughan Pratt 2151: 2150:Boolean logic 2147: 2143: 2142: 2140: 2136: 2132: 2126: 2125: 2120: 2119: 2118: 2117: 2113: 2109: 2082: 2066: 2007: 1988: 1952: 1946: 1940: 1929: 1926: 1922: 1918: 1912: 1906: 1899: 1898: 1897: 1858: 1852: 1827: 1802: 1799: 1790: 1789:definition. 1788: 1780: 1778: 1777: 1773: 1769: 1761: 1758: 1754: 1753: 1752: 1751: 1748: 1739: 1735: 1732: 1731:Michael Hardy 1728: 1727: 1726: 1725: 1722: 1705: 1702: 1694: 1688: 1683: 1669: 1663: 1660: 1642: 1638: 1632: 1629: 1617: 1615: 1614: 1611: 1610:Michael Hardy 1604: 1601: 1600:Michael Hardy 1597: 1593: 1592: 1591: 1590: 1587: 1579: 1575: 1571: 1567: 1563: 1562:Michael Hardy 1559: 1558: 1557: 1556: 1553: 1545: 1543: 1542: 1538: 1534: 1528: 1527: 1524: 1515: 1511: 1508: 1504: 1500: 1497:I redirected 1496: 1495: 1494: 1492: 1484: 1476: 1471: 1468: 1464: 1460: 1459: 1455: 1451: 1447: 1441: 1440: 1436: 1432: 1426: 1425: 1421: 1417: 1413: 1407: 1406: 1403: 1394: 1390: 1386: 1385: 1384: 1380: 1373: 1371: 1370: 1366: 1362: 1341: 1338: 1330: 1324: 1319: 1314: 1310: 1306: 1300: 1297: 1294: 1291: 1288: 1282: 1275: 1274: 1273: 1270: 1253: 1250: 1247: 1244: 1236: 1230: 1215: 1211: 1203: 1202: 1198: 1195: 1194: 1193: 1187: 1185: 1183: 1179: 1176: 1173: 1170: 1166: 1163: 1160: 1157: 1151: 1149: 1148: 1144: 1140: 1139:Michael Hardy 1136: 1133:(and so also 1132: 1128: 1124: 1120: 1116: 1108: 1104: 1100: 1084: 1064: 1061: 1053: 1049: 1048: 1047: 1046: 1043: 1039: 1035: 1034:Michael Hardy 1031: 1027: 1023: 1019: 1015: 1011: 1007: 991: 988: 980: 974: 969: 965: 956: 955: 954: 940: 937: 931: 925: 916: 906: 905: 898: 894: 890: 886: 867: 861: 855: 839: 833: 827: 819: 818: 817: 816: 812: 808: 803: 801: 796: 782: 776: 770: 764: 748: 742: 736: 727: 721: 717: 713: 709: 705: 701: 700: 699: 696: 692: 688: 684: 680: 667: 663: 659: 655: 651: 610: 605: 601: 577: 574: 571: 562: 556: 550: 530: 499: 496: 476: 446: 416: 383: 380: 372: 368: 367: 366: 365: 361: 347: 342: 338: 314: 311: 308: 302: 296: 287: 286: 282: 279: 278: 274: 271: 270: 266: 263: 258:Inconsistency 257: 242: 238: 237:High-priority 232: 229: 228: 225: 208: 204: 200: 199: 191: 185: 180: 178: 175: 171: 170: 166: 162:High‑priority 160: 157: 154: 150: 137: 133: 127: 124: 123: 120: 103: 99: 95: 94: 89: 86: 82: 81: 77: 71: 68: 65: 61: 56: 52: 46: 38: 37: 27: 23: 18: 17: 4029: 4026: 4001:source check 3980: 3974: 3971: 3944: 3941: 3810: 3724: 3703: 3701: 3684: 3578: 3574: 3557: 3340:The formula 3339: 3307:81.98.35.149 3296: 3250:— Preceding 3219: 3198:— Preceding 3195: 3192: 3189: 3186: 3182: 3118:The section 3117: 3095: 3089: 3085: 3081: 3077: 2951: 2855: 2837: 2829: 2713:— Preceding 2709: 2691: 2688: 2625: 2615: 2598: 2575: 2551: 2507: 2503: 2492: 2487: 2479: 2475: 2473: 2434: 2395: 2391: 2387: 2383: 2379: 2375: 2371: 2353: 2296: 2292: 2253: 2208: 2193: 2173: 2083: 2008: 1975: 1791: 1786: 1784: 1764: 1743: 1621: 1607: 1583: 1549: 1529: 1519: 1488: 1470: 1462: 1442: 1427: 1408: 1398: 1381: 1377: 1357: 1271: 1268: 1191: 1181: 1180: 1177: 1174: 1171: 1167: 1164: 1161: 1158: 1155: 1134: 1130: 1126: 1122: 1114: 1113: 1051: 1029: 1025: 1021: 1017: 1013: 1009: 1005: 907: 903: 902: 807:Quiet photon 804: 799: 797: 728: 725: 677:— Preceding 672: 648:— Preceding 363: 362: 288: 284: 283: 280: 276: 275: 272: 268: 267: 264: 261: 236: 196: 131: 91: 51:WikiProjects 34: 3088:) = dPr / d 2964:) / P(x = x 2960:) = P(x = x 2869:) / P(x = x 2865:) = P(x = x 2719:37.60.184.8 2431:Terminology 2417:Jason Quinn 1747:MisterSheik 1596:convolution 899:"Intuitive" 212:Mathematics 203:mathematics 159:Mathematics 4051:Categories 4038:Report bug 3708:Hanator123 2830:References 2805:Derivative 2108:Winterfors 1781:Generality 1463:References 1402:Ralf-Peter 107:Statistics 98:statistics 70:Statistics 4021:this tool 4014:this tool 2970:Duoduoduo 2576:See also 2415:concern. 2197:Autopilot 1768:Ageofmech 1594:It's the 1533:Ageofmech 1099:Suprahili 1052:intuitive 273:and then 39:is rated 4027:Cheers.— 3787:Agreed. 3252:unsigned 3220:positive 3200:unsigned 3163:Jowa fan 3133:Jowa fan 2715:unsigned 2528:Melcombe 2456:Melcombe 2328:, where 2052:or even 1896:so that 1580:addition 1450:Raiteria 1162:Hi all: 691:contribs 679:unsigned 662:contribs 650:unsigned 3951:my edit 3890:Density 3031:) / P(x 2956:) / f(x 2921:) / P(x 2861:) / f(x 2616:density 2499:density 2488:defined 2177:Boris B 1787:general 1586:Jackzhp 1552:Jackzhp 1523:Albmont 1178:Partho 885:Theodds 683:Fryasdf 654:Fryasdf 239:on the 134:on the 41:B-class 2152:.) -- 2131:Zaqrfv 1740:Reals? 1721:Gp4rts 47:scale. 3876:Steve 3789:McKay 3775:Steve 3560:Garfl 3101:pasha 2797:that. 2631:pasha 2557:pasha 2513:pasha 2476:often 2250:as... 1874:over 1361:Worik 1024:, as 703:math. 28:This 3924:talk 3916:here 3906:talk 3880:talk 3866:talk 3852:talk 3834:talk 3818:talk 3793:talk 3779:talk 3760:talk 3745:talk 3734:talk 3712:talk 3693:talk 3633:< 3592:< 3564:talk 3326:talk 3311:talk 3284:talk 3260:talk 3235:talk 3208:talk 3167:talk 3153:talk 3137:talk 3065:talk 2974:talk 2820:talk 2743:talk 2723:talk 2702:talk 2673:talk 2653:talk 2605:talk 2586:talk 2532:talk 2484:find 2460:talk 2450:See 2441:talk 2421:talk 2404:talk 2374:and 2361:talk 2201:talk 2181:talk 2158:talk 2135:talk 2112:talk 1772:talk 1757:Icek 1570:talk 1537:talk 1454:talk 1435:talk 1420:talk 1365:talk 1298:< 1292:< 1143:talk 1103:talk 1038:talk 889:talk 811:talk 712:talk 687:talk 658:talk 231:High 3995:RfC 3965:to 3577:of 3094:// 3078:any 2624:// 2299:to 1501:to 1320:7.8 1315:4.3 1301:7.8 1289:4.3 1169:=1 1152:pdf 1016:to 429:on 126:Top 4053:: 4008:. 4003:}} 3999:{{ 3926:) 3908:) 3896:, 3892:, 3882:) 3868:) 3854:) 3836:) 3820:) 3795:) 3781:) 3773:-- 3762:) 3747:) 3739:. 3714:) 3695:) 3614:… 3566:) 3509:⋯ 3477:∂ 3469:∂ 3446:∑ 3424:⋯ 3381:⋯ 3352:∫ 3328:) 3313:) 3286:) 3262:) 3237:) 3210:) 3169:) 3155:) 3139:) 3105:» 3098:st 3067:) 3047:ϵ 3039:+ 3019:ϵ 3011:+ 2991:ϵ 2976:) 2937:ϵ 2929:+ 2909:ϵ 2901:+ 2881:ϵ 2822:) 2745:) 2725:) 2704:) 2696:. 2675:) 2655:) 2635:» 2628:st 2607:) 2588:) 2580:. 2561:» 2554:st 2550:… 2534:) 2517:» 2510:st 2506:… 2462:) 2443:) 2423:) 2406:) 2396:dx 2392:dx 2390:) 2380:dx 2378:+ 2363:) 2293:ƒf 2203:) 2183:) 2160:) 2137:) 2129:-- 2114:) 2006:. 1989:⊆ 1930:∈ 1923:∫ 1803:∈ 1774:) 1673:¯ 1664:− 1651:∞ 1646:∞ 1643:− 1639:∫ 1630:σ 1572:) 1539:) 1456:) 1437:) 1422:) 1367:) 1311:∫ 1224:∞ 1219:∞ 1216:− 1212:∫ 1145:) 1135:dx 1115:dx 1105:) 1040:) 1022:dx 1020:+ 1010:dx 1008:) 966:∫ 953:? 921:Ω 917:∫ 891:) 813:) 800:is 714:) 693:) 689:• 664:) 660:• 606:∗ 566:Pr 506:→ 503:Ω 477:ν 417:μ 390:→ 387:Ω 360:. 343:∗ 300:↦ 4040:) 4036:( 4023:. 4016:. 3922:( 3904:( 3878:( 3864:( 3850:( 3832:( 3816:( 3791:( 3777:( 3758:( 3743:( 3732:( 3710:( 3704:g 3691:( 3685:n 3671:V 3668:d 3648:. 3645:y 3642:d 3639:+ 3636:y 3630:) 3625:n 3621:x 3617:, 3611:, 3606:1 3602:x 3598:( 3595:g 3589:y 3579:n 3575:g 3562:( 3543:V 3540:d 3530:2 3526:) 3520:n 3516:x 3512:, 3506:, 3501:1 3497:x 3493:( 3485:j 3481:x 3472:g 3461:n 3456:1 3453:= 3450:j 3440:) 3435:n 3431:x 3427:, 3421:, 3416:1 3412:x 3408:( 3405:f 3397:) 3392:n 3388:x 3384:, 3378:, 3373:1 3369:x 3365:( 3362:g 3359:= 3356:y 3324:( 3309:( 3282:( 3258:( 3233:( 3206:( 3165:( 3151:( 3135:( 3090:x 3086:x 3084:( 3082:f 3063:( 3037:2 3033:2 3009:1 3005:1 2972:( 2966:2 2962:1 2958:2 2954:1 2927:2 2923:2 2899:1 2895:1 2871:2 2867:1 2863:2 2859:1 2818:( 2741:( 2721:( 2700:( 2671:( 2651:( 2603:( 2584:( 2530:( 2458:( 2439:( 2419:( 2402:( 2388:x 2386:( 2384:f 2376:x 2372:x 2359:( 2339:x 2336:d 2316:x 2313:d 2310:+ 2307:x 2297:x 2279:x 2276:d 2271:) 2268:x 2265:( 2262:f 2238:x 2235:d 2230:) 2227:x 2224:( 2221:f 2199:( 2179:( 2156:( 2133:( 2110:( 2093:R 2067:n 2062:R 2039:R 2017:X 1993:X 1985:A 1957:X 1953:d 1950:) 1947:x 1944:( 1941:f 1934:A 1927:x 1919:= 1916:) 1913:A 1910:( 1907:P 1883:X 1862:) 1859:x 1856:( 1853:f 1832:X 1828:d 1807:X 1800:x 1770:( 1706:x 1703:d 1698:) 1695:x 1692:( 1689:f 1684:2 1680:) 1670:x 1661:x 1658:( 1633:= 1568:( 1535:( 1452:( 1433:( 1418:( 1363:( 1354:. 1342:x 1339:d 1334:) 1331:x 1328:( 1325:f 1307:= 1304:) 1295:x 1286:( 1283:p 1254:1 1251:= 1248:x 1245:d 1240:) 1237:x 1234:( 1231:f 1141:( 1131:x 1127:x 1125:( 1123:f 1101:( 1085:x 1065:x 1062:d 1036:( 1030:A 1026:x 1018:x 1014:x 1006:x 992:x 989:d 984:) 981:x 978:( 975:f 970:A 941:x 938:d 935:) 932:x 929:( 926:f 887:( 871:) 868:x 865:( 862:F 856:x 852:d 846:d 840:= 837:) 834:x 831:( 828:f 809:( 783:. 780:) 777:x 774:( 771:F 765:x 761:d 755:d 749:= 746:) 743:x 740:( 737:f 710:( 685:( 656:( 632:R 611:P 602:X 581:] 578:x 575:= 572:X 569:[ 563:= 560:) 557:x 554:( 551:f 531:X 510:R 500:: 497:X 457:) 452:A 447:, 442:X 437:( 395:X 384:: 381:X 348:P 339:X 318:] 315:a 312:= 309:X 306:[ 303:P 297:a 243:. 138:. 53::

Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.

Index