Talk:Probability density function

message: I got two problems with this section.
First is, that I don't see (because it is not explained) why : f(x) = \frac{\mathrm{d}}{\mathrm{d}x} F(x) .
should hold, i.e. why F is differentiable in the first place.
Second, the title of the section is not explained. What is a univariate distribution? 
Please fix this. Quiet photon (talk) 17:01, 2 March 2010 (UTC)
replies: text: Yeah, it shouldn't be true in general that  f(x) = \frac{\mathrm{d}}{\mathrm{d}x} F(x). 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. --Theodds (talk) 18:52, 29 June 2010 (UTC)
thread_title:  Contiunous univariate distributions  

message: Intuitively, if a probability distribution has density f(x), then the infinitesimal interval [x, x + dx] has probability f(x) dx.
Arrgh, now you tell me this has something to do with \int_\Omega f(x) dx?
: Certainly.  An integral \int_A f(x)\,dx is intuitively thought of as the sum of infinitely many infinitely small quantities ƒ(x) dx, each equal to the area below the graph of ƒ above the interval from x to x + dx, as x runs through the set of all numbers in the set A.  That applies to integrals generally, not just those in probability theory. Michael Hardy (talk) 00:50, 12 June 2008 (UTC)
replies: text: I completely agree that this view is not intuitive if one is used to the proper definition, because then dx is just some syntactical delimiter capturing the variable x, 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?). Suprahili (talk) 22:14, 19 May 2009 (UTC)
thread_title:  "Intuitive" 

message: 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 
* f(x) is greater than or equal to zero for all values of x 
* The total area under the graph is 1. Refer to equation below.
: \int_{-\infty}^\infty \,f(x)\,dx = 1  
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 . 
For example: the variable x being within the interval 4.3 < x < 7.8  would have the actual probability of  
:p(4.3<x<7.8) = \int_{4.3}^{7.8} f(x)\,dx.
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 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. Worik (talk) 02:25, 26 April 2010 (UTC)
thread_title:  Simple English 

message: 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 does it have a "real-world" correlate?  How should "density" be interpreted? --anon
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
: One more answer: If you have a random variable, 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. Oleg Alexandrov 15:03, 2 Jun 2005 (UTC)
thread_title:  what is a probability density? 

message: I don't have time to correct it now, but the page Probability Density links to Probability Amplitude, which is about quantum mechanics. I think that should be a disambiguation page. --anon
: I redirected Probability Density to Probability density, which I made into a disambig. Oleg Alexandrov 18:04, 18 August 2005 (UTC)
thread_title:  Probability Density 

message: 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 :-) Albmont 18:14, 9 November 2006 (UTC)
thread_title:  Self Inconsistency 

message: if we know the joint density function f(x,y), how to get the marginal density function fY(x) & fX(y)? Jackzhp 01:37, 1 December 2006 (UTC)
replies: text: I think you integrate in x to get fX and integrate in y to get Fy. But I am not  sure. Ask Michael Hardy, he should know. Oleg Alexandrov (talk) 16:17, 1 December 2006 (UTC)
thread_title:  marginal density function

message: 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. Jackzhp 18:00, 1 December 2006 (UTC)
replies: text: It's the convolution of the two densities. Michael Hardy 19:44, 5 December 2006 (UTC)
thread_title:  addition 

message: It would be helpful to add the standard deviation formula for completeness  \sigma = \sqrt{\int_{-\infty}^\infty \,(x-\bar x)^2 f(x)\,dx}     Gp4rts 19:02, 5 December 2006 (UTC)
replies: text: I've added this at the bottom (the variance, not the SD, but close). Michael Hardy 19:44, 5 December 2006 (UTC)
thread_title: Standard deviation

message: 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?
MisterSheik 21:34, 27 February 2007 (UTC)
replies: text: That is covered in the section "Probability function associated to multiple variables". -Icek 05:47, 17 April 2007 (UTC)
thread_title: Reals?

message: 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.  Boris B (talk) 03:29, 13 April 2008 (UTC)
thread_title:  Uses of PDF vs. distribution function 

message: 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 so.  One that I wasn't sure how to fix was "ƒf" in Further details.  How should that render? --Autopilot (talk) 21:40, 28 December 2008 (UTC)
thread_title:  Math mode versus html encodings 

message: Loosely, one may think of f(x)\, dx as the probability that a random variable whose probability density function is ƒf is in the interval from x to x + dx, where dx is an infinitely small increment.
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).
Richard Giuly (talk) 18:35, 18 February 2009 (UTC)
replies: text: I find that deletion unfortunate.  The probability of being between x and x + dx is infinitely small, and f(x) dx is also.  When BOTH approach zero, their ratio may approach a finite nonzero value.  This is standard use of the dx notation in calculus, except that the way calculus has been taught to freshmen in recent years has been influenced by undue squeamishness. Michael Hardy (talk) 00:43, 20 May 2009 (UTC)

text: 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 concern. Jason Quinn (talk) 23:03, 22 April 2011 (UTC)
thread_title:  Loosely, one may think of <math>f(x)\, dx</math> as... 

message: Are "probability distribution function" and "probability density function" synonymous?
To me a "probability distribution function" is the distribution function, not the probability density function..
131.175.127.242 (talk) 08:52, 21 October 2009 (UTC)
replies: text: See Probability distribution function ... this seems correct in saying that different sources use different meanings. Melcombe (talk) 09:06, 21 October 2009 (UTC)
thread_title:  Terminology 

message: Why would somebody put incorrect information into the lead and then insist to keep it there? I'm talking about the “… often referred to as the probability distribution function” 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 find any published source which would have defined probability distribution function as density, although some of them seem to use the term in the “density” sense without ever defining it.
The confusion seems to originate from physics, where the term “distribution function” was used by Maxwell to describe the probability density function of gas particles multiplied by the physical density of those particles.
Stating in the very first sentence that “probability density function” is the same as “probability distribution function” is at least misleading.  … stpasha »  20:27, 26 November 2009 (UTC)
replies: text: 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 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." Melcombe (talk) 10:57, 27 November 2009 (UTC)
replies: text: 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 separate article for it, quite distinct from both the pdf and cdf.  … stpasha »  20:05, 27 November 2009 (UTC)
thread_title:  Probability distribution function 

message: See also Wikipedia talk:WikiProject Mathematics#Codomain of a random variable: observation space?. Boris Tsirelson (talk) 16:54, 27 March 2010 (UTC)
thread_title:  Codomain of a random variable: observation space? 

message: "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 [a; b] (for example, a variable being worth 1 if X is in [a; b], and 0 if not)." — Does anyone understand it? I do not. Boris Tsirelson (talk) 07:43, 20 September 2010 (UTC)
replies: text: 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 density, without the quotation marks) that is a weighted sum of Dirac delta functions. It is clear the entire section has to be rewritten to make it comprehensible to a person who does not know that already.  // stpasha »  16:28, 20 September 2010 (UTC)
replies: text: 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. Boris Tsirelson (talk) 17:27, 20 September 2010 (UTC)

text: Anyway, I have deleted it. Boris Tsirelson (talk) 18:55, 23 September 2010 (UTC)
thread_title:  Link between discrete and continuous distributions 

message: 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 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(x1) / f(x2) = P(x = x1) / P(x = x2)? No, since the right side of this is zero over zero.  So the intended meaning must be something about the limit as \epsilon goes to zero of P(x1 < x < x1 + \epsilon) / P(x2 < x < x2 + \epsilon), or something like that. That interpretation is unfamiliar to me (albeit intuitive as a counterpart to the interpretation of discrete probability functions). 
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(x1) / f(x2) = P(x = x1) / P(x = x2); and (2) put it later in the article, with a careful statement of what is meant and with a citation.  Comments, anyone?  Duoduoduo (talk) 15:38, 14 November 2010 (UTC)
replies: text: Yes, one may say that it is about the limit as \epsilon goes to zero of P(x1 < x < x1 + \epsilon) / P(x2 < x < x2 + \epsilon). 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.) Boris Tsirelson (talk) 16:12, 14 November 2010 (UTC)

text: 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 any 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 f(x) = dPr[ x < X < x+dx ] / dx, 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.  // stpasha »  18:08, 14 November 2010 (UTC)
thread_title: "Relative probability"

message: The section Probability_density_function#Families_of_densities 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 these is a domain in the sense of Domain (mathematical analysis).  I thought on a first reading that should be "the domain of a family of densities" the domain of a function--specifically the function sending the parameter values to the corresponding distribution--but this doesn't seem consistent with what follows.  Can anyone clarify this? Jowa fan (talk) 12:08, 9 November 2012 (UTC)
replies: text: 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" 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. --Boris Tsirelson (talk) 14:09, 9 November 2012 (UTC)
replies: text: 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. Jowa fan (talk) 05:19, 10 November 2012 (UTC)
thread_title:  Families of densities and domains? 

message: The formula
: \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
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. Garfl (talk) 15:46, 16 October 2013 (UTC)
replies: text: Jacobian? No, you cannot make Jacobian out of a single function g of n 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 y<g(x_1,\dots,x_n)<y+dy. The formula is correct as far as the reader understands that dV is an infinitesimal element of (n-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. Boris Tsirelson (talk) 17:06, 16 October 2013 (UTC)
thread_title:  Dependent variables and change of variables : Multiple variables 

message: This recent edit  is very long and, I suggest, tedious. I propose that it be reverted. Isambard Kingdom (talk) 18:08, 16 November 2016 (UTC)
replies: text: Or to Wikiversity? Boris Tsirelson (talk) 18:52, 16 November 2016 (UTC)
replies: text: 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. --Steve (talk) 22:48, 16 November 2016 (UTC)
replies: text:  Agreed. McKay (talk) 02:40, 17 November 2016 (UTC)
thread_title:  Recent long (and possibly tedious) addition on the derivation of a pdf. 

message: 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, is unhelpful.  There's probably some room for improvement over the way it is now, but it should still convey the same overall gist.
Deacon Vorbis (talk) 00:31, 2 December 2016 (UTC)
replies: text: 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 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. Boris Tsirelson (talk) 06:14, 2 December 2016 (UTC)

text: Also, "relative likelihood" sounds like "likelihood ratio"; 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. Boris Tsirelson (talk) 07:00, 2 December 2016 (UTC)

text: On the other hand I agree that saying "the density is a function that describes the local density" is unhelpful. Boris Tsirelson (talk) 07:04, 2 December 2016 (UTC)
replies: text: "...is a function that quantifies how likely it is for this random variable be very close to a given value" or something like that?? --Steve (talk) 19:28, 3 December 2016 (UTC)
replies: text: I wonder, why this density is described so differently from, say, Density, Charge density, Energy density. On the level of lead, they could (or should?) look very similar. Boris Tsirelson (talk) 20:43, 3 December 2016 (UTC)

text: I like a text in “The elements of continuum biomechanics” by Marcelo Epstein, quoted here (pages 50-51). Regretfully, it is too long for a lead... Boris Tsirelson (talk) 20:49, 3 December 2016 (UTC)
thread_title:  Reverted change to opening sentence 

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