Talk:Random variable

Definition is not correct

The text says that there are two types of random variables - discrete and continuous; which is unfortunately not true. If a random variable is not discrete, which means that the set of its realizations is not countable, it does not necessary mean that it has a density. For instance, if {1,2, [3;4]} represents the set of the realizations of a random variable, such that {1}, {2}, [3;4] occur with non-zero probability, then it is not possible to construct a continuous distribution function although the realizations are not countable. —Preceding unsigned comment added by Bcserna (talk • contribs) 12:43, 21 October 2009

Wouldn't your example be considered a "mixed" type? The "mixed" type is mentioned in the last sentence of that same paragraph. (Perhaps, we should clarify that there are actually 3 types, instead of 2?) Jwesley78 (talk) 12:54, 21 October 2009 (UTC)

A variable for which there does not exist a countable set with probability 1 is a continuous variable, whether or not there exists a probability density (with respect to Lebesgue measure), and whether or not there exist individual points with non-zero probability. In connection with the example given by Bcserna, it is true that the distribution function is discontinuous, but "continuous random variable" is not synonymous with "random variable with a continuous distribution function". It is true that most commonly when people refer to a "continuous random variable" they have in mind an absolutely continuous distribution with a probability density, but this is not a requirement. However, the present wording of this section of the article is faulty in more than one respect, for example because it asserts that in the continuous case the probability of any one value is always zero, and because it asserts that in the discrete case the probability of one value is never zero. I shall try rewording it in an attempt to clarify the issue. JamesBWatson (talk) 20:51, 22 October 2009 (UTC)

But the article also asserts that: "This categorisation into types is directly equivalent to the categorisation of probability distributions". This is contradictory with the given definition of a continuous random variable. --Tomek81 (talk) 18:30, 18 March 2010 (UTC)

I think the point in the above example is that one of the possible outcomes is, in some sense, interval-valued. Melcombe (talk) 10:01, 23 October 2009 (UTC)

I don't understand that. What do you mean by "outcomes"? For any random variable on a subset of the real line events can be intervals, whether the variable is continuous or discrete). The point of the example, as I understand it, is that it is possible to have a distribution in which part of the probability is concentrated on individual points (as in a discrete variable) and part spread out over an individual (as in a typical continuous distribution). Such a distribution, as Bcserna correctly realized, does not have a continuous distribution function. Nevertheless the expression "continuous random variable" includes variables having such distributions. JamesBWatson (talk) 13:58, 26 October 2009 (UTC)

My reading of the OP was that [3;4] represented an interval, and that a possible outcome/observation is the interval [3;4], so that the value of the ransom variable would be the interval [3;4]. Of course there would be ways of data-coding that could indicate that and lead to a discrete distribution. It is unfortunate that the article doen't start by saying that it starting with/only about scalar, real-valued random variables. I have recently added a "see also" to multivariate random variable, but it is not good as an extension of what is here either. Melcombe (talk) 16:56, 26 October 2009 (UTC)

I think the paragraph is fine as it stands, does anyone other than the original poster disagree with that? 0¹⁸ (talk) 18:51, 18 March 2010 (UTC)

Yes, I disagree. The definition of continuous random variables is not standard. Plus, with the given non-standard definition, Example 1 under Functions of random variables is incorrect, since that example assumes P(X=c)=0 for continuous variables (which should have been used as the definition, but it's not).--216.239.45.4 (talk) 19:00, 18 March 2010 (UTC)

So you think that the reference (Rice's textbook) is non-standard? Rice is, as far as I can tell, the most common intro stats text book. What is the sense of standard if Rice is non-standard? 0¹⁸ (talk) 22:33, 18 March 2010 (UTC)

How a definition can be incorrect when we don't have a definition? Lead section contains just some introductory notes. Anyways, there are 3 “pure” types of random variables: discrete, (absolutely) continuous, and singular. Any random variable is representable as a sum (or mixture) of these three. And the sum means in the sense of probability distributions: $F_{X}=\alpha F_{D}+\beta F_{C}+(1-\alpha -\beta )F_{S}$. Reference: Lukacs (1970). Characteristic functions. London: Griffin.  // stpasha »  19:13, 18 March 2010 (UTC)

Are singular random variables a pathological example? i.e. we usually don't state things in introductory articles like, any function with countable singularities... maybe that would be better for a "see also." 0¹⁸ (talk) 22:49, 18 March 2010 (UTC)

Singular cdf is continuous but non-differentiable. This is indeed “pathological example” in the sense that such rv's cannot be observed “in practice”. And that’s whe i’m saying that we need a proper definition session, because this stuff cannot be discussed in the lead.

The OP of this thread is correct: the claim in the lead is incorrect, and if it is supported by a reference then the reference is incorrect (or misinterpreted).  // stpasha » 

I cited a paragraph in the reference (by page). Why don't you read it and decide what you think. If it is misinterpreted, lets change it. Otherwise, I think we have to pick our reference and I'd argue for Rice's book over Lukacs and I think we could count references or count classes that use it as a basis for choosing. 0¹⁸ (talk) 23:12, 18 March 2010 (UTC)

I don't have either Rice(1999) or Lukacs(1970) books right now, however an authoritative reference would be something like Billingsley(1995) (which incidentally i don't have either) — so it seems a trip to the library is in order. But regardless of what the correct definition is, i think the entire discussion of discrete/continuous/mixed topic should be move out of the Lead section into probably its own. The topic is just too subtle and perhaps too complicated for the lead.

As of right now, the lead says only about where the random variables are used, but fails to mention what a random variable IS. I see this as a gargantuan drawback.  // stpasha » 

Actually, I just reread Rice, and I don't think it really comments on the question at hand. I agree with you that the issue of the definition is a little problematic. Part of the problem is that the definition of a random variable is not intuitive so I think the idea is to not come out and say it because it will confuse and require lots of explaining. BTW, I'm not defending this as a good writing style. 0¹⁸ (talk) 00:26, 19 March 2010 (UTC)

The introduction contradicts itself. It claims that "There are two types of random variables: discrete and continuous." Later it claims that "For a continuous random variable, the probability of any specific value is zero." It is not true that for every non-discrete random variable, the probability of a specific value is zero. Later in the same paragraph such "mixed" variables which are neither discrete nor continuous are mentioned, which contradicts the statement that there are only discrete and continuous variables. Tomek81 (talk) 20:04, 21 November 2010 (UTC)

In order to have a random variable there should be something random about it. A density function (usually called a pdf, although a density function does not have to have anything in common with probability) is extant everywhere within its support with magnitude equal to a unit area at any point within that support. Such functions are not randomized in any sense of the word random. I would prefer a non-controversial definition because to do otherwise creates considerable confusion. For example, a discrete convolution of two RV's is the paired sums of coincident random outcomes. If one confuses this with convolution integral pdf's there is no pairing of random outcomes to sum and we are stating a null. CarlWesolowski (talk) 06:08, 19 February 2019 (UTC)

"there should be something random": like it or not, formally, there is nothing random about it.

Like the alligator pear that is neither an alligator nor a pear and the biologist’s white ant that is neither white nor an ant, the probabilist’s random variable is neither random nor a variable.

(Alligator pear = avocado; white ant = termite.)

Similarly, mathematics describes motion by a function. You may say: something should be changing... But no, nothing is changing in the (eternally static) mathematical universe. Mathematical model of reality is never identical to the reality. See also #Confusing sentence below.

About density functions: the article does not say that a random variable is its density function (nor its distribution); it says that a random variable is a function from a probability space to a sample space. Right?

Boris Tsirelson (talk) 07:14, 19 February 2019 (UTC)

References

Functions of random variables

This section was removed. You may check the old revision just before the removal.

Confusion between "variables" and "functions"

The definitions given in this article, including the mathematical one, call a random variable a function. Yet the lead paragraph also states that it is a variable (as its name suggests). To the best of my understanding, a variable is not the same as a function. A variable is a quantity that may vary. A function may be used to describe how one variable relates to another variable, but the function itself does not vary, of course. This confusion should be cleared up: is a random variable a function, the output of a function, or something else? JudahH (talk) 14:40, 2 May 2019 (UTC)

"Like the alligator pear that is neither an alligator nor a pear and the biologist’s white ant that is neither white nor an ant, the probabilist’s random variable is neither random nor a variable." (See User:Tsirel#Oddities of mathematical terminology.)

(Alligator pear = avocado; white ant = termite.) Boris Tsirelson (talk) 17:24, 2 May 2019 (UTC)

Well and good, but the very first sentence of this Wikipedia article begins, "In probability and statistics, a random variable, random quantity, aleatory variable, or stochastic variable is a variable" [emphasis mine]. Clearly, there is some confusion here that ought to be cleared up. JudahH (talk) 02:24, 5 May 2019 (UTC)

On one hand, you are right. But on the other hand, see what is written in the linked article "Dependent and independent variables" (Section "Mathematics"): A symbol that stands for an arbitrary input is called an independent variable, while a symbol that stands for an arbitrary output is called a dependent variable. And Subsection "Statistics" there... Boris Tsirelson (talk) 03:58, 5 May 2019 (UTC)

I read that, but I still don't understand what this article means by calling a random variable both a "variable" and a "function". A independent variable is the input to a function. A dependent variable is the output of a function. The function that links the two is not a variable.

Can you clarify for me any further? JudahH (talk) 16:31, 5 May 2019 (UTC)

In a math encyclopedia, of course, it would be a function, nothing else. (Well, maybe an equivalence class of functions.) (For example, here.) The problem is, that Wikipedia is not intended for mathematicians. Most readers of the "Random variable" article are not mathematicians. For them a random variable is not a mathematical notion, but a natural phenomenon (or something like that). The mathematical notion is rather a mathematical model of this phenomenon. We do not have two different words, one outside math, the other inside math. This makes troubles (and not only for "random variable"). I see no escape from these troubles. Do you? And do not forget, Wikipedia cannot establish a new terminology; we must follow the widely used terminology. Boris Tsirelson (talk) 17:27, 5 May 2019 (UTC)

I believe that the article should do one of two things. (1) If indeed the phrase "random variable" is commonly used in two distinct, albeit related senses, (in the mathematical sense: a function; in the colloquial sense: the output of that function), then these senses should be careful distinguished from each other. (2) Alternatively, if the article is meant to explain one particular sense of "random variable", then that sense, and that sense alone should be carefully explained. What the article shouldn't do, but does at present, is give a confused muddle of explanations which alternate between senses of the word without ever making it clear that two different senses are involved here.

For what it's worth, an earlier version of the article contained the following text: Contrary to its name, this procedure itself is neither random nor variable. Rather, the underlying process providing the input to this procedure yields random (possibly non-numerical) output that the procedure maps to a real-numbered value.

Ladislav Mecir deleted the above text and replaced it with the following:

It is a variable (specifically a dependent variable), in the sense that it depends on the outcome of an underlying process providing the input to this function, and it is random in the sense that the underlying process is assumed to be random.

It appears to me that the earlier text, while accessible to the non-mathematician, clearly defined a random variable as a function, distinguishing it from a "variable" and pointing out that its name was therefore a misnomer. The revised text muddies the water by insisting that it is in some sense a "dependent variable", even though, if a random variable is a function, its output is the dependent variable. JudahH (talk) 19:33, 5 May 2019 (UTC)

As for me, you are right. Do it, if you want. But note that (a) I am not a dictator here, everyone may disagree and revert; and (b) you may be asked, where are the sources of all your claims (which is often a problem in such cases). Boris Tsirelson (talk) 19:54, 5 May 2019 (UTC)

@JudahH: you asked "Can you clarify for me any further?" What you seem to insist on, is that the 'random variable' term may be, in a formal way, inappropriate or inaccurate. That is rather shortsighted. Indeed, a random variable is formally defined as a function of a certain type, as you correctly note. Note, however, that formalizing a random variable X as a function, you would have to write X(ω) = 1, which is not done. Instead, the 'X' symbol is used also as a value of the output - see the literature, where this is described, just one textbook citation for all: "Notationally, we shall use the same symbol for the random quantity and its output value." That is done because the original goal was to formalize the notion of a random variable, not the notion of a function. This convention (which can be characterized as a misuse of the notation by formalists) results in a formula X = 1 being used instead of the formally correct formula X(ω)=1. Ladislav Mecir (talk) 07:14, 6 May 2019 (UTC)

@Ladislav Mecir:, I am not trying to insist on a particular definition of "random variable". All I would like is for the meaning of random variable—whatever it be—to be spelled out clearly in this article. Owing to the confused terminology that prevails, I have frequently felt uncertain about what "random variable" meant. Drawing attention to the 'formal inaccuracy' was thus not meant to be critical of its users, but rather to clarify a point that might otherwise escape the reader.

I gather from the quotation you give above that the term "random variable" is conventionally used to denote the function, not its output; however the symbol for this function, a capital letter, is also used to denote the output of the function (presumably, context is what determines which way X is being used). Is this correct? If so, I submit that the definition given for "random variable" should carefully describe it as a function, not as the output, but additionally note the point about the interchangeable symbol. If the term "random variable" is also used interchangeably for the function and its output, then this should be explicitly stated. Does that sound reasonable to you? JudahH (talk) 17:55, 6 May 2019 (UTC)

On one hand, yes, I confirm, I usually write (in research papers and lectures) the conventional $P(X=1)$ instead of the formal $P(\{\omega \mid X(\omega )=1\})$ or $P{\big (}X^{-1}(\{1\}){\big )}.$ On the other hand, this is just a notational convention; in words, writing "the random variable $X$" I always mean that $X$ is the function, not its value. Boris Tsirelson (talk) 18:00, 6 May 2019 (UTC)

But a harder problem persists: mathematicians usually think about a mathematical model; others usually think about the modeled phenomenon; and all use the same word. Boris Tsirelson (talk) 18:03, 6 May 2019 (UTC)

See also Talk:Central_limit_theorem#There_is_a_mix-up_here_between_an_observation_and_a_random_variable. Boris Tsirelson (talk) 18:13, 6 May 2019 (UTC)

I do not think this is a "mathematicians" versus "others" terminology issue. Let me illustrate it on real numbers. For one mathematician, a real number may be an element of a Dedekind-complete ordered field, while for another, a real number may be an equivalence class of Cauchy sequences of rational numbers. These two might not even agree that a real number must necessarily be a set. The commonly used definition of random variables defines them as functions, but that should not hide the fact that, for example Pafnuty Chebyshev, used the notion of a random variable earlier than Kolmogorov gave his measure-theoretic model of probability, which allowed random variables to be defined as measurable functions. I see it as myopic to insist that random variables are not random or that they are not variables when that is the original meaning of the notion, while the definition using measurable functions is merely a set-theoretic model (albeit the most commonly used one at present) of the original notion that can as well be modelled differently (for example as equivalence classes of functions as Tsirel pointed out), or axiomatized as members of certain algebraic structures, etc. Ladislav Mecir (talk) 07:20, 7 May 2019 (UTC)

I agree... but I appear to agree with both of you... so what? I am still not sure what to write in the article, and how to source it... Boris Tsirelson (talk) 07:38, 7 May 2019 (UTC)

The mainstream dictionaries that I've looked at indeed define a random variable as a kind of variable (the OED, for instance, writes: "Statistics. a variable whose values are distributed in accordance with a probability distribution; a variate"). As you say, they seem to have been originally conceived as variables. On the other hand, I gather that modern mathematicians regard them as functions. So it's complicated. But using a mishmash of competing terminology to explain them in a single article ("...[A] random variable...is a variable.... More specifically, ... a function") is a recipe for confusion.

Please understand that I'm not trying to insist on any particular definition or to brush others under the rug. All I would like is for the article to be clear. Perhaps the introductory section of the article could describe what random variables are used for (e.g. "modeling uncertainty"), without committing itself to a definition, leaving the niceties to be sorted out by the dedicated definition section? JudahH (talk) 15:30, 7 May 2019 (UTC)

What about saying that

In probability and statistics, a random variable, random quantity, aleatory variable, or stochastic variable is described informally as a variable whose values depend on outcomes of a random phenomenon. More formally, a random variable is treated as a function that maps the outcomes of a random process to numerical quantities, typically real numbers.

, leaving out the subsequent sentence. I think that this formulation may be acceptable for all readers. Ladislav Mecir (talk) 06:47, 8 May 2019 (UTC)

What is mean by "formally" here? Surely not "mathematical", since there is no notion "unpredictable process" in mathematics, nor "numerical quantities". Maybe, "more formally"? And not "defined" (which sounds too mathematical) but "treated"? Boris Tsirelson (talk) 07:37, 8 May 2019 (UTC)

Also, what is mean by "unpredictable"? Unpredictable in principle, or in practice? If a gamer believes in Laplace determinism, for him the future may be predictable in principle but not in practice. If a physicists tosses a coin, he may think that the result is sensitive to trajectories of molecules, but not to quantum uncertancy. Is it predictable then? Beware of deep water. Boris Tsirelson (talk) 07:53, 8 May 2019 (UTC)

Also, given that one got 7 "heads" in 10 tosses of a fair coin, we may calculate the conditional probability of 3 "heads" in the first 5 tosses. In this case, the "random" value is already known to him, but unknown to us. Also in game theory it often happen that something is known to one player but unknown to another; and each one calculates his own conditional probabilities. Boris Tsirelson (talk) 08:03, 8 May 2019 (UTC)

@Tsirel: I adjusted the wording to reflect your suggestions. Check the wording now, please. Ladislav Mecir (talk) 08:08, 8 May 2019 (UTC)

I see, now it is better.

Here is another subtlety. The strong law of large numbers (in contrast to the weak law) is about the result of an infinite sequence of independent random "head/tail"s. Really infinite, not "very large". This transcends any physical reality.

Also, think about a mathematician that uses the probabilistic method, say, he/she proves existence of some astonishing graph by calculating the probability that a random graph fits (and getting a positive probability). Here, the physical reality is rather irrelevant. Boris Tsirelson (talk) 08:20, 8 May 2019 (UTC)

Is the distinction between a variable and a function, then, one of formality? Do you mean that there's no truly formal way to define "variable" at all? Is that the reason that mathematicians moved to use "random variable" to denote a function instead?

I ask because I'd really like to know. One of the things that's been confusing me about this topic is that I haven't understood the motivation for revising the definition of "random variable" in the first place. Why did formalizing the concept of "random variable" result in changing its denotation from a variable whose value is determined by applying some (constant, given) function to a random input, to denoting the function itself instead? Can either of you give me any insight, please? JudahH (talk) 15:14, 8 May 2019 (UTC)

Indeed, there is no notion of "variable" in mathematics. (Just try to define it, as a mathematical object, taking into account that all mathematical objects are sets nowadays.) There is a notion "variable" in metamathematics, mathematical logic, model theory etc.; there, a variable is a part of a formal language; but that is rather irrelevant. Surely, probability theory does not need theory of formal languages. Boris Tsirelson (talk) 16:18, 8 May 2019 (UTC)

Whoever feels able to define "variable", should be ready to answer (or at least understand) such questions as, for instance: how many variables exist (in the whole mathematical universe)? A finite set of all variables? Or countably infinite? Or maybe continuum? Or more? Or maybe all variables are a proper class, not a set? Why? Boris Tsirelson (talk) 16:40, 8 May 2019 (UTC)

The situation is somewhat similar to this: "correlations have physical reality; that which they correlate does not" (David Mermin; see here). In this spirit I can say "functional dependencies have mathematical reality; that which depend does not". Boris Tsirelson (talk) 16:59, 8 May 2019 (UTC)

JudahH, you wrote: 'One of the things that's been confusing me about this topic is that I haven't understood the motivation for revising the definition of "random variable" in the first place.' - this one is actually easy to answer: the notion of a random variable was only informal before it was formalized. So, the motivation was to have a formal description of the random variable notion, similarly as there are formal descriptions of other mathematical objects. E.g. the notion of a set was also informal originally, and I hope that you know some reasons why there was a need to formalize it. More challenging is, why random variables were called random variables. I do not think there is any difficulty with the 'random' adjective, but why 'variable'? I think that the original idea was that the objects in question were assumed to acquire different values (i.e. vary, that is why the 'variable' word, used in a different context than in formal languages) if the experiments are performed repetitively. To present also a different view, note that many Bayesians dislike the notion of a 'variable' in this context. Bayesians, such as Bruno deFinetti, say that there is no need to restrict the notion to contexts involving repeated trials over which the quantity may vary, proposing an alternative notion of a random quantity. Ladislav Mecir (talk) 22:52, 8 May 2019 (UTC)

A linguistic note: in my native language, the most commonly used term is actually a literal translation of the 'random quantity' term. Ladislav Mecir (talk) 00:16, 9 May 2019 (UTC)

About edit of today: "maps the outcomes of a random process to numerical quantities" — the "random process" bothers me. What does it mean? A stochastic process? No; "a stochastic or random process is a mathematical object usually defined as a collection of random variables", therefore a random variable must be defined before a random process. A real-life random process, as part of the physical reality? Hardly so. As noted above, physical reality is not always relevant; and time is not always relevant. What else? Boris Tsirelson (talk) 06:27, 10 May 2019 (UTC)

Maybe, just "maps outcomes to numerical quantities"? Rather unclear, of course; but at least, it transfers the problem to the "Outcome (probability)" article (and there, only a part of relevant meanings is stipulated, for now; the same problem in the "Experiment (probability theory)" article). Boris Tsirelson (talk) 06:50, 10 May 2019 (UTC)

Really, the only appropriate meaning of "random process" here would be "random choice (really performed or only imagined) of a point from a probability space"; but this would be "original research" that satisfies no one. A mathematician would be unhappy of "random choice"; a non-mathematician—of "probability space". Boris Tsirelson (talk) 08:52, 10 May 2019 (UTC)

Then I think that the best approach might be to kill two birds with one stone. The fact is that the lead section currently has seven paragraphs, which, according to the WP:MOS is too much. What about deleting the seventh paragraph of the lead section, replacing the second sentence you do not like anyway by the wording:

The formal mathematical treatment of random variables is a topic in probability theory. In that context, a random variable is understood as a function defined on a sample space whose outcomes are typically real numbers.

Ladislav Mecir (talk) 09:40, 10 May 2019 (UTC)

References

OK. Boris Tsirelson (talk) 09:47, 10 May 2019 (UTC)

Wrong set expression in definition?

Under Definition, the right side of the definition equation seems strange. Shouldn't there be a "such that" (:) instead of "given" (|) inside the set expression? See eg. the first paragraph under Distribution functions, where an example is given for the event set X=2 in the format using "such that", i.e. {ω : X(ω)=2}. — Preceding unsigned comment added by 212.71.89.93 (talk) 09:04, 15 November 2020 (UTC)

I dont know why you give different meanings to ":" and "|". In set-builder notation, these two symbols are clearly presented as synonyms, and this follows the common usage in mathematics. D.Lazard (talk) 09:43, 15 November 2020 (UTC)

2022 comments

In The discrete random variable section the function F is not well defined and we have that every real number has a probabilty which implies that the probablity of the set of real number is not finite which should be equals to one , under that condition the probality function F has a limit of 1 at infinity. — Preceding unsigned comment added by Boutarfa Nafia (talk • contribs) 16:49, 20 March 2022 (UTC)

In the "function of Random variable" section the formula where the g function is non bijective is wrong , take for example for every real number g(x)=x^2-1 and y=0 , we have that P(g(X)<=0)=P(-1<=X<=1) — Preceding unsigned comment added by Boutarfa Nafia (talk • contribs) 03:26, 22 March 2022 (UTC)

In the formula of the density of probabilty of the function P(g) where g is a real function , there is no absolute value on g' the derivative of g , so it should be g' not the absolute value of g'. — Preceding unsigned comment added by Boutarfa Nafia (talk • contribs) 04:09, 22 March 2022 (UTC)