Talk:Random variable

message: See Wikipedia talk:WikiProject Mathematics#Codomain of a random variable: observation space?. Boris Tsirelson (talk) 16:53, 27 March 2010 (UTC)
thread_title:  Codomain of a random variable: observation space? 

message: "Expectation of a random variable"? Or rather, "Expectation of the distribution of a random variable"? The second option may be tempting, but leads to problems, as follows.
*If X≤Y almost surely then E(X)≤E(Y). Can we reformulate it in terms of distributions? Yes I can (can you?) but it becomes more technical and less intuitive.
*E(X+Y)= E(X)+E(Y) (always, not just for independent random variables). Can we reformulate it in terms of distributions? Yes I can (can you?) but it becomes much more technical and less intuitive.
*Also, think about Jensen inequality, first of all, (E(X))2≤E(X2), and many other statements.
Boris Tsirelson (talk) 07:08, 24 October 2010 (UTC)
thread_title:  Expectation of what? 

message: Could anybody explain why in the lead sentence the topic is associated with the probability theory? Since the random variable is a mathematical variable, and used as a pattern in probability theory and statistics, it would be more common to mention only mathematics.--Kiril Simeonovski (talk) 18:56, 13 November 2010 (UTC)
replies: text: Are you sure that "the random variable is a mathematical variable"? I am not. There is no formal notion of a "variable" in mathematics (there are sets, functions etc., but not "variables"), but even informally, a "variable" in mathematics (whatever it reasonably means) has no probability distribution. A random variable has. Boris Tsirelson (talk) 20:22, 13 November 2010 (UTC)
replies: text: Indeed, there are variables in mathematics, described as changeable values. Unlike random variable, other variables do not have probability distribution, but other types of distributions (note the frequency distribution). However, you're right. There is not exact definition about what in general a variable should include.--Kiril Simeonovski (talk) 23:13, 15 November 2010 (UTC)

text:  I agree with Boris, "variable" has a range of meanings within mathematics. In this case, a "random variable" is a particular type of mathematical object, one that (as I think about it anyway) has no value a-prior but that has associated with it a probability distribution function that determines values it could take on. I don't understand what Kiril means by "used as a pattern". I would say there are lots of types of objects in mathematics, one type is a "random variable", which has applications in probability and statistics. While I wouldn't be surprised if they are in some way isomorphic to something with broader applications in math, I think that's stretching it.  —Ben FrantzDale (talk) 17:00, 14 November 2010 (UTC)
replies: text: Thanks. I consider probability theory a branch of mathematics, same as algebra, geometry or calculus. The global usage and importance of the random variable in the field of probability theory made me compare it with another key mathematical terms, such as plane in geometry, or permutation in combinatorics. In the lead sentences of both, plane and permutation, is mentioned "mathematics" (not "geometry" and "combinatorics"), and therefore I used to do the same with the random variable (in general, no probability without random variable, as no geometry without plane). My notion is that the articles concerned with the probability theory are very well covered, which could be used to trim the other mathematical articles.--Kiril Simeonovski (talk) 22:38, 15 November 2010 (UTC)
thread_title:  Mathematics or probability theory 

message: Right now the first sentence says "In probability theory, a random variable, or stochastic variable, is a way of assigning a value to each possible outcome, that is element of a sample space."  This seems very awkward: a rv is a way of assigning? each possible outcome of what? How about changing it to something like "In probability theory [or better, in probability and statistics], a random variable or stochastic variable is a variable that can take on any of various values --  one for each possible situation that could arise (each possible element of a sample space).  Duoduoduo (talk) 21:47, 14 November 2010 (UTC)
replies: text: Hi Duoduoduo, good catch there. "a rv is a way of assigning?" >> "r.v. is a function that assigns a value to each possible outcome" is more accurate.  "each possible outcome of what?" >> It just tells about it in the second sentences, but sure it could be combined with the first sentence for the clarity. ~ Elitropia (talk) 22:19, 14 November 2010 (UTC)
replies: text:  And now I see "a random variable ... is a variable whose value results from a measurement on some random process"; the problem is that technically "random process" is a notion more complicated than "random variable" and in fact defined in terms of random variables. --Boris Tsirelson (talk) 16:20, 19 February 2011 (UTC)
replies: text:  Not only that: there is no "measurement" involved. The first sentence confuses the mathematical notion of random variable, which is well defined, with a supposedly empirical notion of a 'random' or 'stochastic' natural process (in the colloquial sense) that in practice is not defined empirically or otherwise, and is most likely meaningless. Better to stick to the purely mathematical definition. illywhacker; (talk) 21:11, 31 January 2012 (UTC)

text: A random variable is not a variable at all (nor is it random). That should probably clarified. 69.110.145.10 (talk) 16:14, 19 September 2012 (UTC)
thread_title: First sentence

message: I found the first example very confusing. It starts with "the following random variable : X=either head or tail."
However, the definition of a random variable given just below is "a random variable is a measurable function". In the example, where is the function ? What is the variable of the function, what is the outcome depending on the variable ?? I find that very unclear : it seems to define the state space only. - Nicolas. 152.81.114.116 (talk) 15:00, 28 December 2010 (UTC)
replies: text: Thanks for pointing this out.  Hope my revision has answered your objection.    Duoduoduo (talk) 15:33, 28 December 2010 (UTC)
replies: text: Thanks, it's indeed much better. I'm wondering if the notation X(w=head)=1, X(w=tail)=0 would be better ? Or X(w)=1 iff w=head ? It would clearly indicate the state space and would show the random variable as a function. I'm not expert enough to decide by myself.  152.81.114.116 (talk) 15:48, 28 December 2010 (UTC)
replies: text: How does it look now?  Duoduoduo (talk) 20:42, 28 December 2010 (UTC)

text: Thanks. I guess this is better - but I'm not a mathematician! 152.81.114.116 (talk) 21:59, 29 December 2010 (UTC)
thread_title:  Examples 

message: Does "a suitable measure" refer to the probability measure, P? If so, maybe this could be made explicit. If not, why do the reals need a measure here? They're said to serve as the first item of the tuple (E,script E), but no measure was required for (E,script E) where E is not R. Dependent Variable (talk) 16:40, 2 March 2011 (UTC)
replies: text: I agree, and remove the "suitable measure". --Boris Tsirelson (talk) 16:53, 2 March 2011 (UTC)
thread_title:  The real numbers with a suitable measure 

message: The former phrase "Formally, it is a function from a probability space, typically to the real numbers" was correct. The new phrase "and the probability (or probability density for continuous random variables) of each of these values is defined by a probability space, typically restricted to the real numbers" is either incorrect or vague. In which way does the probability space define the probability? What is typically restricted to the real numbers, the probability space?! Boris Tsirelson (talk) 17:06, 8 February 2012 (UTC)
replies: text: I agree that it is not a paragon of clarity but I think they are saying that you could not list out value such that your list would have non-zero measure. Sometimes, you are better off forcing the reader to click on the term in question. 018 (talk) 03:21, 9 February 2012 (UTC)
replies: text: Well, now these are clickable, and "set" replaced with "list". Boris Tsirelson (talk) 06:58, 9 February 2012 (UTC)
thread_title:  "try and improve comprehensibility" 

message: The sample space is the set of outcomes of the experiment. Possible outcomes are the number of eyes shown on the upper side of the die. These are numbers. The rv has as its values these numbers. This is a typical example of rv's of the type X(ω)=ω.Nijdam (talk) 06:58, 21 June 2012 (UTC)
replies: text: No, the possible outcomes are not numbers. They are the possible state of affairs after the random event. State of affairs is something like "there is a 6 shown on the upper side of the die". A clear difference needs ot be made between the outcomes of the random event and the value of the random variable. Else things won't make sense to anyone. --rtc (talk) 17:11, 21 June 2012 (UTC)
replies: text:  On one hand, this makes sense. But on the other hand, we work with mathematical models, not with the reality itself. A mathematical model is built out of mathematical objects. "The set of an apple and an orange" is a common abuse of the mathematical language, but really the mathematical universe contains no fruits; we encode fruits (and other real things) by mathematical objects. Likewise, you cannot send an apple by email, but you can send a picture encoded by bits, – and it is a conventional substitute for the apple... Boris Tsirelson (talk) 20:52, 21 June 2012 (UTC)
replies: text:  This is the classic "platonist" view of mathematics, which is by no means uncontroversial. There are views that hold the mathematical world in fact to be not as separate from the real world as the Planonists claim. Anyway, the state of affairs of a die after rolling it is not the number it shows, and thus they should not use the same mathematical representation even if that's okay as far as theory is concerned. It's just too confusing and does not give readers the right intuition. --rtc (talk) 21:05, 21 June 2012 (UTC)
replies: text:  So, I am not uncontroversial, :-( but classic. :-) Boris Tsirelson (talk) 05:49, 22 June 2012 (UTC)

text: Maybe both approaches could be mentioned. Indeed, the "nonclassic" approach is also not uncontroversial. Boris Tsirelson (talk) 05:49, 22 June 2012 (UTC)
replies: text: The recently added "example" (of a pair a measures related to individuals in a population) emphasises the utility of regarding the sample space as a generic set of items, leaving the "numerical value" of a random variable to be treated as part of the "function" in the "functions defined on a probability space," rather than as part of the sample space. Melcombe (talk) 09:36, 22 June 2012 (UTC)
thread_title:  Die 

message: I feel as if an explicit statement may be needed to extend
:f_Y(y) = \sum_{i} f_X(g_{i}^{-1}(y)) \left| \frac{d g_{i}^{-1}(y)}{d y} \right| .
When working with either a mixed distribution or functions where the determinant of the jacobian (or derivative if working with a single RV) become 0 or undefined (and hence non-invertable), there is a missing term.
:f_Y(y) = \sum_{i} f_X(g_{i}^{-1}(y)) \left| \frac{d g_{i}^{-1}(y)}{d y} \right| + \sum_{ y | \not \exist g_i^{-1}(y) } P\left(X= g_i^{\dagger}(y)) \, \delta(J(g)\right) 
:g^{\dagger}(y) = \inf\left\lbrace x \mid g(x) = y \right\rbrace .
This missing term must have a probability mass (not simply a density) to include. I am not sure if I wrote the math correctly, so someone should double check that. For example consider the example given on the following website (http://www.probabilitycourse.com/chapter4/4_3_1_mixed.php). The derivative turns to zero at y=1/2 AND that event has a probability mass of 3/4.  Mouse7mouse9 00:23, 7 May 2015 (UTC)
replies: text: Zero Jacobian at a point may lead to density (still integrable, but) infinite at the corresponding point; it cannot lead to an atom. If g is constant (that is, Jacobian is zero) in a region (with interior points), this leads readily to an atom. And of course, atoms of X (if any) lead to atoms of Y. Boris Tsirelson (talk) 05:38, 7 May 2015 (UTC)
thread_title:  Pedantics and Functions of random variables 

message: The concept of orthogonality of random variables seems not to be discussed in this article nor in a separate article. Should we include the definition and an example in this article or in a separate article as e.g. for independent random variables and uncorrelated random variables? Another option would be to add the topic to the very general article Orthogonality.
Fvultier (talk) 11:54, 26 June 2017 (UTC)
replies: text: Independence is more important than uncorrelatedness; the latter turns into orthogonality when expectations are zero; otherwise, orthogonality (of random variables) is of little importance. All that is about joint distributions. Probably, these notions should be mentioned here, with links to the two articles mentioned above, and to "Joint probability distribution". Boris Tsirelson (talk) 15:44, 26 June 2017 (UTC)
thread_title:  Orthogonality of random variables 

message: The definitions are the same. Golopotw (talk) 23:50, 8 November 2017 (UTC)
thread_title:  Can we merge the sections "Definition" and "Measure-theoretic definition"? 

message: 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)
replies: text: 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)
thread_title:  Wrong set expression in definition? 

message: 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)
thread_title:  Discussion 

message: 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)
thread_title:  Discussion 

message: 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)
thread_title:  Discussion