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Let X e = (X1, X2, . . . , Xn) be a random sample from the normal distribution with mean µ and variance σ 2 , find the maximum likelihood estimators of µ and σ 2 — Preceding unsigned comment added by 197.220.137.224 ( talk) 08:06, 22 March 2024 (UTC)
"random variable may be attributed to a function defined on a state space equipped with a probability distribution that assigns a probability to every subset ... of its state space..." A function defined on a state space should assign something to points of the state space. If it assigns something to sets, it is rather a SET FUNCTION. Also, if the state space is already equipped with a probability distribution, what is the role of the random variable?
"A random variable then defines a probability measure on the sample space by assigning a subset of the sample space the probability of its inverse image in the state space." Really? The two states are swapped here. On the sample space a measure is given from the beginning; on the state space it appears due to the random variable.
"In other words the probability distribution of a random variable is the push forward measure of the probability distribution on the state space." I do not understand this phrase; once again, on which space the measure is given from the beginning? Boris Tsirelson ( talk) 16:03, 12 December 2008 (UTC)
Perhaps subsection Terminology with its current content is superfluous. Repetition of discrete vs continuous distribution seems confusing (the reader could think he overlooked sth he didn't). The notion of support should appear in the article on measures. Measure, probability and possibly a few other concepts should be referenced at the bottom of the article. More thoughts are needed about organization of the vast legion of important links to related articles.
As a continuous counterpart of formula for probability distribution should be used that with integral using density function (since resembles discrete version very well - just think of density as discrete histogram with dense set of values). After presenting discrete and absolutely continuous distributions in separate sections, there should appear the third section unifying both into the general theory with formula using Lebesgue integral w.r.t. axiomatic probability measure P (this formula which is now horribly serving as definition in the special case of absolute continuous random variable - inacceptable for general purpose encyclopedia). Then some properties, examples and graphs should follow (independently of famous and important distributions described in dedicated articles). Compare e.g. with some good stylistic and logistic ideas applied to the article on expected value. -- Megaloxantha ( talk) 02:37, 31 December 2008 (UTC)
Can we do something sensible about the stuff presently in this section, which says:
In the first bullet point, there is a need to say this for general distributions, not just those which have densities ...unfortunately the article on convolution does not seem to give a sensible formula in terms of cumulative distribution functions. In the second bullet point, there is again the problem of dealing with general distributions but, in addition, the use of the word "cross-correlation" would need to be given the interpretation in the article pointed-to, which is very different from the common one in statistics.
Melcombe ( talk) 10:38, 31 December 2008 (UTC)
Given that this article has had a major revamp, much of the old discussion is irrelevant to the present content. I have reordered the threads a little to put newer stuff towards the end, but I am suggesting that all the stuff now before the section headed "Lead is confusing" be archived. Any thoughts? Melcombe ( talk) 12:15, 31 December 2008 (UTC)
In the formal definition, we have:
"A random variable is defined as a measurable function X from a probability space to its observation space ."
Can we have someone put up a definition of the "observation space"? I'd do it myself but I'm not able enough. -- WestwoodMatt ( talk) 15:41, 20 March 2010 (UTC)
See also Wikipedia talk:WikiProject Mathematics#Codomain of a random variable: observation space?. Boris Tsirelson ( talk) 16:50, 27 March 2010 (UTC)
The term seems self-explanatory when used the way it's used in the article. Someone used the word "unfortunate" above without rigorously defining it. I don't have a problem with that. Michael Hardy ( talk) 18:50, 27 March 2010 (UTC)
Enough is enough. Observation space is gone.
Shouldn't the basic terminology used to discuss/describe a distribution be included here? It would not only fill out the basic presentation of the material but also provide an anchor for references to such terms in other articles. Joja lozzo 03:02, 25 July 2011 (UTC)
As far as I know, "probability distribution" is, in general, a probability measure (rather than this or that function). In some (but not all) cases it can be described by a cumulative distribution function. Sometimes also by the probability mass function; sometimes also by the probability density function. But the lead says "a probability mass, probability density, or probability distribution is a function..." Or is it meant that a measure is also a kind of function (namely, a set function)? But no, this is not written in the sections. Boris Tsirelson ( talk) 05:57, 2 July 2012 (UTC)
Why does this article refer to the Gaussian distribution as "the most important distribution"? Isn't this kind of arbitrary? — Preceding unsigned comment added by 2607:4000:200:13:1A03:73FF:FEB3:B07C ( talk) 05:46, 27 August 2012 (UTC)
Right now there seem to be somewhat strange redirects. What article should I link to for a plain old distribution of occurrences as observed in a finite sample?
Nanite ( talk) 18:45, 21 January 2014 (UTC)
I believe "In contrast, when a random variable takes values from a continuum, probabilities can be nonzero only if they refer to intervals" is incorrect. Imagine a random variable whose image is [0,1]. Let the point 1 have a 50% chance of occurring, and the rest of the probability mass is uniformly distributed amongst [0,1). Is there anything wrong with this counter-example? Note that obviously this counter-example extends to letting any finite number of points of the image of a continuous random variable have non-zero probability. 38.88.227.194 ( talk) 17:23, 23 September 2014 (UTC)
I feel that this article is incomplete because it deals only with 'random variables' (scalar or not). However, the notion of a 'random element' can be applied to any set. Specifically, a probability measure on a functional space is a well defined concept under the Kolmogorov measure theoretic point of view. Nevertheless, there are several items on the article that does not apply easly to the case of random functions. For exemple, the concept of a probability density for random functions does not exist ( [1]); the concept of cumulative distribution function is replaced, in most presentations, by a infinite hierarchy of cumulative distribution functions; etc. Crodrigue1 ( talk) 21:36, 12 October 2016 (UTC)
References
See Talk:Mode (statistics)#Different treatment of discrete and continuous? Boris Tsirelson ( talk) 18:58, 26 June 2017 (UTC)
There is some confusion about the term 'continuous r.v.'. A continuous r.v. is in this article defined as having a continuous cumulative distribution function, hence it doesn't need to have a density. In the article probability density function it is said a density belongs to a continuous r.v. Madyno ( talk) 08:49, 23 July 2017 (UTC)
I would suggest to combine this page with Probability_density_function, because a probability density function is simply the mathematical representation of a given probability function. Mimigdal ( talk) 17:13, 31 January 2018 (UTC)
Why "Measure theoretic formulation" for "Discrete probability distribution", but "Kolmogorov definition" for "Continuous probability distribution"? Boris Tsirelson ( talk) 07:42, 10 May 2019 (UTC)
Could someone please check this section?
Maybe I am somewhat confused, but I think what is stated as axioms of Kolmogorov here is wrong. Condition 2 has to state that the probability to be in the whole space equals $1$.
The conditions given here are, for instance, satisfied by the trivial assignment $P(X\in A)$ identical to $0$ for all considered events $A$. — Preceding unsigned comment added by 2003:DC:DF37:9300:7D0B:9E28:D2F3:91D8 ( talk) 17:59, 21 March 2021 (UTC)
Giving the the fraction of male students in a school as an example of a random phenomenon only makes sense in a coeducational school; otherwise the fractions are fixed! Wprlh ( talk) 05:08, 1 July 2021 (UTC)
For example, the second paragraph of the Introduction has both “throwing a fair die” and “the dice rolls” Wprlh ( talk) 05:15, 1 July 2021 (UTC)
The definition given of a continuous RV is actually of an absolutely continuous RV and is potentially confusing. There can exist an RV for which the CDF is continuous in the classical sense of continuity but is not absolutely continuous. Hence what is called continuous RV should be called as an absolutely continuous RV. This is explicitly done in Valentin V. Petrov's book, 'Limit theorems of probability theory: sequences of independent random variables' on Page 2:
The distribution of the random variable is said to be continuous if for any finite or countable set of points of the real line. It is said to be absolutely continuous if for all Borel sets of Lebesgue measure zero.
Theorem 31.7 in Billingsley's classic book 'Probablity and Measure' proves that the above definition is equivalent to being absolutely continuous. That book does not mention continuous RV at all, it only mentions absolutely continuous distribution function. Rosenthal's book 'A First Look at Rigorous Probability Theory' explicitly talks about absolutely continuous RV and not about continuous RV (pg 1,70). In conclusion, the cited source (ie Ross' book) is sowing some confusion when it uses the term continuous RV.
Hence for this reason, I am being bold and replacing continuous with absolutely continuous. I have added the sentence 'Some authors however use the term "continuous distribution" to denote all distributions whose cumulative distribution function is absolutely continuous, i.e. refer to absolutely continuous distributions as continuous distributions.' to make this confusion (hopefully) clear to the reader. - Abdul Muhsy talk 05:29, 18 March 2022 (UTC)
In the passage starting
"it" stands for "probability function". But if this is a definiton for the "probability function" then it contains the definiendum in the definiens by
i.e. it is a circular definition. Jyyb ( talk) 09:18, 19 July 2022 (UTC)
the probabilities that it will rain or not in 30 days 154.156.43.236 ( talk) 12:32, 22 August 2023 (UTC)
the probabilities that it will rain or not in 30 days from now are 0.5and 0.5 respectivly.fi the probabilities that 30 days time it will either rain or not 154.156.43.236 ( talk) 12:35, 22 August 2023 (UTC)
In this MrOllie removed a link to an open source probability distribution site with an edit summary pointing to Links normally to be avoided. None of the 19 criteria on that list matched so I'd like to learn why the link was removed. (I didn't originally add it, I just verified that it seemed not to fit the list before moving it to External links). Johnjbarton ( talk) 00:14, 25 February 2024 (UTC)
![]() | This ![]() It is of interest to the following WikiProjects: | ||||||||||||||||||||
|
/Archive 1 |
Let X e = (X1, X2, . . . , Xn) be a random sample from the normal distribution with mean µ and variance σ 2 , find the maximum likelihood estimators of µ and σ 2 — Preceding unsigned comment added by 197.220.137.224 ( talk) 08:06, 22 March 2024 (UTC)
"random variable may be attributed to a function defined on a state space equipped with a probability distribution that assigns a probability to every subset ... of its state space..." A function defined on a state space should assign something to points of the state space. If it assigns something to sets, it is rather a SET FUNCTION. Also, if the state space is already equipped with a probability distribution, what is the role of the random variable?
"A random variable then defines a probability measure on the sample space by assigning a subset of the sample space the probability of its inverse image in the state space." Really? The two states are swapped here. On the sample space a measure is given from the beginning; on the state space it appears due to the random variable.
"In other words the probability distribution of a random variable is the push forward measure of the probability distribution on the state space." I do not understand this phrase; once again, on which space the measure is given from the beginning? Boris Tsirelson ( talk) 16:03, 12 December 2008 (UTC)
Perhaps subsection Terminology with its current content is superfluous. Repetition of discrete vs continuous distribution seems confusing (the reader could think he overlooked sth he didn't). The notion of support should appear in the article on measures. Measure, probability and possibly a few other concepts should be referenced at the bottom of the article. More thoughts are needed about organization of the vast legion of important links to related articles.
As a continuous counterpart of formula for probability distribution should be used that with integral using density function (since resembles discrete version very well - just think of density as discrete histogram with dense set of values). After presenting discrete and absolutely continuous distributions in separate sections, there should appear the third section unifying both into the general theory with formula using Lebesgue integral w.r.t. axiomatic probability measure P (this formula which is now horribly serving as definition in the special case of absolute continuous random variable - inacceptable for general purpose encyclopedia). Then some properties, examples and graphs should follow (independently of famous and important distributions described in dedicated articles). Compare e.g. with some good stylistic and logistic ideas applied to the article on expected value. -- Megaloxantha ( talk) 02:37, 31 December 2008 (UTC)
Can we do something sensible about the stuff presently in this section, which says:
In the first bullet point, there is a need to say this for general distributions, not just those which have densities ...unfortunately the article on convolution does not seem to give a sensible formula in terms of cumulative distribution functions. In the second bullet point, there is again the problem of dealing with general distributions but, in addition, the use of the word "cross-correlation" would need to be given the interpretation in the article pointed-to, which is very different from the common one in statistics.
Melcombe ( talk) 10:38, 31 December 2008 (UTC)
Given that this article has had a major revamp, much of the old discussion is irrelevant to the present content. I have reordered the threads a little to put newer stuff towards the end, but I am suggesting that all the stuff now before the section headed "Lead is confusing" be archived. Any thoughts? Melcombe ( talk) 12:15, 31 December 2008 (UTC)
In the formal definition, we have:
"A random variable is defined as a measurable function X from a probability space to its observation space ."
Can we have someone put up a definition of the "observation space"? I'd do it myself but I'm not able enough. -- WestwoodMatt ( talk) 15:41, 20 March 2010 (UTC)
See also Wikipedia talk:WikiProject Mathematics#Codomain of a random variable: observation space?. Boris Tsirelson ( talk) 16:50, 27 March 2010 (UTC)
The term seems self-explanatory when used the way it's used in the article. Someone used the word "unfortunate" above without rigorously defining it. I don't have a problem with that. Michael Hardy ( talk) 18:50, 27 March 2010 (UTC)
Enough is enough. Observation space is gone.
Shouldn't the basic terminology used to discuss/describe a distribution be included here? It would not only fill out the basic presentation of the material but also provide an anchor for references to such terms in other articles. Joja lozzo 03:02, 25 July 2011 (UTC)
As far as I know, "probability distribution" is, in general, a probability measure (rather than this or that function). In some (but not all) cases it can be described by a cumulative distribution function. Sometimes also by the probability mass function; sometimes also by the probability density function. But the lead says "a probability mass, probability density, or probability distribution is a function..." Or is it meant that a measure is also a kind of function (namely, a set function)? But no, this is not written in the sections. Boris Tsirelson ( talk) 05:57, 2 July 2012 (UTC)
Why does this article refer to the Gaussian distribution as "the most important distribution"? Isn't this kind of arbitrary? — Preceding unsigned comment added by 2607:4000:200:13:1A03:73FF:FEB3:B07C ( talk) 05:46, 27 August 2012 (UTC)
Right now there seem to be somewhat strange redirects. What article should I link to for a plain old distribution of occurrences as observed in a finite sample?
Nanite ( talk) 18:45, 21 January 2014 (UTC)
I believe "In contrast, when a random variable takes values from a continuum, probabilities can be nonzero only if they refer to intervals" is incorrect. Imagine a random variable whose image is [0,1]. Let the point 1 have a 50% chance of occurring, and the rest of the probability mass is uniformly distributed amongst [0,1). Is there anything wrong with this counter-example? Note that obviously this counter-example extends to letting any finite number of points of the image of a continuous random variable have non-zero probability. 38.88.227.194 ( talk) 17:23, 23 September 2014 (UTC)
I feel that this article is incomplete because it deals only with 'random variables' (scalar or not). However, the notion of a 'random element' can be applied to any set. Specifically, a probability measure on a functional space is a well defined concept under the Kolmogorov measure theoretic point of view. Nevertheless, there are several items on the article that does not apply easly to the case of random functions. For exemple, the concept of a probability density for random functions does not exist ( [1]); the concept of cumulative distribution function is replaced, in most presentations, by a infinite hierarchy of cumulative distribution functions; etc. Crodrigue1 ( talk) 21:36, 12 October 2016 (UTC)
References
See Talk:Mode (statistics)#Different treatment of discrete and continuous? Boris Tsirelson ( talk) 18:58, 26 June 2017 (UTC)
There is some confusion about the term 'continuous r.v.'. A continuous r.v. is in this article defined as having a continuous cumulative distribution function, hence it doesn't need to have a density. In the article probability density function it is said a density belongs to a continuous r.v. Madyno ( talk) 08:49, 23 July 2017 (UTC)
I would suggest to combine this page with Probability_density_function, because a probability density function is simply the mathematical representation of a given probability function. Mimigdal ( talk) 17:13, 31 January 2018 (UTC)
Why "Measure theoretic formulation" for "Discrete probability distribution", but "Kolmogorov definition" for "Continuous probability distribution"? Boris Tsirelson ( talk) 07:42, 10 May 2019 (UTC)
Could someone please check this section?
Maybe I am somewhat confused, but I think what is stated as axioms of Kolmogorov here is wrong. Condition 2 has to state that the probability to be in the whole space equals $1$.
The conditions given here are, for instance, satisfied by the trivial assignment $P(X\in A)$ identical to $0$ for all considered events $A$. — Preceding unsigned comment added by 2003:DC:DF37:9300:7D0B:9E28:D2F3:91D8 ( talk) 17:59, 21 March 2021 (UTC)
Giving the the fraction of male students in a school as an example of a random phenomenon only makes sense in a coeducational school; otherwise the fractions are fixed! Wprlh ( talk) 05:08, 1 July 2021 (UTC)
For example, the second paragraph of the Introduction has both “throwing a fair die” and “the dice rolls” Wprlh ( talk) 05:15, 1 July 2021 (UTC)
The definition given of a continuous RV is actually of an absolutely continuous RV and is potentially confusing. There can exist an RV for which the CDF is continuous in the classical sense of continuity but is not absolutely continuous. Hence what is called continuous RV should be called as an absolutely continuous RV. This is explicitly done in Valentin V. Petrov's book, 'Limit theorems of probability theory: sequences of independent random variables' on Page 2:
The distribution of the random variable is said to be continuous if for any finite or countable set of points of the real line. It is said to be absolutely continuous if for all Borel sets of Lebesgue measure zero.
Theorem 31.7 in Billingsley's classic book 'Probablity and Measure' proves that the above definition is equivalent to being absolutely continuous. That book does not mention continuous RV at all, it only mentions absolutely continuous distribution function. Rosenthal's book 'A First Look at Rigorous Probability Theory' explicitly talks about absolutely continuous RV and not about continuous RV (pg 1,70). In conclusion, the cited source (ie Ross' book) is sowing some confusion when it uses the term continuous RV.
Hence for this reason, I am being bold and replacing continuous with absolutely continuous. I have added the sentence 'Some authors however use the term "continuous distribution" to denote all distributions whose cumulative distribution function is absolutely continuous, i.e. refer to absolutely continuous distributions as continuous distributions.' to make this confusion (hopefully) clear to the reader. - Abdul Muhsy talk 05:29, 18 March 2022 (UTC)
In the passage starting
"it" stands for "probability function". But if this is a definiton for the "probability function" then it contains the definiendum in the definiens by
i.e. it is a circular definition. Jyyb ( talk) 09:18, 19 July 2022 (UTC)
the probabilities that it will rain or not in 30 days 154.156.43.236 ( talk) 12:32, 22 August 2023 (UTC)
the probabilities that it will rain or not in 30 days from now are 0.5and 0.5 respectivly.fi the probabilities that 30 days time it will either rain or not 154.156.43.236 ( talk) 12:35, 22 August 2023 (UTC)
In this MrOllie removed a link to an open source probability distribution site with an edit summary pointing to Links normally to be avoided. None of the 19 criteria on that list matched so I'd like to learn why the link was removed. (I didn't originally add it, I just verified that it seemed not to fit the list before moving it to External links). Johnjbarton ( talk) 00:14, 25 February 2024 (UTC)