Talk:Markov's inequality

Notation

There needs to be some explanation of notation.

For example, in the equation:

$\Pr(|X|\geq a)\leq {\frac {{\textrm {E}}(|X|)}{a}}$

what is |X|? The absolute value of X? The norm of X? How is it different from just X?

Why isn't this page (at least the probability interpretation part) as clear as http://mathworld.wolfram.com/MarkovsInequality.html —Preceding unsigned comment added by 98.230.173.131 (talk) 06:29, 28 October 2010 (UTC)

98.230.173.131 (talk) 03:52, 1 November 2010 (UTC)Pika

I removed the |.| signs from the proof so that it matches the statement. Also, I expanded on the proof very slightly for easier reading; unlike academic publications, we are not constrained for space on a Wikipedia article. --unkx80 (talk) 12:15, 24 March 2013 (UTC)

Consistency in the General vs. Private Order

In the statement the general case appears before the private case of the probability space. I suggest putting the special case first and then go general, for two reasons:

1. Most readers of this article are only interested in the probability space case.
2. Consistency with the proofs section where the special case appears first.

Stangaa (talk) 09:06, 18 February 2009 (UTC)

The Interval

Last year the interval [0, ∞) was changed to [0, ∞]. I feel that this is not a good change.

The first form specifies the non-negative real numbers; the second form includes the possibility that the function h can map from a real number to ∞. If ∞ is in the range of h, and the preimage of ∞ has non-zero probability under the appropriate probability measure, then clearly E[h(X)] = ∞, which must be greater than any probability!

Most importantly, the latter version of the interval goes against the convention with which I am familiar (which is, of course, to use the former version.)

I have changed it back; if anyone would like to discuss this further please feel free! Ben Cairns 01:27, 24 Jan 2005 (UTC)

The proof provide is too terse for the level of most people with just an undergraduate level understanding of probability. To see an easier proof to understand check the following link: http://mathworld.wolfram.com/MarkovsInequality.html

Now I've reinstated a proof I put here months ago, which someone removed. Michael Hardy 19:50, 6 October 2006 (UTC)

Also: It should be mentioned what is assumed about f and t. That is, f should be a real-valued (complex also?) and measurable function (right?). And t should be strictly positive. Moreover, there should at the very least be agreement on notation between the probabilistic and the measure theoretic proofs; the indicator function should be the same. Superpronker (talk) 03:33, 1 June 2011 (UTC)

There's a small error in the article

In the first general measure theoretic statement, it says that the bound is given by an integral of f. It should be |f|.

No idea who should be told to fix it, but it should be fixed... —The preceding unsigned comment was added by 132.68.1.29 (talk) 13:19, 24 December 2006 (UTC).

Edit: never mind, fixed it.

Diagrams

The explanation of this inequality would benefit greatly from a diagram. Sanchom (talk) 08:03, 26 February 2007 (UTC)

Diagram Correctness

The diagram seems incorrect to me. Is it the graph of a probability density function? If so, surely the region for which the probability is given an upper bound is a range from some point on the x axis off to the right of the diagram {eg. [a,∞)}

It is quite possible that I have not understood the diagram, as I am by no means an expert on the inequality. Even so, if I have misentrpreted it, I am certain others will have, and in this case, could furthur text be given to explain it, or an alternative diagram provided. —The preceding unsigned comment was added by 129.234.4.10 (talk) 19:55, 13 May 2007 (UTC).

Version for a monotone increasing function

Isn't letting $Y=\varphi (|X|)$ adequate?

$\mathbb {P} (|X|\geq a)\leq \mathbb {P} (Y\geq \varphi (a))\leq {\frac {\mathbb {E} (Y)}{\varphi (a)}}={\frac {\mathbb {E} (\varphi (|X|))}{\varphi (a)}}$

I don't think it's worth a separate section, although the statement is somewhat interesting. — Arthur Rubin (talk) 15:48, 13 February 2015 (UTC)

It's also much more precise than the statement of the original theorem. If we're going to do that, we need to state the original theorem in as precise language, or state the new version in less precise language:

If φ is a monotone non-decreasing function from the nonnegative reals to the reals, X is a (real) random variable, and a is a positive real, then:

$\mathbb {P} (|X|\geq a)\leq {\frac {\mathbb {E} (\varphi (|X|))}{\varphi (a)}}$

This is a trivial change in the statement of the theorem, but a similar clarification of the definition of the original statement should also be present. — Arthur Rubin (talk) 16:22, 13 February 2015 (UTC)

I am fine with a reformulation. One could, of course, try to figure out which way of formulating is best absorbed by the human brain. But I don't see the solution yet (it may also depend on the brain; hence one should write for the largest possible number of brains, or some other sensible norm). In the long run I anyway had intended to read a lot about presenting maths (e.g. from sources like this), and maybe it would be beneficial to collect guidelines for good writing and compile them somewhere where they are easily accessible to other Wikipedia mathematics writers.

I also wonder whether there is something in Wikipedia:Manual_of_Style/Mathematics which is relevant; unfortunately, today I can't read it due to lack of time.

About the section thing, I thought that maybe this would help the reader to sort out what belongs to the 'classical' statement and what is sort of an addition; i. e. I thought it would ease the classification process. But I'm ready to listen to arguments. --Mathmensch (talk) 19:52, 13 February 2015 (UTC)

I think it would be adequate to then summarize the proof of Chebyshev's inequality from the main result as

$\mathbb {P} (|X-\mathbb {E} (X)|\geq a)=\mathbb {P} \left((X-\mathbb {E} (X))^{2}\geq a^{2}\right){\overset {\underset {\mathrm {MI} }{}}{\leq }}{\frac {\mathbb {E} \left({(X-\mathbb {E} (X))}^{2}\right)}{a^{2}}}={\frac {\operatorname {Var} (X)}{a^{2}}}$

and then note that the "monotone" result can be proved using

$\mathbb {P} (|X|\geq a)\leq \mathbb {P} (Y\geq \varphi (a)){\overset {\underset {\mathrm {MI} }{}}{\leq }}{\frac {\mathbb {E} (Y)}{\varphi (a)}}={\frac {\mathbb {E} (\varphi (|X|))}{\varphi (a)}}$

Wikipedia:WikiProject Mathematics/Proofs has guidelines for when proofs should be included; my interpretation is that a proof should be included only if it, itself, is "significant" (or, as would commonly be stated, "notable" or "noted"), or it demonstrates a method of particular interest. Neither is true of the proof of the "monotone" result. The proof of Chebyshev's inequality does meet the first criterion. In other words, I think the "monotone" result should be listed, without a subsection, but possibly with a theorem name (and reference); the proof should then be done by analogy with the proof of Chebyshev's inequality.

And, if it's not "classical", it needs a reference. (Actually, much of the article needs references, although many of the statements can be properly sourced to the textbook listed below.) — Arthur Rubin (talk) 20:29, 13 February 2015 (UTC)

What about deriving Chebyshev's inequality from the "monotone" result? One could prove the monotone result and then use that $X-E[X]$ is a random variable and $x\mapsto x^{2}$ restricted on $(0,\infty )$ is a monotonely increasing function. This is how my prof did it. --Mathmensch (talk) 20:36, 13 February 2015 (UTC)

I mean, of course 'your' version probably would comply better with the guidelines, but 'my' version would have the advantage that everything is proven (which always gives me a good feeling, don't know how it is for others). --Mathmensch (talk) 20:39, 13 February 2015 (UTC)

Got a source now, it's corollary 4.7.2 (p. 89) of this book. --Mathmensch (talk) 20:46, 13 February 2015 (UTC)

BTW I guess the properties "notable" or "significant" depend somewhat on the person. At least I don't recall an objective criterion (apart from the number of locigal conclusions or theorems applied) measuring those. --Mathmensch (talk) 20:54, 13 February 2015 (UTC)

"Notable" is in the eye of the beholder, but the monotone version is less noted than the other two. I'm on a smartphone, so complex statements involving copying are difficult. More, later. — Arthur Rubin (talk) 01:14, 14 February 2015 (UTC)

That's true, and clearly you are right that your proposal would match the guidelines much better. But based on what Terry Tao has written here (especially about the bit that learning a theory well might free up mental space), I think that one also should consider the benefits it might have to prove everything, namely that our brain puts a label on that bit saying 'completely understood' and hence is not unconsciouly busy figuring out that stuff, thus freeing mental space. Now I'm not sure if our brain works that way, but I'm also not sure that it doesn't. --Mathmensch (talk) 18:11, 14 February 2015 (UTC)

The "Intuitive" section under Proofs

This section is confusing because the "a bar" term is undefined, and I don't know how the formulas were derived. — Preceding unsigned comment added by 146.142.1.10 (talk) 16:58, 23 January 2020 (UTC)

Changes made to intuitive section

Hi, I am a new user who has edited the intuitive proof section today as I feel that the previous proof was erroneous and confounding for I feel that the idea of a bar and a representing all of the values which X can take as implied by the sentence is confusing and nonsensical, not to mention that in my opinion the E(X) equation in that particular proof was incorrect rendering the entire proof wrong as a result.

The proof which I added is similar to one of the proofs on the Maths Stack Exchange website, albeit I did not know it was hosted on said site until I found it through a Google search.

Inconvenient notation

In the Section "Statement" the measure space and the random Variable are both introduced as $X$. This might be confusing to the reader. — Preceding unsigned comment added by Guenterino (talk • contribs) 09:22, 18 October 2020 (UTC)