Re: [MLUG - DISCUSSION] the Rearrangement Inequality

[Mike Miller <EMAIL:PROTECTED>]

On Wed, 2 Aug 2006, Jonathan King wrote:

> On 8/1/06, Stephen Montgomery-Smith <EMAIL:PROTECTED> wrote:
>
> [Concerning the Rearrangment Inequality and Jensen's Inequality]
>
> > Both these inequalities are the bread and butter of my research.  They 
> > are both great inequalities, but I use them so much that they have 
> > become almost like 1+1=2 to me.
>
> At lunch today I read a short tutorial style article the Rearrangment 
> Inequality and some of the most basic stuff you can do with it, of which 
> the most powerful was to provide a trivial proof of Chebysev's 
> Inequality.  So now I guess that proof for you would be sort of like n + 
> n = 2n. :-)


I had a funny stat professor years ago who told us "Some people will spell 
it 'Chebyshev' and others will spell it 'Tchebysheff' or 'Tchebyshoff' but 
there is really only one correct spelling!"  He then promptly wrote it on 
the board in cyrillic characters.  Maybe it's an old joke -- see the last 
stanza of this poem:

http://www.math.tamu.edu/~boas/courses/math696/spelling-lesson.html

I'm used to the probabilistic form of the inequality:

http://en.wikipedia.org/wiki/Chebyshev's_inequality
http://mathworld.wolfram.com/ChebyshevInequality.html

Apparently, it can be written like this...

http://planetmath.org/encyclopedia/ChebyshevsInequality.html

...and then proved like this:

http://planetmath.org/encyclopedia/ProofOfChebyshevsInequality2.html

But the usual way seems to use Markov's inequality:

http://planetmath.org/encyclopedia/ProofOfChebyshevsInequality.html

What was your proof like, Jon?

Interestingly, I also do not recall learning these useful inequalities but 
they seem familiar enough that I probably just forgot about them:

http://mathworld.wolfram.com/MarkovsInequality.html
http://planetmath.org/?op=getobj&from=objects&name=KolmogorovsInequality

Mike

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