User:GFauxPas/Sandbox

Welcome to my sandbox, you are free to play here as long as you don't track sand onto the main wiki. --GFauxPas 09:28, 7 November 2011 (CST)

Normal Distributions
There are a bunch of definitions and theorems I plan on putting up regarding normal distributions. However, as Brace/Brace's book does not assume knowledge of Calculus, I'm going to have to make some calls. Regarding the definite integral:


 * $\displaystyle \int_a^b \frac 1 {\sqrt{2\pi}} \exp \left({-\frac 1 2x^2 }\right) \, \mathrm dx$

I can approach it two ways. As a function of two variables $a,b$ or as a function of intervals of the form $[a..b]$. The first option is I think easier, but the second option might closer represent the underlying sample space of all real intervals. Or am I arguing semantics? Also, if the second way, what's the best way to represent "the set of all real intervals"?

Also, it's often used as a real function:


 * $x \mapsto \displaystyle \int_{\to -\infty}^x \frac 1 {\sqrt{2\pi}} \exp \left({-\frac 1 2t^2 }\right) \, \mathrm dt$

Should I give that guy a transclusion, or put them both on one page? --GFauxPas (talk) 15:10, 12 December 2012 (UTC)


 * The first stage is for the raw indefinite integral itself to be extracted into a separate page (it might already be, as the Gaussian integral) - not trivial to solve of course.


 * You can then treat the def int just as a standard instance of a definite integral: I would treat it as a function of two variables a and b because the interval-ness of it is completely subsumed by the integral-ness of it. Once you've established it's a d.i. you don't need to establish that it's an interval - you know it is because it's an integral.


 * The second is then a function of one variable, $x$. Give these two babes separate pages to start with, both linking to the same indef int (see above) and then if it's appropriate we can merge them with a transclusion. Maybe transclude both def ints into the one main indef int page. That will set us a precedent for all calculus pages: make the indefinite integral the main page and transclude the def int as a subpage. I confess I haven't addressed integral calculus since this new paradigm evolved, so none of this has been even thought about. Feel free to play. --prime mover (talk) 16:24, 12 December 2012 (UTC)


 * I hunted around a bit and have found no sources that deal with the above as an indef integral - all sources skip right to the definite integral. I'll deal with the definite first, because I'd like to put up published theorems and definitions before I wax adventurous. --GFauxPas (talk) 13:52, 20 December 2012 (UTC)


 * IIRC the indefinite integral usually goes by the name "error function", $\operatorname{erf}$. --Lord_Farin (talk) 14:24, 20 December 2012 (UTC)

Continuous Random Variables
Boo, there's not enough foundation for continuous random variables to do stuff on normal distributions. That means more work for me -_-


 * $\displaystyle \operatorname{var} \left({X}\right) = \int_{x \in \Omega_X} \left({x - \mu}\right)^2 \operatorname{pdf}\left({x}\right) \, \mathrm dx$

Do we have stuff up about density functions? I couldn't find anything. --GFauxPas (talk) 15:21, 13 December 2012 (UTC)


 * The relevant articles are Definition:Probability Distribution and Definition:Probability Mass Function. Most of the abstract measure-theoretic foundation for analysis has been laid down, but indeed the application to random variables is still (largely) missing. --Lord_Farin (talk) 15:31, 13 December 2012 (UTC)

To do
I'll need this theorem in order to solidify some results with the definite integrals above.

Let $f$ be a differentiable real function. Then:


 * $\dfrac {f(x+h) - f(x-h)} {2h} \to f\,'(x)$ as $h \to 0$ --GFauxPas (talk) 22:28, 19 December 2012 (UTC)


 * Is the "2" in the denominator extraneous? --GFauxPas (talk) 22:31, 19 December 2012 (UTC)


 * Not unless $f'(x) = 0$. Try multiplying both sides with $2$. --Lord_Farin (talk) 23:30, 19 December 2012 (UTC)


 * Oh, okay, right. So to prove this I'm going to use that $f\,'_+ = f\,'_-$, from Limit iff Limits from Left and Right, right? And then what's the next step, can I have a hint?


 * It may be worthwhile to add definitions for "derivative from the left" and "derivative from the right". This also allows for neater structuring of Definition:Differentiable/Real Function/Interval.


 * Furthermore, it appears that the following equivalent formulation of differentiability is not covered yet (it's rather trivial that they're equivalent but needs proof nonetheless):


 * Let $x \in \R$ be a real number.


 * Let $f$ be a real function whose domain is a neighborhood of $x$.


 * Then $f$ is said to be differentiable at $x$ iff the following limit exists:


 * $f' \left({x}\right) := \displaystyle \lim_{h \to 0} \frac {f \left({x + h}\right) - f \left({x}\right)} {h}$


 * If this is the case, $f' \left({x}\right)$ is called the derivative of $f$ at $x$.


 * With this definition, we observe that your expression is (one half times) $\displaystyle \lim_{h \to 0} \frac {f \left({x + h}\right) - f \left({x}\right)} {h} + \frac {f \left({x - h}\right) - f \left({x}\right)} {-h} = 2 f' \left({x}\right)$, using that $-h \to 0$ iff $h \to 0$. --Lord_Farin (talk) 08:16, 20 December 2012 (UTC)


 * Equivalence of Definitions of the Derivative, Definition:Right-Hand Derivative, Definition:Left-Hand Derivative --GFauxPas (talk) 12:10, 20 December 2012 (UTC)

Is there a such thing as a one-sided (real valued) definite integral? --GFauxPas (talk) 13:54, 20 December 2012 (UTC)


 * The closest thing I can think of is Definition:Stieltjes Function of Measure on Real Numbers. --Lord_Farin (talk) 14:24, 20 December 2012 (UTC)

Mistake
Consider the integral:


 * $\displaystyle \int \frac 1 x \, \mathrm dx$

From $N$th Derivative of Reciprocal of $M$th Power:


 * $\displaystyle \frac {\mathrm d} {\mathrm dx} \frac 1 x = - \frac 1 {x^2}$

From Derivative of Identity Function:


 * $\displaystyle \frac {\mathrm d} {\mathrm dx} x= 1$

From Integration by Parts:

--GFauxPas (talk) 15:44, 10 January 2013 (UTC)