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)

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?

By the way, where this is going, is the definition of a density function for a continuous random variable $X$:


 * $\text{pdf}(x) := \displaystyle \lim_{\epsilon \to 0} \frac {\Pr (x - \frac \epsilon 2 \le X \le x + \frac \epsilon 2)} \epsilon$

where the cdf is not necessarily differentiable. --GFauxPas (talk) 04:24, 20 December 2012 (UTC)