gtMath

Hello Mr. Troderman,

First off, big fan of your work and I hope you continue to keep it up! I count myself as one of those persons who isn't (necessarily) sick at math...

My request is for a topic on object set problems that come up often on the GRE: permutations and combinations. I know the basics, like combination=order doesn't matter & permutation=order does matter, and the former is reduced in number from the latter. Where it gets dicey for me is when it comes to situations of repetition vs. no repetition.
So for example with a combination with repetition (a store sells shirts that come in 6 different colors and I have a coupon to get 3 shirts of whatever color, how many variations of that can I make?) what is the best way to solve for that when you can have two or more colors be the same? I know there are formulas for each scenario that you can use, but I guess I'm curious about the basic proof behind the formulas (especially the combination with repetition.)

Thanks!

Great request- here's the post: http://www.gtmath.com/2015/11/stars-and-bars.html

Dear Mr. Troderman,

I come to you today with a problem that has been gathering like a dark cloud over my head for some time now. So much so that I haven't been able to fully enjoy the refreshing taste of a light beer, the euphoria of a poetry reading, or the presence of a nice girl lately.

Getting to the point of my frustration, here it is: I use Monte Carlo simulations all the time in my work. Or, I should say, I input financial data (i.e., letters) into fancy software and it generates cool graphs and stuff. While I am able to make great application of these results, I feel as if my understanding of the underlying mechanics of the Monte Carlo Method could use some deepening. I get that the basic idea is to average many different possible outcomes that are going to be effected by numerous uncertain variables. I even get that the method requires the analyzing of massive numbers of these possibilities (the more massive, the better- I hear). Perhaps my question is too general, but is there an easier-ish way to mechanically breakdown Monte Carlo, especially as it relates to financial products and the predictive quality of the different Monte Carlo simulators on the market?

The following Wikipedia articles are about as helpful as a monkey at a traffic stop on a Sunday morning in Arkansas: https://en.wikipedia.org/wiki/Monte_Carlo_method#Finance_and_business, https://en.wikipedia.org/wiki/Monte_Carlo_methods_in_finance.

Please let me know.

Hi guys, I already posted this question under the "Convergence of Sequences of Functions" article but thought I'd also paste a copy here so that if anyone is interested they'll be more likely to see it. See my original inquiry as follows:

Troderman,

I'm enjoying the blog! This post got me thinking about specific examples of metric spaces generated by norms and interesting properties that can arise. It might be out in the weeds but I never figured out the proof relating the lp(x) to lq(x) given in the theorem below. The case becomes trivial when p=q=2, since you get the added structure of a Hilbert Space. It could be an interesting question to pursue, there's a lot going on with it! (And i'd love to see a solution explained.)

Let lp be a subspace of the space of all sequences of scalars such that ∑n|x|p<∞ equipped with the usual norm ∥x∥p=(∑n|xn|p)1p

Theorem: The duel space of lp(x) for 1<p<∞ has an isomorphism with lq(x), where q is such that 1p +1q=1

Dear Mr. Troderman,

I continue to commend you for your excellent work. Your response to my previous RR ("stars and bars") was wonderfully presented and very insightful.

So now on to my newest request. This is in the area of a statistics, which I'm doing a lot of now for my Masters in Public Health. In one of our first biostats classes, we went over measures of variability and learned when looking at the sample mean taken from a target population, this is the formula for the sample variance:

sample variance = sum of(Xi - Xavg)^2/n-1
*sorry my subscripts aren't working, I hope you get what formula I'm getting at.

Ok, this "n-1" in the denominator...I understand we divide by n-1 degrees of freedom because it gives an unbiased estimate of the true population variance. If used just n, we would only estimate the variance for that sample we selected from, not the real population of interest. But from what I understand, while this formula may be an unbiased estimate of variance, the standard deviation/s.d.-which is simply the square root of the formula…won’t that end up being biased? I was reading a little about your explanation of concave function and Jensen’s Inequality, which makes me feel like make the s.d. bias would be…downwards? So I guess my question is: do you know a way to calculate an unbiased s.d. of a population?
(Obviously with a large n, this is inconsequential, but I am curious if there is a better formula that might fix the square root issue of the sample standard deviation.)

Appreciate any solution you can offer!