Ask MetaFilter questions tagged with Math

Math/Stats: help me analyze a data set and determine the values that created it

intermod — Wed, 23 Dec 2009 12:58:21 -0800

Mathematics / Statistics Filter: I have some pairs of numbers that are the result of a process. Given just that data set, and a rule that relates them, can you determine the integer values that could have resulted in those sets? Apologies for the phrasing of the FPP -- I know it doesn't make much sense. Hopefully some mathematics / statistics types will click through and see this longer version.

I have some sets of numbers, shown below. I'm trying to reverse engineer the numbers that could have resulted in these sets, based on some known mathematical relationships between them.

In general, a given Device(n) consumes a resource in integer Quantities at a floating point Rate(n), resulting in a total Cost for that consumption run. What I have is pairs of Device/Cost, for several different Device for several runs each, and I'm trying to determine the floating point Rate. The Rate is constant for a given Device. The Quantity consumed is different for each run, but the one key here is that I know that the Quantity values are integers.

So for a given Device run, we have:

Quantity(int) x Rate(float) = Cost(float)

All I have is Cost data for each Device, but I have multiple sets of these and am hoping there's some sort of numeric analysis that can tell me the likely Quantity values that fit.

Here's a sample of the data:

Device / Cost
Device1 / 1235
Device1 / 988
Device1 / 1003
Device1 / 1526
Device2 / 3652
Device2 / 1207
Device2 / 1729
Device2 / 518
Device3 / 745
Device3 / 2115
Device3 / 1415
Device3 / 334

So, for example, using the Device1 / 988 and Device1 / 1003 set, I could eyeball it and see that the Cost difference of 15 is due to 1 unit of Quantity difference in the runs. Thus the first run consumed 66 x 14.97 = 988 and the second run consumed 67 x 14.97 = 1003 . (Alas, the Rate values should be more in the 30-50 range, so 14.97 doesn't make much sense) But I'm hoping that with a larger population of data, there's some analysis I can do that will give a more confident answer.

Perhaps this can even be solved without ensuring that the Quantity values are integers, but it's a constraint that the data is supposed to have so I thought I'd mention it.

I'll monitor this thread for the next couple hours to answer any questions. And will add better tags! I used the science category because this seems like the kind of math that a lab scientist might be familiar with, trying to analyze a data set to work out the conditions that created it. I'm especially hoping for a statistic analysis that produces some sort of confidence measure, because a couple of these data points might be outliers, screwing up what might otherwise be a closed solution.

Note: this is not homework filter, or even do-my-job filter. It's just something I'm trying to reverse engineer.

If it does exist, I imagine the school crest is something like duct tape, a paper clip and a mullet

rileyray3000 — Tue, 22 Dec 2009 08:09:30 -0800

Is there a REAL MacGyver School in the world? Is there the equivalent of a MacGyver School anywhere in the world? Say tomorrow I wanted to learn, for lack of a better term, “Improvisational Engineering”…where would I go? Is there a real class anywhere out there or a major or even a specific school that teaches these things? Even if it’s an informal collective of engineers that occasionally takes apprentices, that’s fine.

Conditional Probability

chrisamiller — Wed, 16 Dec 2009 17:25:54 -0800

Stats-filter: Given a binary matrix, if I know the total number of ones in a given row and a given column, can I calculate the probability that a given position contains a one? I have a binary matrix, like so, where every value is either 1 or 0. So, if the first column contains 2 ones, and the first row contains 1 one, what's the probability that position A contains a one?

Example:



    2

    ________

 1 | A|  |  |

   |__|__|__|

   |  |  |  |

   |__|__|__|

   |  |  |  |

   |__|__|__|

   |  |  |  |

   |__|__|__|

(not homework-filter)

Ask a Stats Nerd

sonofslim — Tue, 15 Dec 2009 08:18:52 -0800

I need to ruin everyone's fun by adding a rigorous mathematical scoring system to the company bake-off. Li'l help? We just had an employee bake-off and while the outcome was satisfactory to all, I think the scoring methodology (average of ratings on a 1-10 scale) was too arbitrary to be statistically meaningful.

How would you build an objective and mathematically sound ranking system based on the following criteria?

* Entries will be judged by all participating employees in each of three categories: taste, presentation, and creativity
* There will be a winner within each category, as well as an overall winner
* Not everyone has to vote on every entry

We make software, so yes, the methodology by which we judge pie is quite critical.

How many songs do I have to listen to before I start again?

jacquilynne — Sat, 12 Dec 2009 08:49:09 -0800

Can you help me figure out this math question related to probability? I feel like my college course in probability should have enabled me to figure this out for myself, but clearly, it did not.

I have an iTunes playlist with 500 songs on it. Once I've listened to a song, it disappears from the list, and is replaced by a new song from my library. Songs that have been on the list once can not reappear on the list, and for the purposes of this question, you can assume the library from which the replacement songs are drawn is infinite.

The list is selected randomly and then sorted in alphabetical order, and I start at the top and play my way through it alphabetically until I reach the 500th song. But because of the replacing of songs, which might appear in the list after the song I just listened to (and thus I will have to listen to them before I get to the end of list) or might appear in the list before the song I just listened to (and thus will not get listened to before I get to the end of the list) I actually listen to way more than 500 songs on the way through. What I'm trying to figure out is how many songs I'm likely to listen to in a single pass.

I know that when I'm listening to the first song, there's relatively high odds that it's replacement will belong later in the list. I assume this to be 499/500 -- if the replacement song gets slotted into the list pretty much at random, it could end up in any of 500 slots, and one of them is before the point I'm at in the playlist, while the other 499 after. And when I'm done listening to the 373rd song, then, the odds should be down to 227/500 that I'll have to listen to the replacement song. So, I can figure out the odds that I'll have to listen to any given replacement song.

But I can't get from there to an overall expected number, because I can't figure out how to account for new songs I've already had to listen to vs. the ones I haven't. Say I've listened to 100 songs, and 90 of their replacements ended up later in the list than where I was at the time. That means that while I've listened to 100 songs, I'm only at position 10 on my list -- a long way from the 100 I'd be at if I wasn't replacing the songs.

This is obviously not earth shattering, and I don't need to know this. But it's been nagging at me because I can't figure it out, and I feel like I should be able to.

Teach me about this thing you call "math"...

incomple — Tue, 08 Dec 2009 05:03:16 -0800

Here's an obnoxious school question that's been nagging me for days: Suppose that the height (at the shoulder) of adult African bull bush elephants is normally distributed with µ = 3.3 meters and ∂ = .2 meter. The elephant on display at the Smithsonian Institute has a height 4 meters and is the largest elephant on record. What is the probability that an adult African bull bush elephant has height 4 meters or more? I know the answer is .0002, but how can I prove this? I'm not a math guy, and I can't find any "quick and easy"—or even vaguely comprehensible—formulas to help me pound this into my thick skull... Just having the answer doesn't matter to me—I wish to understand!

Other questions that have been sticking in my craw...

The lifetime of a certain brand of tires is normally distributed with mean µ = 30,000 miles and standard deviation ∂ = 5000 miles. The company has decided to issue a warranty for the tires but does not want to replace more than 2% of the tires that it sells. At what mileage should the warranty expire?

Again, I know the answer is 19,750 miles, but I'm completely failing to grasp exactly how one comes to that answer.

In American roulette, the probability of winning when betting "red" is 9/19. What is the probability of being ahead after betting the same amount 90 times?

Answer is .2745. Grr!!

The probabilities of failure for each of three independent components in a device are .01, .02, and .01, respectively. The device fails only if all three components fail. Out of a lot of 1 million devices, how many would be expected to fail? Find the probability that more than three devices in the lot fail.

The first question is easy for me—0.01 x 0.02 x 0.01 = 0.000002, and 0.000002 x 1000000 = 2. But why is there a .1469 probability of more than three devices in the lot failing?

I flunked math in high school, and now I feel like everything in math is set up to make guys like me look dumb.

Reverse my percentage?

JulianDay — Sun, 06 Dec 2009 19:25:28 -0800

How do you find a number that minus a percentage of itself is a certain number? What would be the formula (in a spreadsheet like pages.app) to find that 100 - 20% = 80, if you only know the percentage and the result (in this example 20% and 80)

I've tried googling this but I don't think I know the name or term of what I'm looking for, reverse percentage doesn't seem right...

Calculate Possible Combinations

rahnefan — Tue, 01 Dec 2009 16:38:19 -0800

How many combinations are possible in this grid? I have a grid with four cells in it; two rows by two columns. Each cell has a symbol in it; either Sun, Moon, Star, or Egg. Each symbol has four potential orientations; 0°, 90°, 180°, & 270°. An example layout might be:

Sun-90° Moon-180°
Star-90° Egg-270°

How many unique layouts are possible, if each symbol must appear once and only once in each unique layout? Also, by unique, I mean do not rotate the entire grid -- for example, the below would be unique because it is not the same as the example above, even though the "markup" here has been "rotated:"
Star-90° Sun-90°
Egg-270° Moon-180°

But the example below would not be unique:
Star-180° Sun-180°
Egg-0° Moon-270°
Because it is the same as rotating the entire grid in the first example.

RDRR. Get it?

bondcliff — Tue, 01 Dec 2009 07:14:36 -0800

What are some cool math and number facts that would blow the mind of a seven year old? Yesterday at the dinner table my son says "Hey dad, did you know 111,111,111 x 111,111,111 = 12,345,678,987,654,321?"*

He thought this was awesome. I've previously told him the trick about multiplying by 9 (2x9 = 18, 1+8=9, 3x9=27, 2+7=9 and so on up the times table) and a couple of other fun facts.

I'd like to present him with some other neat numerical properties and math facts. He'd need to be able to grasp the concept, so something like the non-palindromic properties of 196 might be a bit beyond him. He's in (the equivalent of) 2nd grade at a Montessori school and math is his favorite subject. He's currently doing 2-digit division and he's pretty much mastered multiplication. I'm not afraid to go a bit beyond what he's learning but I don't want to go too far beyond it.

What can you tell him about other interesting numbers and how to get to those numbers?

Calculator tricks might be fun as well. I'm sure if I teach him the S8008 trick he'll be the King of the Playground. Also tricks that require props such as playing cards, Legos, or anything else are welcome, as long as they demonstrate the type of thing I'm looking for.

I'll take book suggestions as well. Christmas is coming...

*interestingly, the first Google hit for 12345678987654321 is a MeFi thread.

Re-enter PhD program after many years -- recommendations?

lakshmi_mefi — Mon, 23 Nov 2009 15:31:39 -0800

Applying to enter a math PhD program after working in industry. Question about recommendations. I am thinking of applying to re-enter a math PhD program, maybe in applied math.

My brief background: a masters degree in Mechanical Engineering from U. of Toledo and another in Math from Auburn U. 16 years ago. Was briefly in the math PhD program at Purdue immediately afterward, but regrettably left it after 2 years - I squandered it.

Forward to now. I plan to take my GRE general soon and the math test in April.

I am stuck with getting recommendations however. Being a web developer, my current work is not too relevant to math, but still I can ask my boss for references. For another reference I could try asking my former boss from the previous company I worked for, but I can't think of whom to ask for a 3rd letter.

Any suggestions for what I can do for references? Do I have a chance of being accepted with 2 letters attesting to my web development skills? My GPAs from the MS degrees: 3.4 and 3.7, and I scored full on the math GRE in '93, and I was a good student at Auburn and considered potential for a PhD.

Thanks.

Business optimization eek!

nomnomnom — Sun, 22 Nov 2009 03:55:24 -0800

I've got these 2 math problems that are driving me crazy. Business optimization. Please help. I just can't grok this concept! Help me figure it out?

Q#1 A business consultant advises a client that by spending extra on advertising, the client will increase profits. For an increase of $x in advertising, profit will increase by 20 x²e^-.5x. What increase in expenditure will yield the greatest increase in profit? (x is the amount in 100s of dollars.)

Q#2 Fred has a business that makes two kinds of dining room suites. Model A takes 4 hours to assemble and 2 hours to finish. Model B takes 3 hours to assemble and 4 hours to finish. 3 employees assemble the suites and 2 do the finishing work. All employees work a 36 hour week.

Profit on Model A is $200, on Model B, $260. A minimum of 4 of each model has to be manufactured each week.
What production of each model will maximize profit? What is the maximum profit?

Adaptive computer programs for math practice?

Suciu — Thu, 19 Nov 2009 20:32:22 -0800

Any good computer programs for math practice? Impoverished children need your help. I'm a developing country watching some consultants pitch a computer program that helps students learn basic math. Essentially, students are asked questions and then moved to higher/lower difficulty questions based on their responses. If you do well with simple addition, you are moved to double-digit addition. And then double-digit addition with carryover. If they mess up on addition, then they get sent back down to basic counting. You know.

They've made their business model tremendously complicated. Despite never updating the software, they have the program run online and bill you for the program on a cash per student per year basis. That's whack.

I was wondering if anyone knew about educational software that covers a broad variety of math topics (the broader the better) with this sort of adaptive questioning without this ridiculous payment model. Computers might be easy to get, but this program would be crazy-expensive over time and internet access is no easy thing to provide (expensive and also hard to get in most of the country).

I understand that this kind of program is not going to repair math education, but it's not really going to be a harmful teaching aid (okay, I guess there's a bit of debate there), especially given the inexperience of most teachers in the country. These consultant are trying to sell a bunch of bullshit on how this software pinpoints where students are failing, but the truth is that it just offers personalized practice. A good thing, to be sure, but one that doesn't require their software. Right?

HY=EY to you too, buddy

ailouros08 — Tue, 17 Nov 2009 08:00:44 -0800

I'm being outsmarted by a personalized license plate (I think)! I was driving through Durham, NC last Saturday when I saw a truck with a personalized license plate reading HY=EY. The truck also had some math bumper stickers on it. What, if anything does HY=EY mean?

Paired T Test sounds like parakeet test to me

amethysts — Fri, 13 Nov 2009 11:37:29 -0800

Hi, I have a specific question about paired t-tests! On the paired t-test, the spreadsheet looks like this
Let's say I'm doing a a paired t-test for timed trials.
The spreadsheet looks like this:

Subject A Subject B Difference
5 3 2
4 4 0
5 6 -1

I don't understand why the "Difference" column allows for negative numbers. That would imply that there was a meaningful difference in the order of the two columns. If the order of the two columns isn't meaningful (it doesn't matter what data goes in col.1 and what data goes in col.2, since the trials were not completed in a set order) then shouldn't the "Difference" column transform the difference into the positive number? (For example, the -1 above would become a 1). Or is the order of the columns meaningful somehow that it would need to be preserved through the negative number?

Math Geek Filter: Where to find and what to call these

hipersons — Tue, 10 Nov 2009 15:37:42 -0800

Back in middle school and high school we'd play these fun math games where you'd have to draw a chart and figure out who had what. Example, there are five people, joe jon julie mike and amy. Each has a favorite color. Mike doesn't like red, but amy doesn't like blue, etc etc. What are these called and where can I get more? They were a blast!!

GMP math friendly web host?

wmeredith — Tue, 10 Nov 2009 10:02:13 -0800

Sys Admin filter: I'm looking for cheap web hosting that is out of the box GMP math friendly. Does anyone have a good recommendation?

What industries are relevant to my mathematical interests?

alligatorman — Mon, 09 Nov 2009 10:13:23 -0800

I've just obtained a BS in Mathematics and am planning to go to grad school, but not before I work for a few years. What industries should I be looking at, and what entry level jobs should I aim for? Details inside. My interests are in discrete math, probability, and related fields. I hope to return to school, but I would like to work first, and I'd like to work in a field even slightly relevant to my interests.

What industries should I be looking at, and what entry level jobs are out there?

I can program, though not well enough for software development. (And I have no interest in software development.)

I also don't want to apply for any position that requires a security clearance, so the obvious NSA is out of the question.

What's out there? Operations Research comes to mind, but I'm not sure what types of companies do this kind of work.

Any advice would be greatly appreciated.

I know I won't be doing any kind of real math work with just a BS.

Help me learn to typeset equations like it was my job. (Why? Because it is.)

nebulawindphone — Wed, 04 Nov 2009 12:56:56 -0800

I am looking for a math typesetting style guide. By this I don't mean the kind of stylesheet for journal submissions that says "Be sure to use the blah-blah-blah LaTeX package and the XYZ equation environment, and our army of editorial assistants will tie up the loose ends and knock off the rough edges." (Why not? Because my advisor is involved in starting a new journal, and suddenly my labmates and I are that army of editorial assistants.) I am less interested in the technical details of mathematical typesetting. We've got our fonts chosen already, we're committed to using LaTeX and AMSMath which I speak pretty fluently, and we're distributing online so anything having to do with print is Not An Issue. In particular, I am not looking for another LaTeX user's manual — although if the advice I need happens to be buried in one, I'm okay with that. I'm also not particularly interested in simple questions of usage ("bigger parentheses or square brackets?" "~ or ¬ for negation?"), especially since a lot of those boil down to taste and convenience anyway.

I'm more interested in what you might call the visual semantics of it. (F'rinstance: How do you set a long equation so as to reveal its structure quickly and easily to the reader? How can spacing, line breaks, alignment and so on be used to produce that sort of clarity, and what other tricks are there that I'm not thinking of? What about a sequence of equations? A derivation or proof? How do you set a nonstandard symbol — an operator, function, etc. defined by the author;we get this a lot in my field — so that it's clear what its role in the equation is? This isn't a complete list of questions, but it's questions like that that I want to learn how to answer.) Aesthetic details — good spacing, good line breaks and page breaks, all-around symmetry and tidiness — are also important. The goal is to make these thorny and technical articles as easy and joyful to read as I possibly can.

Bonus points for a guide with good advice on the odd situations that come up in formal semantics and mathematical logic. (For instance, I've been unable to find any advice for laying out expressions in lambda calculus, or ones containing multiple quantifiers, and both of those are frequent sources of difficulty here.) But if that's asking too much, then I'm looking for general best practices that I can apply to the edge cases when they come up.

Grapher: How do I animate a parameter?

b1tr0t — Sun, 01 Nov 2009 11:33:28 -0800

How do you animate a parameter in the OSX Grapher utility? There is an example called "Variable Parameter" that has a time line where T varies from 0 to 10. When you move the slider, the graphs are redrawn accordingly.

I'd like to do something similar, with different equations. But I can't seem to animate any parameters. I can create a system of equations with discrete steps (using curly brace notation) or do a parametric equation that varies continuously, but none allow me to animate a parameter.

Simple example:
I'd like to create an animation for y=sin(x+t) or even y=t*sin(x). How do I do that?

Countdown

Blazecock Pileon — Fri, 30 Oct 2009 13:31:34 -0800

Is there an enumeration of complete chess games? Has anyone made an accurate (ideally, "closed form") count of possible chess games?

I found an older article via Google that does a very rough guess-timate of how many complete chess games are possible.

I'm curious if anyone has since tried to count all possible games, given standard FIDE rules, played to completion. Presumably, for example, it is not exponential because some moves will result in an end state ("checkmate" or "stalemate") faster than other moves.

Please do not offer speculation about the answer
Please do not link to Google search results (unless you have found something that answers the question as specifically framed)
Thanks!

Math Symbol Mystery

mazniak — Sun, 25 Oct 2009 20:19:54 -0800

I've never seen this mathematical symbol used this way before. Can anyone tell me what it means? Ok, my ignorance is on display for the world to see…

Please look at this image .

I'm referring to the parallel bars between the two resistances at the top. Obviously anyone who's taken 9th grade geometry knows that it is a symbol for describing a relationship as being parallel. But how does it fit here? How is it used to derive the final value of the voltage gain (3.82)?

The twenty most important numbers

bigmusic — Fri, 23 Oct 2009 14:13:51 -0800

In a talk (at TED) by Brian Greene on string theory he says that there are "there appear to be about 20 numbers that really describe our universe..." He lists a few in his talk, but what are the rest of of those numbers? The pertinent excerpt from the talk:



And that question is this: when we look around the world, as scientists have done for the last hundred years, there appear to be about 20 numbers that really describe our universe. These are numbers like the mass of the particles, like electrons and quarks, the strength of gravity, the strength of the electromagnetic force -- a list of about 20 numbers that have been measured with incredible precision, but nobody has an explanation for why the numbers have the particular values that they do.

Help me understand homogenous coordinates

tickingclock — Thu, 22 Oct 2009 19:37:11 -0800

Help me understand homogenous coordinates! I encountered homogenous coordinates in computer graphics, where I understand that they let us translate vertices. But this is a rather shallow level of understanding, and I don't really get why, for instance, the position of the light source in OpenGL is necessarily specified with (x, y, z, w). Is it kind of like how there is an infinite number of points that can correspond to a single point on a 2D image at (x, y), but by specifying a third dimension, we can disambiguate between the different points?

I've tried reading Googling and I've tried reading Wikipedia -- but to no avail. I'm really hoping that somebody in the hive mind can explain it at a level that I can understand, and also why homogenous coordinates are important on a more broad level. Thanks!

How do I calculate how many square kilometers country's populations take up?

ejfox — Thu, 22 Oct 2009 15:45:13 -0800

I want to compare the area of some countries against the area of their people (assuming 1 person takes up 2 square feet). Help me work out the math! I suck at math, I am sorry and grateful to those of you who don't.

Here's an example of what I am trying to do:

China is 9,596,961 square kilometers.

It has a population of 1,333,670,000.

Google tells me 2 (square feet) = 1.8580608 × 10-7 square kilometers - which works out to 0.00000018580608

So what I am doing is calculating how much area the people take up by multiplying the population by 2 square feet, but in kilometers.

1,333,670,000 * 0.00000018580608 = 247.803995 sq / km

Am I to believe that if we gave everyone in China 2 square feet we could huddle them together in an area of 247 sq / km ?

The problem with this is I want to make a graphic comparing the area of the people to the area of the country, and it is a bit hard to show data so disparate. Let me know your thoughts.

Thanks!

Given the rate of a fish upstream, what is the speed of that fish in still water. Eh?

sidd.darko — Thu, 22 Oct 2009 08:46:09 -0800

InterviewPrepFilter: I need to brush up on my numerical reasoning. I'm looking for ways to master things like word/story problems (probability, work-rate, distance), and some math review. I have good logic skills, but basic math can be confusing for me, esp. under pressure. Tutoring is not really an option and anything that is cram-friendly is also good. Any books, resources, tips, general guidance from math-y people ?