Wikipedia:Reference desk/Archives/Mathematics/2007 August 13

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What is the inverse function of f(x) = x + sin(x) if x is in radians. Thanks, *Max* 01:04, 13 August 2007 (UTC)

I may be wrong, but unless you restrict yourself to a small domain, it doesn't seem quite possible for there to be any -- the slope of the function is 0 at a number of points, meaning that the inverse function would have to have infinite slope at the corresponding points. Graph the function y = sin(x) + x, then flip it over the line y=x. The flipped function would be the inverse, but there are points where the slope is infinite, making it impossible for there to be a function that draws that curve. I could be completely wrong, though -- someone please confirm this. Gscshoyru 01:24, 13 August 2007 (UTC)

Actually, having zero derivative at multiple points is no barrier to having an inverse (unless it's always zero). In fact, since the derivative is f'(x) = 1 + cos(x), this is always positive, so f is a monotonic function and thus guaranteed to have an inverse. Unfortunately, it's not an inverse that can be written in terms of elementary functions, to my knowledge. Confusing Manifestation 02:54, 13 August 2007 (UTC)

Just because there is infinite slope at certain points doesn't necessarily mean that the function being considered isn't a function.

You sure? The inverse has to be continuous, as the normal function is. Can you name a continuous function with infinite slope? At the least, you can't do it with elementary functions.

Plus, the derivative would have to be infinite at the point -- which means the integral containing that point would be infinity, meaning at that point the inverse would be at infinity, which isn't true. At least, I think that's how it works. Gscshoyru 04:13, 13 August 2007 (UTC)

Have you actually tried to construct an example? Give x^3 a shot. Its slope at x=0 is 0, right? Then, the slope of its inverse (x^1/3) is infinite/undefined at x=0. The cube root of x, however, is without question an elementary function, continuous at all points. In answer to the original question, I don't know what its inverse would be called. It might be possible to work out a power series for it. Black Carrot 04:26, 13 August 2007 (UTC)

I had tried to construct an example, but failed. But that one works. So much for me -- I have no clue what the inverse function is, then. Gscshoyru 04:28, 13 August 2007 (UTC)

Fair enough. One other thing - a curiosity, really, more than an answer. If the function is y = x + sin(x), then the inverse of that is x = y + sin(y), solved for y. Solved partway, that's y = x - sin(y). You could argue that it's also y = x - sin(x-sin(y)), though, or y = x-sin(x-sin(x-sin(x-sin...))). It's one limiting behavior of the system k_n+1 = x - sin(k_n) as n->inf. Black Carrot 05:10, 13 August 2007 (UTC)

Oooh... pretty patterns. I wonder if you can get a nice taylor series or something out of that. Gscshoyru 05:19, 13 August 2007 (UTC)

The first few terms of the power series are

${\frac {x}{2}}+{\frac {x^{3}}{96}}+{\frac {x^{5}}{1920}}+{\frac {43x^{7}}{1290240}}+{\frac {223x^{9}}{92897280}}+\cdots$

I don't think anyone has ever cared to give a name to the function.  --Lambiam 05:39, 13 August 2007 (UTC)

I hope you don't mind, I shifted your text. How did you derive that? Black Carrot 05:45, 13 August 2007 (UTC)

The Lagrange inversion theorem is a general technique for finding power series expansions of inverses.

Suggestion for Gscshoyru - think about integrals such as $\int _{0}^{1}{\frac {dx}{\sqrt {x}}}$ and $\int _{0}^{\infty }x\sin(x^{3})\ dx$, which are both perfectly finite. -- Meni Rosenfeld (talk) 13:46, 13 August 2007 (UTC)

So they are. Usually my mathematical intuition is pretty good... but it failed me this time. Thanks for helping me out here. Gscshoyru 19:24, 13 August 2007 (UTC)

Carrying White Man's Burden

An White Man explorer wishes to cross a barren desert that requires 6 days to cross, but one coolie (black dude hired to carry oods for a mere pittance) can only carry enough food for 4 days. What is the fewest number of coolies required to help carry enough food for him to cross?

each person needs 6 days of food

There is 1 white man and X coolies

Each coolies can only carry 4 days of Food

6 * (1 + X) = 4 * X

6 + 6 X = 4 X

6 + 2 X = 0

3 + X = 0

X = -3

ANSWER He need -3 coolies to cross the desert,

Somehow, I think there is something wrong with my answer but I just can't put me finger on it. Where did I go wrong? 220.239.110.87 08:03, 13 August 2007 (UTC)

I think something may be missing from the question (or from my understanding, or your interpretation). If each coolie can carry only 4 days worth of food, but each coolie needs 6 days worth of food, you can easily see that with each coolie you are bringing in a 2 day deficit of food. This is why you got the answer -3, because you need -3*-2 = 6 days food for the original explorer, which is correct mathematically, but pretty useless in the real world. Now you need to work out where you are interpreting the question wrong, and re-explain the question for us :) Capuchin 08:37, 13 August 2007 (UTC)

Right, one coolie can't even make it across the desert by himself, much less carry any food for the white man. Each extra coolie only worsens the problem (an extra mouth to feed and not enough food), hence the nonphysical negative solution. If we're interpreting the question correctly, the answer is "The white man cannot make it across the desert, no matter how many coolies he employs.". —Keenan Pepper 08:46, 13 August 2007 (UTC)

Your answer is correct. Substitute X=-3 into you equation, and find it satisfied: 6 * (1 + (-3)) = 4 * (-3). Do you have problems getting a negative number of coolies? Then you must change strategy and establish sufficient stores of food in the desert. One coolie can do that before the white man enters the desert. Bo Jacoby 08:57, 13 August 2007 (UTC).

It may be a correct solution to the equation, but it's a nonsensical and incorrect solution to the original word problem. If I were a math teacher, I'd use questions like this all the time to make sure my students knew how to interpret confusing numerical results. —Keenan Pepper 09:12, 13 August 2007 (UTC)

And if I were a math teacher, I'd be fired for this. Somehow, I don't think the question is about math. --Lucid 09:34, 13 August 2007 (UTC)

Someone trolling around with racial epithets, more likely.

I'm pretty sure there is a solution, and it requires this hint: what if (some of) the coolies only go part way, then hand over some food and go back home? I haven't worked it all the way out, but I think that's the way to do it (that, or let them die in the desert while he carries on, but I'm pretty sure it isn't that kind of question). Confusing Manifestation 10:24, 13 August 2007 (UTC)

1 White Man with Infinite number of coolies can only travel 3 days into the desert. If he tries to travel 4 days, his group would run out of food. So maybe if he travels one days, create a food depot and travel back. This way he can create food depot for each day journey distance into the desert.

So the absolute minimum number of coolies must be 2 since travel 1 day in and 1 day out is 2 days and White Man + 2 coolies requires 6 units of food and 2 coolies can carry 8 units of food. So each time they can increase the amount of food in the depot by 2 units.

220.239.110.87 12:03, 13 August 2007 (UTC)

I went for the hardcore leaving people to die tactic (White men don't want to be going back and forth for weeks when he can just leave some coolies to die): I think I have an answer, It has 2 food unused though but he gets across in 6 days. If he sets out with 8 coolies, then he has 32 food. After the first 2 days he has 14 food left. He sends one coolie home with 2 food and leaves 4 to die (or leaves 5 to die and has a bit of a feast that day.. you decide). He carries on with 3 coolies and 12 food. After another 2 days he has 4 food left. He leaves another 2 coolies to die. He makes the final 2 days of travelling with the last coolie carying the last 4 bits of food. And the coolie lives. Is this the minimal number of coolies to set out with assuming a 6 day crossing? I don't know. Capuchin 11:12, 13 August 2007 (UTC)

One coolie carries four day-parcels of food into the desert for a day and leave 3 parcels there, eating the fourth parcel and resting at night. He marks the camp with a flag so that it can be found. He returns the second day to the base to eat and rest. The third and fourth day he repeats the trip, and so on until the supply at camp one is sufficient. Then he walks out to the supply and starts transferring food another days journey further into the desert until the supply at camp two is sufficient. And so on. When finally he has left 1 parcel at each of the camps 1, 2, 3, 4 and 5, and returned to the base, he informs the white man that the road is now prepared for him to walk across the desert. The detailed calculation of how much food is sufficient is left to you, but one coolie is sufficient, and that was the question. Bo Jacoby 12:02, 13 August 2007 (UTC).

I wonder where the OP has got this question from? Whether he thought it up himself or was asked it at school, there is something very wrong here. Since there is nonetheless an interesting mathematical problem beneath, I will rephrase it. An artillery battery needs to cross a desert urgently. To complete this 6-day journey, they bring along some groups of fuel tanker trucks. Each group has only 4 days worth of fuel, but enough supplies of food and water to survive being stranded in the desert until being rescued. How many groups of tankers are needed?

Of course, the optimal strategy will be to jounrey until one group's reserves are depleted, leave it in place and continue. Let $T_{n}$ be the maximal distance possible with n groups. $T_{0}=0$ and $T_{n}=T_{n-1}+{\frac {4}{n+1}}$, so $T_{n}=4\sum _{k=2}^{n+1}{\frac {1}{k}}\approx \ln n-0.423$. We conclude that 6 groups of tankers are needed, $T_{6}\approx 6.37$. -- Meni Rosenfeld (talk) 14:16, 13 August 2007 (UTC)

I'm wondering if there's a book of "Mathematics problems for white people" out there. It reminds me of all my mathematics problems that involved characters like Osama and Hadji, out of political correctness. Capuchin 14:53, 13 August 2007 (UTC)

• Please note that to my knowledge the use of the term 'coolie' is deemed derogatory, and is thereby technically racist. Given that this is a mathematical question where the term is not needed, it should be left out. Furthermore, beyond the derogatory inclination of ther term 'coolie', your mathematical story itself hold racist connotations - the mere fact that the black people have to carry the food for the white man - not exactly false in certain historical contexts - the fact that the story specifically points them out as "black and white" and the use of the word 'coolie' makes for, overall, a rather racist problem. That is not to say you are racist, etc., or that anyone is yet offended (myself included - I am caucasian), but it is absolutely worth pointing out - Wikipedia should not be a place where racism -- even at its mildest levels -- is tolerated. Rfwoolf 01:03, 14 August 2007 (UTC)

We have an article about coolie which has apparently changed use. The description of an explorer having to bring many days of food on foot sounds like something happening long ago. But solutions with helpers left to die sound unpleasant in all cases. What if: All people must start at the same time, all helpers must survive either by crossing or going back (and cannot bring out more food after coming back), and the explorer must cross in 6 days. PrimeHunter 01:20, 14 August 2007 (UTC)

You seem to be confused about what racism is. The OED gives "The theory that distinctive human characteristics and abilities are determined by race" or "Belief in the superiority of a particular race leading to prejudice and antagonism towards people of other races..."; Merriam-Webster gives "a belief that race is the primary determinant of human traits and capacities and that racial differences produce an inherent superiority of a particular race" or "racial prejudice or discrimination". The word you're looking for may be "offensive". There is no word such that simply uttering it makes or proves the speaker racist or prejudiced, although there are many words that offend merely by being spoken. And while I agree this problem gratuitously invokes the particulars of the situation, be careful of whitewashing history to ease your racial discomfort. "Black" people having to carry food for the white man is not just "not exactly false in certain contexts"; it is an emblematic symbol of Western colonialism which occurred on every continent and continued for centuries. Let's not forget it. Tesseran 19:15, 15 August 2007 (UTC)

Just wanted to say I'm glad people are also addressing the more fundamental problem with this question. I was just reading today about the math professor H. Hanani of the Technion, whose long-time argument that the school needed a humanities program fell on deaf ears. One day, he gathered a hundred students for an exam consisting of a single question: what technical details are necessary for the construction of a pipeline to carry blood from the city of Eilat to the port of Ashdod? The students asked questions about the length of the line, the topography, whether there were physical factors that would cause the blood to clot, and other technical questions. Not a single student asked for what purpose the blood needs to be piped. Hanani took the results to the university's senate (ruling body of faculty), who were shocked, and thus granted his proposal. nadav (talk) 05:57, 14 August 2007 (UTC)

I have to say I don't think that's entirely fair to the students. Generally students are expected to assume the conditions of a problem as a given. "Thinking outside the box" is a good talent to develop but a poor test-taking strategy. Also, many instructors give slightly "transgressively" worded questions simply as a way of breaking up the monotony (I recall a problem in my undergrad economics class that asked us to assume our utility was a certain function of our cocaine and marijuana consumption). This (blood piping) problem is a bit more Goth than that one but there's still no hint that the usual "this has nothing to do with real life" rules don't apply. --Trovatore 09:18, 14 August 2007 (UTC)

OK, based on the assumptions that (1) the explorer can carry as much food as a local (I'll call them locals now to avoid any possible racism), (2) the journey must be completed in 6 days, so no caches of food can be left, (3) everyone has to live, I have a solution found by a sort of greedy algorithm-type thinking, working backwards:

On day 6, the explorer arrives at his destination with all locals who didn't turn back. Frankly, it seems that having *any* locals with him is excessive, since any food they carried would be theirs anyway. So assume no locals finish the journey with him.

The most economical way for him to manage this would be to have a full supply of food at day 2, so he can just go straight to the end with no further complications. So on day 2, the explorer and n locals part ways, the explorer with 4 days of food and the locals with 2 (enough to get them home).

Now, if there were no exchange of food on day 1, then everyone who started the journey would make it to day 2 - but with only 2 days of food left, that's only enough to head back home and not give any to the explorer. So, they must have figured a way to fill their packs on day 1, meaning that by day 2 everyone has 3 days of food with them. So, $3(n+1)=4+2n$ (i.e. the food they have by day 2 is enough to give the explorer 4 days of food and still leave them 2 days each, like I said before). => n = 1.

By the same reasoning, if there are m extra locals who only go one day in, then at the end of day 1 the total amount of food is $3(m+n+1)=3(m+1+1)=3m+6$, and since it needs to be split up with the explorer and his "day 2" local with full packs and the other m with just 1 day of food, we have $3m+6=m+4(n+1)=m+4(1+1)=m+8$ => m = 1.

So, the explorer needs just 2 locals to carry food for him. You can double-check this by tracking their journey - start them with 4 days of food, then a day later have the first local split his remainnig food between the three of them and head back home, then the next day have the second local give the explorer the last ration he needs. You can also prove this is the minimal solution by simply showing that he can't make it with just one local. Confusing Manifestation 04:43, 14 August 2007 (UTC)

I had the impression from the statement of the problem that the white guy didn't carry any of his own food. If he did, it seems like it would specify how much he could carry. Black Carrot 05:27, 14 August 2007 (UTC)

This problem suffers from a lack of certain definitions. For example, does the white man carry any food? I don't think so, as the question was worded thus "What is the fewest number of coolies required to help carry enough food for him to cross?" - the question presumes the white guy doesn't carry any food. More of a problem is the statement that "one coolie [...] can only carry enough food for 4 days" -- it isn't clear if that "4 days" is for the coolie and the white man, or just the white man, or for that matter just the coolie. If I may give my own interpretation, I would say each coolie carries 4 days worth of food for the coolie and the white man, therefore, 8 days worth of food for the coolie alone or the white man alone.

Therefore, each coolie carries a surplus of 2 days worth of food -- when feeding him alone. The white man requires 6 days worth of food (made up of the surplusses, 6 / 2 = 3 coolies).

Of course if the definition is changed to say "each coolie only carries enough food for 4 days if only he eats it" then answer is "There is no answer that would solve the problem given the constraints provided".

So I'm saying the answer is 3. PS> I still think the question would offend certain readers Rfwoolf 09:05, 14 August 2007 (UTC)

(edit confilct) Another variation of the problem is that the battery does not have a significant fuel reserve, and the tankers are expected to arrive safely at either the origin or the destination. Then, let $a_{n,n}$ be the maximal distance we can reach with m full tankers if we started with n. The recursion will then be $a_{n,n}=0$ and $4=(m+2)a_{m-1,n}-(m+1)a_{m,n}$, giving $a_{m,n}={\frac {4(n-m)}{n+2}}$. The maximum distance possible with n tankers is then $\max _{m}\{a_{m,n}+{\frac {4m}{m+1}}\}$. For $n=18$ this is 6, obtained for both $m=3$ and $m=4$. -- Meni Rosenfeld (talk) 09:24, 14 August 2007 (UTC)

Rfwoolf: The problem statement certainly doesn't imply that everyone needs to reach the destination, so there is no need to assume that each can carry 8 days worth of supplies. -- Meni Rosenfeld (talk) 09:27, 14 August 2007 (UTC)

You are correct, the questions asks "... carry enough food for him to survive". This leaves room for death of the coolies. It also follows that this isn't an ethical question - the goal is to get the white man across at all costs. However surely you understood my rationale, that if one coolie carries 4 days' worth of food for him and the white man, then it follows that two coolies carry 5 1/3 days's worth of food for each of them, three coolies carries 6 days' worth of food for each of them. etc. But going by the unethical perspective you mentioned, then he would simply require two coolies (if each coolie = 4 days for one person) or one coolie (if each coolie = 8 days for one person). Rfwoolf 10:05, 14 August 2007 (UTC)

Actually, if you look more closely at my last post, you will see that nobody has to die either (which I agree is unethical, and this is why I prefer to work with the rephrased problem) - it is also possible for some to return "home" after a while. Of course, this way it is impossible to travel 8 days or more. -- Meni Rosenfeld (talk) 10:18, 14 August 2007 (UTC)

If locals who return home can leave food depots for locals who return later then any distance is possible without deaths (assuming the food doesn't rot). PrimeHunter 15:43, 14 August 2007 (UTC)

I think you end up eating the coolies. Gzuckier 17:00, 14 August 2007 (UTC)

Nodal properties in Bifurcation theory

What is the exact meaning of the term "Nodal properties" in Bifurcation theory? deeptrivia (talk) 17:58, 13 August 2007 (UTC)

I don't know much about bifurcation theory, but in spectral geometry and PDE, "nodal" means related to the null sets of eigenfunctions of some operator. It seems like you are reading some paper/monograph in analysis (geometric PDE, functional analysis, dynamical systems maybe?). Do you have a reference? I know some analysis, and it will be easier to help if I can see the paper.Phils 22:37, 13 August 2007 (UTC)

Thanks Phils! Please check out "Large Buckled States of Nonlinearly Elastic Rods under Torsion, Thrust and Gravity" by Antman and Kenney. Regards, deeptrivia (talk) 23:46, 13 August 2007 (UTC)

Well, the nodes are the points where bifurcation occurs (for example, the value of a parameter at which a single solution becomes two), so presumable the "nodal properties" are the properties of that node that allow you to identify the type of bifurcation (e.g. Saddle-node bifurcation vs Pitchfork bifurcation etc.). They would include various equations describing the equilibria of the system in terms of the paramater(s). Confusing Manifestation 04:24, 14 August 2007 (UTC)

You can have nodes without having a bifurcation. In the Saddle-node bifurcation there are two nodes at t=-1, a saddle point (Crunode), and an atractor/repellor (Acnode). These are stable phenomena. As t approaches zero the two nodes come together and form a Cusp or Spinode, which is a more degenerate type of node. The two nodes annihilate and for t=1 there are locally no nodes. Catastrophe theory describes some of the basic types of node and transitions which occur. --Salix alba (talk) 09:26, 14 August 2007 (UTC)

Thanks! deeptrivia (talk) 21:58, 17 August 2007 (UTC)

For the Love of MATH?

I'm currently a college student who has finished the math courses required for my degree, which sadly isn't a course with a very high number. Anyway, my question in short is: Does anybody know of a good book, or a few good books, that help us wordy people understand math? I think part of it is probably that all the symbols and numbers intimidate me and I fail to learn the principles, which I think would help me a lot. I've always hated math since early high school when it turned out I wasn't very good at it, however, I value it highly as a method of thinking/perspective and because I only see more math in the future of the world so I'd like to keep working at math without dragging my GPA into the dumpster. Please help. --preferably books for math levels around college algerbra or above. Thanks!

One of the difficulties with answering such a question is that mathematics is enormous in both depth and breadth. There are certainly books which can help in understanding one topic or another, but finding one that can help understand mathematics in its entirety is difficult indeed! Perhaps others know of a book which can achieve this lofty goal, so you should wait for other replies. In the meantime, I can recommend to start with the book "The Mathematical Universe" by William Dunham (not the songwriter) - it probably won't help you with your college courses directly, but I think it can certainly help you get your thinking right. -- Meni Rosenfeld (talk) 19:43, 13 August 2007 (UTC)

I'll second the Mathematical Universe recommendation - and Dunham's Journey Through Genius is also worth reading. Ian Stewart has written clear and accessible books on a variety of different topics in mathematics. He also edited an updated version of the classic What Is Mathematics? by Richard Courant and Herbert Robbins - this is a bit more "hard core", so it might be worth borrowing a copy before buying it, to see if it suits your taste. If you want something that is more hands-on, try You Are A Mathematician by David Wells, or one of Raymond Smullyan's books on mathematical logic, such as The Lady or the Tiger?. Gandalf61 09:53, 15 August 2007 (UTC)

I am personally a fan of neither What is mathematics nor Stewart's "update" of it. You just can't trust a man who writes that NP stands for "non-polynomial". I'm making a deal out of this because this represents a fundamental misunderstanding, and one shouldn't write about what he understands so little. -- Meni Rosenfeld (talk) 13:10, 15 August 2007 (UTC)

It depends on whether you want to become a famous researcher with name to be remembered 100 years after your death, or whether you want to make a lot of money more than the boss of Microsoft. If it is the first case, I am happy to make a new friend. My email address can be found from my signature. twma 00:26, 14 August 2007 (UTC)

Take a look at the books by Robert Kaplan, particularly The Art of the Infinite. You might also enjoy In Code which gives a gentle introduction to the mathematics behind cryptography. If you care to look, I have a log of the mathematics books that I've read at [1]. Donald Hosek 17:32, 14 August 2007 (UTC)

You asked for books, but I propose that the better alternative is people. Since you are a college student, there will be professors and other students who know and love mathematics. That's a more valuable resource than books, and the reason may surprise you. A strange tradition in mathematics writing transforms a very human and informal process into something elegant and austere. As well, a book must make assumptions about what you know and what you will understand; a person can talk with you personally.

Mathematics comes in diverse flavors: the linear algebra and differential equations used by engineers, the combinatorics and abstract algebra used by computer scientists, the game theory and statistics used by economists, the group theory and differential geometry used by theoretical physicists, and the panoply of highly specialized topics studied by research mathematicians.

Like a good novel, mathematics begins with inspiration from human experience, but spins new worlds, new languages, new cultures to explore. If we are lucky, these worlds begin to live and breathe on their own, and inspire other worlds. We like to view a topic from different perspectives; we like to explore its relationships with other topics.

Someone who knows and loves mathematics can be your tour guide, introducing you to the breath-taking view, the little café favored by locals, the dangerous neighborhood to avoid on your own, the personalities and titillating bits of history.

Don't be discouraged by your early high school experiences, which are often primitive calculations taught by uninspired conscripts. Is "See Spot run" literature? Compare:

• "The sun did not shine. It was too wet to play. So we sat in the house All that cold, cold, wet day." — Dr. Seuss, The Cat in the Hat
• "You don't know about me without you have read a book by the name of The Adventures of Tom Sawyer; but that ain’t no matter." — Mark Twain, Adventures of Huckleberry Finn
• "He was an old man who fished alone in a skiff in the Gulf Stream and he had gone eighty-four days now without taking a fish." — Ernest Hemingway, The Old Man and the Sea
• "Okonkwo was well-known throughout the nine villages and even beyond." — Chinua Achebe, Things Fall Apart
• "My name was Salmon, like the fish; first name Susie. I was fourteen when I was murdered on December 6, 1973." — Alice Sebold, The Lovely Bones
• "In the country of Westphalia, in the castle of the most noble Baron of Thunder-ten-tronckh, lived a youth whom Nature had endowed with a most sweet disposition. His face was the true index of his mind. He had a solid judgment joined to the most unaffected simplicity; and hence, I presume, he had his name of Candide." — Voltaire, Candide
• "I have lost count of the days that have passed since I fled the horrors of Vasco Miranda’s mad fortress in the Adalusian mountain-village of Benengeli; ran from death under cover of darkness and left a message nailed to the door. And since then along my hungry, heat-hazed way there have been further bunches of scribbled sheets, swings of the hammer, sharp exclamations of two-inch nails. Long ago when I was green my beloved said to me in fondness, 'Oh, you Moor, you strange black man, always so full of theses, never a church door to nail them to.' (She, a self-professedly godly un-Christian Indian, joked about Luther’s protest at Wittenberg to tease her determinedly ungodly Indian Christian lover: how stories travel, what mouths they end up in!) Unfortunately, my mother overheard; and darted, quick as snakebite: ‘So full, you mean, of faeces.’ Yes, mother, you had the last word on that subject, too: as about everything." — Salman Rushdie, The Moor's Last Sigh

Even for youngsters Dr. Seuss is better reading than Dick and Jane; while Rushdie may be appealing for adults but perhaps unsuitable for the kiddies. So it is with mathematics. It can be beautiful, fun, difficult, surprising. And, like literature, the more you know, the richer the meaning. --KSmrq^T 00:12, 16 August 2007 (UTC)

Baseball probability

I should remember how to do this from stats class, but I'm an idiot. What is the probability a team would score 4 runs per game over 30 games if we know their true ability is to score 5 runs per game (with a standard deviation of 2 runs per game)? What is the formula to use for this kind of problem? Thanks for all the help! Alex 23:24, 13 August 2007 (UTC)

There are many different probability distributions that have an arithmetic mean of 5 and a standard deviation of 2, and not all will result in the same answer. However, when averaging 30 outcomes of a random variable, the central limit theorem kicks in; therefore we can give a very good approximate answer. Even though the distribution of the score for a single game need not be normal, that of the average will be nearly normal. A normal distribution is characterized by two parameters: μ (the mean) and σ (the standard deviation). Let μ₁ and σ₁ be the mean standard deviation for a single game (so it is given that μ₁ = 5 and σ₁ = 2). Then, for the average over N games, μ_N = μ₁ and σ_N = σ₁/N^1/2 – in words, divide by the square root of N. So we have μ₃₀ = 5 and σ₃₀ = 2 / √30 = 0.365... . An observed value x can be normalized by replacing it by (x – μ) / σ; this transforms it into an outcome of a (still normally distributed) random variable with mean 0 and standard deviation 1, that is, its distribution is the standard normal distribution. Applying this to the observed value of x = 4 gives z = (4 – 5) / 0.3651... = –2.7386... .

That was the easy part. Now we need to decide what it is we want the probability of, exactly. If the average over 30 games was 4, the total was apparently 120. But the chance of getting exactly 120 runs in 30 games is of course pretty small. Even if the team had kept to their "true" average of 5, the likelihood of getting precisely 150 runs in 30 games, and not 149 or 151, say, is not large. So a better question is: what is the probability that a random variable Z with a standard normal distribution is at most –2.7386...? This is a question you can only answer by looking this up in a table, or using statistical software that can answer such questions. In the usual notation, this is

Pr(Z ≤ z) = Φ(z),

and it is the case that Φ(–2.7386...) = 0.003..., or about 0.3% Not much.

However, it is quite plausible that the question is asked because the observed average is markedly less than the expected average (the "true" ability, however that is determined). If the team had scored consistently better than an average of 5, you'd also be inclined to wonder what the probability was. Therefore it is more appropriate to ask for the probability of a deviation of the average that is at least as large. This is then

Pr(|Z| ≥ |z|) = 2Φ(z),

which raises the answer to about 0.6%. Still not much – it may be time to start wondering if 5 runs is truly the team's true ability.  --Lambiam 03:15, 14 August 2007 (UTC)