Wikipedia:Reference desk/Archives/Mathematics/2009 August 29

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August 29

Why use Poisson distribution as an approximation to binomial?

I understand that the Poisson distribution can be used as an approximation to a binomial distribution (if n is sufficiently large and p sufficiently small). I've seen it demonstrated, and I think I get the math involved. But why would you want to use this approximation instead of just using the binomial? What is the advantage of using the Poisson? —Preceding unsigned comment added by 118.241.57.8 (talk) 02:03, 29 August 2009 (UTC)

One could just consider it an out-of-date notion, relevant for a time when calculation was more difficult.Julzes (talk) 02:16, 29 August 2009 (UTC)

One reason is you might have no clue how big n is, or p, but you may still have a good estimate of λ = np. Michael Hardy (talk) 02:32, 29 August 2009 (UTC) ...and besides, obviously it's simpler. Michael Hardy (talk) 02:33, 29 August 2009 (UTC)

The last answer is better than mine.Julzes (talk) 03:56, 29 August 2009 (UTC)

• As Michael said, the Poisson distribution is much simpler. Why should you (the mathematician, not the computer) do calculations involving two parameters, and think about the problem in terms of two parameters, when all the relevant information is contained in one?
• The faster computers become, the more difficult the calculations people use them for. Calculating just one value of a distribution is of course easy, but it's not unthinkable a problem would require billions of such calculations. In such cases there may be a significant difference between evaluating a binomial distribution, which requires 3 factorials, and Poisson which only requires one.

-- Meni Rosenfeld (talk) 21:43, 29 August 2009 (UTC)

The binomial distribution is (a limiting case of) the hypergeometric distribution for large populations, and the poisson distribution is (a limiting case of) the binomial distribution for large samples. You may like to study Cumulant#Cumulants_of_some_discrete_probability_distributions to see that the relationship between binomial and poisson distributions are like the relationship between ellipses and parabolas. The parabola can be used as an approximation to an ellipse. But why would you want to use this approximation instead of just using the ellipse? Bo Jacoby (talk) 22:47, 29 August 2009 (UTC).

The prospective user of a quick approximation to something where an exact calculation is possible but may take more time needs to keep in mind that the trade-off may not be a good one because of the need to keep track of errors. If the approximation really is good enough on its face, that's one thing; but where there is uncertainty it may be necessary in very complicated cases to do some sort of prior study of the nature of the trade-off before one decides whether to use the quick approximation.Julzes (talk) 04:00, 30 August 2009 (UTC)

Some folk here might be interested to check out Stein's method, which is one way of generalizing the binomial -> poisson limit. HTH, Robinh (talk) 08:29, 30 August 2009 (UTC)

The Tychonoff theorem

There's every reason to believe that the product space X is (in terms of sets) different from all the spaces X(alpha) in the family. Now take the product space X of the collection of all compact spaces. X is compact hence it belongs to the collection, and so X is different from X (in terms of sets). Is this where proper class come in? Standard Oil (talk) 12:35, 29 August 2009 (UTC)

Right. Within the category of (compact) topological spaces you are only allowed to do products of families of spaces indicized by sets. Indeed what you proposed is just a topological version of a paradox that you may repeat in all versions: the sup of all ordinals is not an ordinal; the union of all sets is not a set.. &c. This does not mean that you can't endow a proper category with a structure that mimes the usual "small" structures (a group structure, a topological structure and so on).--pma (talk) 13:10, 29 August 2009 (UTC)

Question about sets

Hey, I've got a logic question relating to sets that I havn't been able to quite figure out. Maybe someone here could help me. OK, here goes:

There are 6 hypothetical sets (some of them are real sets, but some of them are non-standard). These are N, W, Z, Q, I, and R.

${\sqrt {17}}$ belongs to sets R and I,

$0.86$ belongs to sets R and Q,

${\sqrt {64}}$ belongs to sets R, Q, Z, W, and N,

$-10/2$ belongs to sets R, Q, and Z.

I've gathered that set R are all real numbers, set I are irrational numbers, set Q are rational numbers, and set N are natural numbers. I am have some difficulty with sets W and Z, however. Can anyone help? Thanks in advance, --Anonymous —Preceding unsigned comment added by 做工器 (talk • contribs) 19:59, 29 August 2009 (UTC)

$\mathbb {Z}$ is standard notation for the integers (it comes from German). I've never come across "W" as a standard name for a set before and can't guess what it would be from the information provided. --Tango (talk) 20:07, 29 August 2009 (UTC)

Ah, Blackboard bold says it could be "whole numbers". I guess whoever is using this notation uses W to refer to whichever of {0,1,2,...} or {1,2,3,...} isn't being called the natural numbers (definitions vary, you see - some people consider 0 natural, some don't). --Tango (talk) 20:09, 29 August 2009 (UTC)

The sets in the order presented are contained within the next one - so N would be {1,2,...} and W is {0,1,2,...}. Acb314 (talk) 10:55, 30 August 2009 (UTC)

... maybe, but Q (whatever it is) cannot be contained in I (whatever it is) since Q contains at least three given members that are not in I. Logically, there is no way to infer the entire contents of these hypothetical sets from a few members. We can, however, say that a minimal solution is R={√17, 0.86, 8, -5}; I={√17}; Q={0.86, 8, -5}; Z={8, -5}; W=N={8}. Gandalf61 (talk) 12:06, 30 August 2009 (UTC)

I think "the order presented" means the order in the statement "${\sqrt {64}}$ belongs to sets R, Q, Z, W, and N". That makes Acb314's argument as good as any, but it's of course true that we can't know much about these sets. —JAO • T • C 12:17, 30 August 2009 (UTC)

That was lack of reading comprehension on my part -indeed Q will not be contained in I! Assuming N, Z, Q, R have the usual meanings, which they seem to, N is inside Z inside Q inside R, so my logic was that N would also be inside W from the order they were written in - not mathematically rigorous I know. I throws a spanner in the works unfortunately, although I is "bigger" than Q and contained within R. I agree that there isn't enough information in the examples to tell what N and W are just from those. Acb314 (talk) 14:31, 30 August 2009 (UTC)

Average Value of a Function

Hello, someone asked me a question about this and I have been thinking about it ever since. Working only in the reals, the question is to find the average value of a function over a given set. Obviously, if the set has a finite number of points, then using the definition of average, I can just evaluate at each point and then take the average. If the set is a finite interval, then we just integrate over the interval and then divide by the length of the interval. Extending this, if the set is a bounded region in R^n, then I just integrate over the set and then divide by the volume of that region. Here are the questions:

1.To find the average, it is okay to divide by the Lebesgue measure of the underlying set over which we integrate, right? Is it correct to say, for a bounded set, that to find the average value of a function over that set, just integrate over that set and divide by its Lebesgue measure, assuming that its Lebesgue measure is nonzero?

2.If this is correct, then how do we deal with sets that have measure zero? If the set is finite, we are good. But what if the set has an infinite number of points with measure zero, like the Cantor set or the rationals in [0,1]? I don't want to be dividing by zero but I can't simply evaluate at each point and take the average.

3.In addition, how do we deal with sets with infinite Lebesgue measure? I am sure I can't also "divide" by infinity after the integration. For example, what is the average value of $f(x)=e^{(-x^{2})}$ over $[0,\infty ]$?

4.Furthermore, what is meant when we are asked to find the average value of a function without specifying the region? Is it safe to assume that the entire domain of the function is meant or is there some other standard? Thanks!-Looking for Wisdom and Insight! (talk) 20:52, 29 August 2009 (UTC)

#1 is correct not only for bounded sets, but for any set whose measure is finite. In #3, the way it's usually done involves conditional convergence, so you might get different answers depending on how the region of infinite volume is approached. The idea is: take an average of a subset of finite measure, then take a limit as as that subset in some way approaches the whole domain. There's no non-uniqueness problem if the function is everywhere nonnegative, nor if the positvie and negative parts both have finite integrals; it's only when both are infinite that there is such a problem. Simple example:

$\lim _{a\to \infty ,\ b\to -\infty }\int _{b}^{a}{\frac {x\,dx}{1+x^{2}}}.$

If b = −2a, then the limit is (1/2)log(1/4), but if b = −a, then the limit is 0. See also Cauchy principal value. Michael Hardy (talk) 21:19, 29 August 2009 (UTC)

[ec] I don't think the term "average value of a function" is generalizable to the kind of cases you are thinking about. For these cases it may be more useful to consider the expected value of a function of a random variable, where the latter follows a given distribution.

1. This seems correct.
2. There is probably no generic solution. For a countable set you may choose to assign a weight for each point, or choose an enumeration and take $\lim _{n\to \infty }{\frac {1}{n}}\sum _{i=1}^{n}f(q_{i})$. For a non-countable set you may want to choose a countable subset of "key points" (for the cantor set, a natural choice is the set of interval endpoints).
3. You can choose a weighting function with a finite integral. In some cases (e.g. periodic functions) it is appropriate to consider $\lim _{t\to \infty }{\frac {1}{t}}\int _{0}^{t}f(t)\;dt$ or $\lim _{t\to \infty }{\frac {1}{2t}}\int _{-t}^{t}f(t)\;dt$.
4. It seems safe to assume the entire domain.

-- Meni Rosenfeld (talk) 21:25, 29 August 2009 (UTC)

With periodic functions, I'd just take the average value over one period. Michael Hardy (talk) 02:20, 30 August 2009 (UTC)

Fun little problem

Here's a fun little problem I saw yesterday. Which is bigger $e^{\pi }\,$ or $\pi ^{e}\,$? No calculator, computer, or search engine allowed; just your brain and some paper. Try to find a proof, i.e. no estimations allowed. It took me a little while to think of a proof, but in the end it's not that difficult. I would like to see if anyone can come up with a proof that uses a different method to mine. If this is too easy for you, then after you've posted a proof (which must use a different method to any others that yours follows) then please turn your attention to my serious problem above. ~~ Dr Dec (Talk) ~~ 23:13, 29 August 2009 (UTC)

Consider the inequality $\scriptstyle e^{x}>x^{e}$, that is $\scriptstyle x>e\ln x$. So we're interested in the sign of $\scriptstyle f(x):=x-e\ln x$, which is nicely continuous and differentiable for $\scriptstyle x>0$. $\scriptstyle f'(x)=1-e/x$ so the only solution to $\scriptstyle f'(x)=0$ is $\scriptstyle x=e$. $\scriptstyle f(e)=0$, of course, and it's readily seen that this is a minimum. Hence, $\scriptstyle e^{x}>x^{e}$ for all positive $\scriptstyle x$ (except $\scriptstyle x=e$). —JAO • T • C 23:40, 29 August 2009 (UTC)

Please. It's $\scriptstyle e\ln x$, not $\scriptstyle e\,lnx$. The backslash BOTH prevents italicization and provides proper spacing. Michael Hardy (talk) 06:24, 30 August 2009 (UTC)

Oops, of course. I thought my spacing problems were somehow due to \scriptstyle, so I just thought it weird instead of thinking myself stupid. Thanks for setting me straight on that; I fixed those uglies now. —JAO • T • C 08:59, 30 August 2009 (UTC)

Essentially the same as the above. Taking logs we see we are comparing $\pi$ to $e\log \pi$. Now, differentiate $x/\log x$ to compare it to $e$. Phils 23:46, 29 August 2009 (UTC)

But I'd like to see a non-log proof, if at all possible. ~~ Dr Dec (Talk) ~~ 23:59, 29 August 2009 (UTC)

It is easy to re-write Jao's proof to not use logs. Let $f(x)=e^{x}x^{-e}$ for x > 0. We have $f'(x)=e^{x}x^{-e}(1-e/x)$. This is zero exactly when x = e, so f has a unique minimum at e. Proceed as before. Eric. 98.207.86.2 (talk) 01:18, 30 August 2009 (UTC)

Declan, you've been posting some interesting topics, but they really aren't Reference Desk questions. Have you thought about starting a blog? 67.122.211.205 (talk) 06:16, 30 August 2009 (UTC)

Really? I know that this one is boarderline, but all of my other threads have been perfectly valid reference desk question. My latest questions have been The real numbers and subsets of the real numbers, Commutivity of addition, Associativity of composition and Line bundles and cohomology. I've just read them all again, and they seem perfectly valid reference desk question. I don't know, what do other people think of these five questions? ~~ Dr Dec (Talk) ~~ 09:54, 30 August 2009 (UTC)

I think Dr Dec's questions are entirely appropriate for the Mathematics Reference Desk, and I welcome him as a new regular contributor here. But this is off-topic so if any further discussion is required. let's take it to Wikipedia talk:Reference desk. Gandalf61 (talk) 11:55, 30 August 2009 (UTC)

I agree. I'll admit that this thread is a very unusual RD thread, posing a problem the OP has already solved. But as the goal was not to see if others can solve it, but rather to see if there are other ways to solve it, I took it as a sincere fragment of a pursuit of deeper understanding. Which, after all, is what the RD is intended for, isn't it? (Dec's questions have also certainly been more interesting than the typical homework questions, but I don't think that is disputed.) —JAO • T • C 12:04, 30 August 2009 (UTC)

Fun little macro with a (math) issue...

I posted this here because this is more a math question than a computer question, though it involves both...

Im making a little macro for fun with AutoIt v3 which opens up MsPaint and draws. There are currently values called xMomentum and yMomentum which guide the pen. Basically the program starts with xMomentum = 0, yMomentum = 0, then each step it generates a random number between -2 and 2 and adds that to xMomentum , and does it again seperately to yMomentum, then the mouse is moved that many pixels.

What i am seeming to get is momentums that build up, so the pen starts to fly across the screen in a (non straight) line. Although a few straight bits are cool, id like more changes to happen. My thought is this:

I could have variables xTop and xBottom, where the random number is no longer between -2 and 2 its between xTop and xBottom. I would adjust these based on xMomentum so that the "faster" the pen goes in either X direction, the more likely it is to change direction slightly. (of course, i would do the same for corresponding y values.)

I dont know how to do this mathematically though, any ideas? All thoughts are welcome! Thanks! :)

97.112.117.236 (talk) 23:38, 29 August 2009 (UTC)

There are a lot of ways you could do this. One way would be to subtract k times the momentum from the momentum at each step where k is a small positive constant. This is basically like introducing a frictional force proportional to the velocity.

Another way would be to just cap the momentum. If at any step it exceeds the cap, reduce it to the cap. Rckrone (talk) 00:19, 30 August 2009 (UTC)

I spoke shortly with my father, and he unearthed a problem. Lets say i add a random integer in the set [-2, 2] as i did before. this is added to a momentum, so there are 3 integers which do not decrease the speed, and 2 that do. Thus 0 needs to be removed from my random integers.

I sort of like the frictional thing.... but can i use that and still get the possibility of both increase or decrease? Maybe i should add my random number (-2, -1, 1, or 2) and then do this subtraction? this would have the possibility of speeding up or slowing down, but add the friction as well. am i correct?

97.112.117.236 (talk) 00:26, 30 August 2009 (UTC)

Yeah I meant add the friction part in addition to what you're already doing. With the friction alone it would be pretty dull: just slowing to a stop and then sitting still. Using the set {-2, -1, 0, 1, 2} doesn't cause any problems, although {-2, -1, 1, 2} works fine too. As long as the average is 0 the momentum won't favor one direction or the other. Rckrone (talk) 00:43, 30 August 2009 (UTC)

Perhaps so. I have opted to eliminate 0 just in case :) 97.112.117.236 (talk) 00:49, 30 August 2009 (UTC)

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