Wikipedia:Reference desk/Archives/Mathematics/2010 March 23

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March 23

How to parametrise distribution given a few CDF points

Hi. Let's say I have a random variable, X (that represents a certain loss level) and I have three points for the CDF. So I know that
F(0.5) = k₁
F(0.9) = k₂
F(0.95) = k₃
X has a normal distribution.
Where F is the CDF and the k's are known.

My questions are:

A) How do I find the parameters for the distribution?
B) What if X is lognormal (or some other distribution) - is it still possible to find the distribution?
Thank you! Mudupie (talk) 08:02, 23 March 2010 (UTC)

A) Use a table or a computer to find the standard scores corresponding to the CDF values. For example, if $k_{1}=0.6$ and $k_{2}=0.8$, you have $z_{1}=0.2533,\ z_{2}=0.8416$ so $0.5=\mu +0.2533\sigma ,\ 0.9=\mu +0.8416\sigma$. Solve for $\mu$ and $\sigma$.

B) For lognormal specifically, you know that $\log X$ is normal so you solve problem A with $F(\log 0.5)=k_{1},\ F(\log 0.9)=k_{2}$. If you have some other parametric family, you just plug in the values for the CDF in terms of the parameters and solve. Whether the solution will have a nice closed-form expression, or you will have to solve it numerically, depends on the family. -- Meni Rosenfeld (talk) 10:34, 23 March 2010 (UTC)

nth iteration of a koch snowflake

For a koch snowflake in which iteration 0 (i.e. the triangle) has a side length of 1, is it true that iteration 3 has an area of ${\frac {94}{81{\sqrt {3}}}}$? Also, is the area of the nth iteration $A_{n}=A_{n-1}+{\frac {4^{n-2}{\sqrt {3}}}{3\times 9^{n-1}}}\quad n>1\,,A_{0}={\frac {\sqrt {3}}{4}}$, and if so, is there a non-recursive formula for the area of the nth iteration? --220.253.247.165 (talk) 08:02, 23 March 2010 (UTC)

Yes, it is a geometric series. Bo Jacoby (talk) 08:57, 23 March 2010 (UTC).

what's the best area for contributing something practical

what should i study, graph theory or what to contribute to actual projects later graf theory being microchips —Preceding unsigned comment added by 82.113.121.34 (talk) 18:23, 23 March 2010 (UTC)

Your question isn't clear. Can you rephrase it? -- Meni Rosenfeld (talk) 09:24, 24 March 2010 (UTC)

If you're interested in the design of integrated circuits specifically, it doesn't have too much to do with graph theory, it's more a matter of computer science and electrical engineering. Learning to program is probably the first step. Paul Stansifer 12:38, 24 March 2010 (UTC)

Distribution of a subset of a distribution

Suppose we have a variable x, drawn from some known probability distribution over [l,h], l<h, both real. Suppose then we're interested in y, which is the set of all x less than some constant m, l<m<h -- what is the distribution for y? I suspect that it's just the normalized distribution for x<m, but I'm not super good at statistics so I'd appreciate confirmation or correction as needed. 71.70.143.134 (talk) 18:29, 23 March 2010 (UTC)

It is. Algebraist 18:30, 23 March 2010 (UTC)

You're talking about the conditional probability distribution given the event that y < m. Michael Hardy (talk) 18:43, 23 March 2010 (UTC)

how much do words impede mathematics

a lot of times you see people struggling with concepts because they don't know what the subbranch of mathematics is called, or has been explored, they're just trying to do it ab ovo. Isn't this a case of words impeding mathematics, where if it were more systemized with elementary symbols, people could just see if this property has been explored before? there is no way to "guess" words, you can't just guess words like normal, perfect, complete, etc, when they have nothing to do with English usage. 82.113.121.38 (talk) 19:14, 23 March 2010 (UTC)

I fixed the title ("to" -> "do"). StuRat (talk) 19:28, 23 March 2010 (UTC)

You can't create a usable notation that covers everything. You have to invent new notations for each new branch of maths and those new notations are no easier to guess than words. --Tango (talk) 20:35, 23 March 2010 (UTC)

The general idea that words (or other symbolic expressions like musical or mathematical notation) shape thought (and vice versa) is known as the Sapir-Whorf Hypothesis. The degree to which this is really true is unclear; the current consensus seems to be roughly "maybe, a bit". In another vein, I just finished reading The Strangest Man, a biography of physicist Paul Dirac. It covered the different ways that different mathematical physicists thought about quantum theory - some used physical intuition, some thought about it algebraically, some with abstract timeline diagrams, and Dirac thought about it with a kind of projective geometry. So different mathematical minds used different approaches to understand the same shared concept; surely this will hold for ordinary students of maths too - some will benefit from diagrams, some from measurement, some from algebra and logic, some from thinking about the real-world meaning of the problem. There are some deeply-formal expressions of mathematics, such as Principia Mathematica, but they're deeply unsuitable for struggling learners. -- Finlay McWalter • Talk 20:54, 23 March 2010 (UTC)

I think it does happen, and has not much to do with words. There's a wide dependency graph between mathematical concepts, and if you're struggling with some problem in area X, even if you're relatively knowledgeable, it can take quite a bit of effort to find out that there's a body of theory in area Y that applies to your problem. Being around a big department with lots of different kinds of specialists to talk to helps. The Internet (including Wikipedia) also helps, since it makes surfing between topics much faster than was possible in the era of having to read a paper and then separately chase down each relevant citation from it in the library. 66.127.52.47 (talk) —Preceding undated comment added 01:02, 24 March 2010 (UTC).

But what the original poster seems to want is some way to standardize mathematical discourse, so that (to paraphrase maybe a bit reductively), you wouldn't have to think to see that, oh, this concept from general topology is kind of like that concept from algebra; it would just be obvious from the naming itself. That's not going to happen. To come up with such a standard you'd have to be able to predict what concepts mathematicians in all fields would think of and find useful. --Trovatore (talk) 01:17, 24 March 2010 (UTC)

Good mathematical notation does do that, to some extent—it's full of metaphors that suggest similarities between different things. For example, Leibniz's notation in calculus suggests that derivatives are like fractions, which suggests such properties as the chain rule and leads to the idea of differentials. The Cartesian product of sets is written with multiplicative notation such as × and ∏, suggesting similarities with the ordinary concept of multiplication. The set inclusion symbols ⊆ and ⊇ are analogous to the inequality symbols ≤ and ≥, which suggests that set inclusion induces a (partial) ordering. Similarly, the join and meet symbols ∨ and ∧ are similar to the union and intersection symbols ∪ and ∩, which suggests that join and meet can be thought of as generalized unions and intersections. Exclusive or is often denoted ⊕, because (thinking of true as 1 and false as 0) it can be thought of as addition in the finite field $\mathbb {F} _{2}$. Of course, all of this notation was invented by somebody who first saw the similarities between the concepts—it would be impossible (or at least highly unlikely) for someone to invent notation for a mathematical concept and only then see a similarity with another concept because it was suggested by the notation. —Bkell (talk) 10:36, 24 March 2010 (UTC)

A taxonomy of mathematical concepts, like giving chemicals names. Now there's an interesting idea. I suppose one could make a start with category theory. I think what I'd do for something like this is not actually give names to the concepts but tag them with types and attach them to linked concepts, and have a computer look for similar structures. A bit like matching up separate trees made by different people in one of those ancestry programs on the web that look for similar birthdays or names and which can be mapped partly to each other. Could be interesting, I'm not sure it would help much but occasionally things like that do pan out and are quite useful sometimes in ways you never expect. Dmcq (talk) 10:05, 24 March 2010 (UTC)

Questions about Birkhoff Normal form

I have an area preserving map on the plane with an elliptic fixed point. My questions are:

• where can I find the procedure to compute the coefficients of the Birkhoff normal form?
• are there sufficient conditions which implies that there is at least a coefficient different from 0?

--Pokipsy76 (talk) 21:27, 23 March 2010 (UTC)