Wikipedia:Reference desk/Archives/Mathematics/2010 September 15

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September 15

3rd Axiom of Probability

There is a line in the solution to a homework problem that I don't get.

It reads

Noting that $A=(A\cap B)\cup (A\cap B^{c})$ and $B=(A\cap B)\cup (A^{c}\cap B)$ we can use the third axiom of probability to write

$\mathbb {P} (A)=\mathbb {P} ((A\cap B)\cup (A\cap B^{c}))\mathbb {P} (A\cap B)+\mathbb {P} (A\cap B^{c})$ and $\mathbb {P} (B)=\mathbb {P} ((A\cap B)\cup (A^{c}\cap B))\mathbb {P} (A\cap B)+\mathbb {P} (A^{c}\cap B)$

I don't see how the 3rd axiom is being used to get from one bit to the next, and I can't see where the second bit comes from —Preceding unsigned comment added by 130.102.158.15 (talk) 03:21, 15 September 2010 (UTC)

You're missing a few equality signs. It should be

$\mathbb {P} (A)=\mathbb {P} ((A\cap B)\cup (A\cap B^{c}))=\mathbb {P} (A\cap B)+\mathbb {P} (A\cap B^{c})$ and $\mathbb {P} (B)=\mathbb {P} ((A\cap B)\cup (A^{c}\cap B))=\mathbb {P} (A\cap B)+\mathbb {P} (A^{c}\cap B)$

Where you have clearly used the fact that the probability of a union (of disjoint events) is the sum of probabilities. -- Meni Rosenfeld (talk) 06:49, 15 September 2010 (UTC)

It looks like you're missing a couple of equals signs - $\mathbb {P} (A)=\mathbb {P} ((A\cap B)\cup (A\cap B^{c}))=\mathbb {P} (A\cap B)+\mathbb {P} (A\cap B^{c})$, where the first equality is just using the substitution for A. The second equality is then where the 3rd axiom comes in - since the substitution we're using for A breaks it into two mutually exclusive events (the event where both A and B happen and the event where A happens but B doesn't), the probability of their union (i.e. either "A and B both happen" or "A happens but B doesn't") is the sum of their individual probabilities. The expression for P(B) is similar. Confusing Manifestation(Say hi!) 06:50, 15 September 2010 (UTC)

Polynomial Achieving Infimum

Studying analysis the following question came up. Let $p:\mathbb {R} ^{2}\rightarrow \mathbb {R}$ be a polynomial function of two real variables. Suppose that $p(x,y)\geq 0$ for all x and y real. Does every such function attain its infimum? Prove or disprove. My intuition tells me that it is true (because you can't have something like a decaying exponential which will always decrease and get arbitrarily close to zero). A polynomial must have some zeros (maybe not real). But I don't have any idea how to even begin. Help? 174.29.63.159 (talk) 04:25, 15 September 2010 (UTC)

Here's how I would do it, but there may be better ways to approach it. First it should be clear how to show a positive polynomial in one variable achieves it's infimum over R. Consider the all the lines through the origin in R². On each of these lines L, p can be expressed as a polynomial in one variable, so it attains its infimum b_L. Let f: S¹ → R map each point x on the circle to b_L where L is the line that passes through x. Then show that f achieves its infimum. Rckrone (talk) 05:24, 15 September 2010 (UTC)

Outside a sufficiently large closed disk, {(x,y) : x²+y²≤R²}, you have p(x,y)>p(0,0). The disk is a compact set and a continuous function on a compact set attains its infimum. Bo Jacoby (talk) 08:33, 15 September 2010 (UTC).

I don't think the first claim is true. For example consider p=x²+1. I'm also having serious doubts about the strategy I suggested. I thought it would be relatively easy to show that the function f I constructed is continuous, but the more I think about it, the less obvious it is. Rckrone (talk) 12:56, 15 September 2010 (UTC)

$p(x,y)=(x^{2}y^{2}-1)^{2}+y^{2}$ does not attain its infimum. (It took me quite a while to find; on retrospect, it looks like something I would have seen sometime). -- Meni Rosenfeld (talk) 14:15, 16 September 2010 (UTC)

Ah, the Rabinowitsch trick! How could I miss that (I spent some time trying to fix the arguments above). BTW, $(xy-1)^{2}+y^{2}$ also works.—Emil J. 14:36, 16 September 2010 (UTC)

Wait....can someone explain how the above example(s) work? How does it not attain an infimum? What's happening in the above function? I chucked it into Wolfram Alpha to see what was happening but a 3D plot is none too enlightening... Zunaid 18:31, 16 September 2010 (UTC)

Oh okay...I worked it out. Choose y as close to zero as you like (so that y^2 term is minimized), then choose x=1/y to make the first term zero. You can therefore make the entire expression as close to zero as you like, however you can't actually attain zero because once you set y=0 the first term automatically jumps back up to 1. Hence the infimum is zero but can never actually be attained. Clever! Zunaid 18:36, 16 September 2010 (UTC)

Exactly. Also, Alpha can give a better visualization if you help it a bit (note that I interchanged x and y). -- Meni Rosenfeld (talk) 19:35, 16 September 2010 (UTC)

Box-Muller Transformation

On a completely different note, I am trying to show that the Box-Muller transform takes two independent uniform random variables (from [0,1] each) to two independent standard normal random variables. So when I was trying to invert the transformation, I came across the expression ${\frac {X_{2}}{X_{1}}}=\tan(2\pi U_{2})$. I have a problem at this point taking the inverse tangent of both points. My reasoning is this. Since U2 can take on any value in (0,1], 2*pi*U2 can take on any value in (0,2*pi] but tangent is not invertible in this domain (as a function). So how/why can we do this? I took the forward Jacobian and I know that it is never zero on (0,1]x(0,1]. So the entire transformation is invertible. But when I was trying to invert it, I couldn't convince myself that this is valid. How does this work? Thanks! 174.29.63.159 (talk) 04:36, 15 September 2010 (UTC)

Your expression is correct but not sufficient to define the transformation which is (quoting the Box-Muller transform article that uses Z₀, Z₁ rather than X₁,X₂)

$Z_{0}={\sqrt {-2\ln U_{1}}}\cos(2\pi U_{2})\,$

$Z_{1}={\sqrt {-2\ln U_{1}}}\sin(2\pi U_{2}).\,$

Bo Jacoby (talk) 08:17, 15 September 2010 (UTC).

Riemann Zeta Function

On yet another completely different note, I am doing some reading on Riemann and his zeta function and the prime number theorem. I am just curious where (what journals, good papers, books, or websites) I can find some up to date info on this. I don't know much about complex analysis so something that is easy to follow would be appreciated (like the most recent list of the computed zeros or something). I am just wondering what the most cutting edge research is in this field. Maybe a popular book for the general (slightly math oriented) audience? Thanks! 174.29.63.159 (talk) 04:41, 15 September 2010 (UTC)

How about Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics by John Derbyshire (ISBN 978-0452285255). --Qwfp (talk) 08:41, 15 September 2010 (UTC)

The last paragraph of the introductory section of the Riemann hypothesis article actually gives a great list of sources. -- Creidieki 10:57, 15 September 2010 (UTC)

Synthetic division

please why the synthetic division can work? the article uses too many terms I have not gotten to in class yet. THank you. —Preceding unsigned comment added by 24.92.78.167 (talk) 23:02, 15 September 2010 (UTC)

Dividing a polynomial by x-a amounts to multiplying that polynomial by

1/(x-a). Consider expanding 1/(x-a) in a geometric series as follows:

${\frac {1}{x-a}}={\frac {1}{x}}{\frac {1}{1-{\frac {a}{x}}}}={\frac {1}{x}}\sum _{k=0}^{\infty }\left({\frac {a}{x}}\right)^{k}$

If you multiply your polynomial by this and consider the coefficients of the largest power of x, the next largest power etc. etc. you'll see the synthetic division method emerge. Count Iblis (talk) 00:26, 16 September 2010 (UTC)

I am not surprised that you are having difficulty understanding our synthetic division article - it is a very poor explanation of the process. Synthetic division is really quite straightforward. Here is an example. Suppose we want to divide $3x^{4}+x^{3}+6$ by $x^{2}+x+1$:

1. To match the $3x^{4}$ term we need to start by multiplying $x^{2}+x+1$ by $3x^{2}$
2. This gives us $3x^{4}+3x^{3}+3x^{2}$. Subtract this from $3x^{4}+x^{3}+6$ and we are left with $-2x^{3}-3x^{2}+6$.
3. So $3x^{4}+x^{3}+6=3x^{2}(x^{2}+x+1)+(-2x^{3}-3x^{2}+6)$
4. To match the $-2x^{3}$ term we now need to multiply $x^{2}+x+1$ by $-2x$
5. This gives us $-2x^{3}-2x^{2}-2x$. Subtract this from $-2x^{3}-3x^{2}+6$ and we are left with $-x^{2}+2x+6$
6. So $3x^{4}+x^{3}+6=(3x^{2}-2x)(x^{2}+x+1)+(-x^{2}+2x+6)$
7. To match the $-x^{2}$ term we now need to multiply $x^{2}+x+1$ by $-1$
8. This gives us $-x^{2}-x-1$. Subtract this from $-x^{2}+2x+6$ and we are left with $3x+7$
9. So, putting all this together, we have $3x^{4}+x^{3}+6=(3x^{2}-2x-1)(x^{2}+x+1)+(3x+7)$

If you know long division they this should look familiar - it is like long division but without any carries. You can think of it as long division in an "infinite" base.Gandalf61 (talk) 15:22, 16 September 2010 (UTC)