Physics Forums - Set Theory, Logic, Probability, Statistics http://www.physicsforums.com Fundamentals: Set theory, logic and proofs. Probability and statistics en Sun, 19 May 2013 17:33:58 GMT vBulletin 60 http://www.physicsforums.com/images/physicsforums/misc/rss.jpg Physics Forums - Set Theory, Logic, Probability, Statistics http://www.physicsforums.com X is an element of Y, such that 2/X is in Y. Is this allowed http://www.physicsforums.com/showthread.php?t=692456&goto=newpost Sun, 19 May 2013 05:24:06 GMT Hey guys, I was working on something today and I was trying to formalize what I was doing, and I tried to write out: M = { x | ( 2 / x )... Hey guys,

I was working on something today and I was trying to formalize what I was doing, and I tried to write out:

M = { x | ( 2 / x ) [itex]\in[/itex] M }

Then I thought about it and wondered whether this is allowed in set theory. Its not clear that there is anything in M at all, and thus its not clear whether any X would be in M. The criterion rests on an infinite number of other criterion. Is this allowed in any system of set theory? And if it isnt, why not?

If it is allowed, how about:

The Natural Numbers = {x | x-1 [itex]\in[/itex] M [itex]\vee[/itex] x = 0 } ]]> Set Theory, Logic, Probability, Statistics Uvohtufo http://www.physicsforums.com/showthread.php?t=692456 calculating lottery probability http://www.physicsforums.com/showthread.php?t=692364&goto=newpost Sat, 18 May 2013 16:05:40 GMT Hello, this lottery problem has been bothering me for months. I set it up here. Help appreciated. Here is the setup: Suppose you have to pick 5... Hello, this lottery problem has been bothering me for months. I set it up here. Help appreciated.

Here is the setup:

Suppose you have to pick 5 numbers from a set of integers ranging from 1 to 39. The probability of matching x/5 (where x can be 1,2,3,4,5) numbers correctly can be found by computing the hypergeometric distribution formula, as found on wikipedia. we have that:

probability of matching x/5 = {5 choose x}*{(39-5) choose (5-x)} / {39 choose 5}

where {a choose b} = a! / (b!*(a-b)!)

Computing for x = 2,3,4,5 yields:

x=2: probability = 59840/575757 ≈ 0.104
x=3: probability = 1870/191919 ≈ 0.00974
x=4: probability = 170/575757 ≈ 0.000295
x=5:probability = 1/575757 ≈ 1.737e-6

Now here is the problem:

Suppose you buy 8 tickets, assume all of them have distinct sets of 5 integers, and play these 8 combinations for 1,000,000 drawings of the lottery. How many tickets in total (out of 8*1,000,000) do you expect to match x out of 5? i.e. in 8,000,000 plays, how many tickets yield 3 out of 5 matched?

I had originally thought that the solution was simply the probability of matching x/5 multiplied by the total number of plays, in this case 8e6, but this isn't the case. Although, this approach works for 2 out of 5 matches but not for 3 or 4...unsure about 5/5.

Any help is well appreciated. ]]> Set Theory, Logic, Probability, Statistics Mugged http://www.physicsforums.com/showthread.php?t=692364 Confidence intervals for two separate variables? http://www.physicsforums.com/showthread.php?t=692055&goto=newpost Thu, 16 May 2013 16:17:21 GMT Hi

I have a certain experiment that I repeat 40 times and get the result:

0.001 +/- 0.004.

Now I've repeated the experiment using a different method (so it is essentially a new experiment) and I get a new value:

-0.002 +/- 0.003

Now, is it true to say there is no statistically significant difference between these two different methods? Even though they lie within each other's standard deviation, I think the fact that I've repeated the experiment 40 times should mean something- it makes me confident that method 2 gives a lower result. I don't know how to translate this confidence into statistical analysis though.

The fact that the means are different is clearly not sufficient to convince anyone... how can I convince someone that method 2 gives a lower result? Let's say I even repeat the experiment another 40 times and get EXACTLY the same means and standard deviations. But I know with more certainty the means are different because I've done more experiments. How do I show this using maths though (without actually having to do the experiments?).

Thanks ]]> Set Theory, Logic, Probability, Statistics mikeph http://www.physicsforums.com/showthread.php?t=692055 Ross-Littlewood Paradox http://www.physicsforums.com/showthread.php?t=691920&goto=newpost Wed, 15 May 2013 20:47:25 GMT Ross-Littlewood vase filling paradox (from Wikipedia): To complete an infinite number of steps, it is assumed that the vase is empty at one minute... Ross-Littlewood vase filling paradox (from Wikipedia):

To complete an infinite number of steps, it is assumed that the vase is empty at one minute before noon, and that the following steps are performed:

The first step is performed at 30 seconds before noon.
The second step is performed at 15 seconds before noon.
Each subsequent step is performed in half the time of the previous step, i.e., step n is performed at 2−n minutes before noon.
This guarantees that a countably infinite number of steps is performed by noon. Since each subsequent step takes half as much time as the previous step, an infinite number of steps is performed by the time one minute has passed.

At each step, ten balls are added to the vase, and one ball is removed from the vase. The question is then: How many balls are in the vase at noon?


To me, it is somewhat obvious that it is ω+ω+ω+ω+ω+ω+ω+ω+ω. (9ω's).

Where am I wrong/what am I missing? ]]> Set Theory, Logic, Probability, Statistics BDV http://www.physicsforums.com/showthread.php?t=691920 Define sets from peano axioms http://www.physicsforums.com/showthread.php?t=691894&goto=newpost Wed, 15 May 2013 17:40:35 GMT Is it possible to define sets from just the peano axioms? Usually when people use the peano axioms as the basis of their math they just assume the... Is it possible to define sets from just the peano axioms?

Usually when people use the peano axioms as the basis of their math they just assume the existence of sets but without axioms regarding sets we technically shouldn't just say they exist.

Oh, also there are two versions of the induction axiom. Obviously you can't use the one that mentions sets. ]]> Set Theory, Logic, Probability, Statistics Dmobb Jr. http://www.physicsforums.com/showthread.php?t=691894 Probability of coin flipping streaks. http://www.physicsforums.com/showthread.php?t=691728&goto=newpost Tue, 14 May 2013 19:42:43 GMT Recently I was discussing hitting streaks with my dad and I said "If you flip a coin a million time you're bound to get a streak of a hundred."

I am not sure if this is actually true and I am having some trouble figuring it out.

The more general question that I would like to be able to answer is what is the probability that you will get a streak of k heads when you flip a coin n times?

I found this article "http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-spring-2005/lecture-notes/l24_spcl_topics.pdf"

It explaines how to solve this however it requires k base cases. For k=100, this method is kind of useless. Can anyone think of a good way to solve this for large k and n?

PS. Using a computer is certainly okay. ]]> Set Theory, Logic, Probability, Statistics Dmobb Jr. http://www.physicsforums.com/showthread.php?t=691728 Fit to (orthgonal?) polynomial function http://www.physicsforums.com/showthread.php?t=691687&goto=newpost Tue, 14 May 2013 16:29:42 GMT Hi all. I need some advice in a project I'm into.

I have some experimental (simulation) data and i need to find a function that fits to it. The experimental data behaviour change when I modify some parameters I have. My goal is, from that single function, been able to predict how the experimental data will change, acording to the parameters: I mean, been able to find an analytical expression that represents all the information I have.

For the characteristics of my problem, I've decided to try with a polynomial function. To been able to see how the factors varies, i've done some fits with mathematica. However, every time i change the polynom degree, not only the factor changes of value, but its signs can also change (for example, the factor of the cuadratic part, in a 4-degree polynom fit is positive, and in a 5-degree polynom fit is negative). I'm aware that there is not a unique function that can represent some data. However, this behaviour is a big problem to my goal since this breaks down any truly generalization attempt of my solution.

As far as I suspect, a possible solution for this problem is trying to make the fit in a set of orthogonal functions: particularly, in orthogonal polynomial functions. However i would like to know your opinion on this. In particular two aspects worries me of this aproach: the first one is if making my fit in orthogonal polynomials (or orthogonal functions in general), would really solve my problem of changing cofficients with the degree of the function that i use to make the fit. The second one is that some of this sets (the few I know), like the Legendre polynomials, have even in their high degree polynomials, terms that include cuadratic dependence: i wonder if this would not be a problem, since I believe that the data have a strong dependence on this term (this is more an observational hunch, nothing rigorous)

I hope I make myself clear. Any suggestion or advice would be really apreciated. Also, any bibliography to develop the orthogonal fit would be nice (if it is still a good idea, of course). ]]> Set Theory, Logic, Probability, Statistics Tojur http://www.physicsforums.com/showthread.php?t=691687 Question about covariance http://www.physicsforums.com/showthread.php?t=691509&goto=newpost Mon, 13 May 2013 14:12:04 GMT Let X1 and Y1 be two random variables. We have Cov(X1,Y1) = 0. Does this extend to any transformation X2 = g(X1) and Y2 = g(Y1), such that... Let X1 and Y1 be two random variables. We have Cov(X1,Y1) = 0. Does this extend to any transformation X2 = g(X1) and Y2 = g(Y1), such that Cov(X2,Y2)? Here, g is a continuous function. For example, if we set X2 = X1^2 and Y2 = Y1^2. Do we then from Cov(X1,Y1) = 0 that Cov(X1^2,Y1^2) = 0? ]]> Set Theory, Logic, Probability, Statistics PAHV http://www.physicsforums.com/showthread.php?t=691509 Statistics is NOT mathematics http://www.physicsforums.com/showthread.php?t=691447&goto=newpost Mon, 13 May 2013 04:27:35 GMT Hi, I was just working on a problem based on the probability of wind or rain. The only way to solve it is to assume they are dependent factors... Hi,

I was just working on a problem based on the probability of wind or rain. The only way to solve it is to assume they are dependent factors when they clearly are not. Assuming independence will lead to the wrong conclusion.

If stats demands one ignore reality to solve a problem, to bend reality into the formula to make it accurate I contend it is nothing but game playing and puzzle solving, at least in probability.

Another great example we have been given on an exam is you have 6 multiple choice questions.. You have an 80% chance of getting any one individual question right. Create a scenario where you can get 100% correct. The solution is some bizarre random number scheme and does NOT give 100% accuracy. Anything less than KNOWING 100% would be pure chance.

I never thought I would miss vector calc but at least you could prove it without changing reality...

W ]]> Set Theory, Logic, Probability, Statistics Whalstib http://www.physicsforums.com/showthread.php?t=691447 musings on set creation and a question about it http://www.physicsforums.com/showthread.php?t=691406&goto=newpost Sun, 12 May 2013 23:56:09 GMT Hello! :) *My background:* MSc in engineering (up to, but not exceeding, multivariable calculus) and the rest is just free time hobby research. I... Hello! :)
My background:
MSc in engineering (up to, but not exceeding, multivariable calculus) and the rest is just free time hobby research. I ponder things because it's fun.

My musings:
I enjoy the idea of sets living within some kind of "universe", so that before a set is used or referred to it must be created. I have not seen anyone refer to set creation before, so let me briefly explain what I mean:

I dislike the phrase "Let S be a set, where blablabla..."
solely because it conjures up the set S out of thin air.
I prefer the phrase "Set S (created from set Q by algorithm blablabla)"
because it provides a solid grounding point for the existence of that set, thus decreasing the number of assumptions made! :)

Basically, I want to change all "If we blindly assume S exists then..." into "In case Q exists then..." by always requireing all sets to be created before they're referred to.

For this, I envision two different ways to create sets: By adding elements to another set (the additive way), or by removing elements from another set (the subtractive way). This creates a chain of dependencies, where the existence of the tail depends on the existence of the head. I have a vague idea that any set should be able to be created by adding elements to the empty set, or by removing elements from the infinite set. Since the additive way should be impossible due to the non-existence of the elements, this leaves us with the subtractive way. The infinite set should be able to refer to parts of itself when listing what elements to remove, to create the new set. Thus the subtractive way should be possible, while the additive should not.

The reason I want this is because I have a gut feeling that it should provide me with better insight into how to resolve certain paradoxes.

My question:
Does this sound like anything you're familiar with? Is there a jargon term for what I'm thinking about? Do you know of any research paper or book that I may read about this? ]]> Set Theory, Logic, Probability, Statistics nQue http://www.physicsforums.com/showthread.php?t=691406 A proof that a computer cannot generate a truly random number? http://www.physicsforums.com/showthread.php?t=691234&goto=newpost Sun, 12 May 2013 03:07:54 GMT I'm thinking about how to do a proof that a computer cannot generate a truly random number.

Attempt. Let Ω = {ω1, ω2, ..., ωn}, a subset of ℝ, be all the numbers represented on a certain machine. A random number generator rand(), because its output is dependent on how many times it has been called, is analogous to a function f:N→Ω (where I'm letting N denote the natural numbers). We need f to have the following properties:

(1) The long term frequency of any wi appearing is 1/n. That is, for any i ε {1, ..., n}, if we let gi(f(n)) = 1 if f(n) = ωi, and gi(f(n)) = 0 if f(n) ≠ ωi, then

ƩnεN gi = gi(f(1)) + gi(f(2)) + ... = 1/n.

(2) We need independence .... I'm trying to think how to formalize this statement. ]]> Set Theory, Logic, Probability, Statistics Jamin2112 http://www.physicsforums.com/showthread.php?t=691234 probability of getting 53 sundays http://www.physicsforums.com/showthread.php?t=691134&goto=newpost Sat, 11 May 2013 14:00:50 GMT In a leap year the probability of getting 53 sundays or 53 tuesdays or 53 thursdays is In a leap year the probability of getting 53 sundays or 53 tuesdays or 53 thursdays is ]]> Set Theory, Logic, Probability, Statistics mia5 http://www.physicsforums.com/showthread.php?t=691134 Need a real life example that satisfies the property? http://www.physicsforums.com/showthread.php?t=690910&goto=newpost Fri, 10 May 2013 06:22:20 GMT I was solving a question which is the following :

Give examples of 3 sets W,X,Y such that W ε X and X ε Y but W doesn't ε Y . I solved the question by taking the following 3 sets:

W = {1,2}
X = { 7 , 8 , W}
Y = { 3 , 4 , X}

looking it from the theory point of you I find that W is not a part of Y but the problem is I cannot come up with an example from real life that satisfies this condition. Kindly help me visualize this.

Thanks in advance. ]]> Set Theory, Logic, Probability, Statistics ankitsablok89 http://www.physicsforums.com/showthread.php?t=690910 <![CDATA[Test Hypotheses with sample of Binomial RV's]]> http://www.physicsforums.com/showthread.php?t=690871&goto=newpost Fri, 10 May 2013 02:06:19 GMT Hi, I am trying to teach myself how to test hypotheses for any distribution, but am having some trouble.

X=number chosen each year
θ=Mean number chosen in the population

H0: θ=.5
h1: θ>.5

The random sample of n=4 is 0,1,3,3

Test the Hypotheses at α≤0.05 assuming X is a binomial(5,θ/5).

This is what I have so far, but I feel I am completely missing something..

Sample average (Xbar = 1.75

So,

Reject H0 if P(Xbar≥1.75, given that X is binomial(5,.1)) ≤ 0.05

Then I figure out 1-P(Xbar≤1.75)=0.08146 which is greater than 0.05 so I reject the null.

I know something is not right... Any help would be much appreciated. Thanks! ]]> Set Theory, Logic, Probability, Statistics LBJking123 http://www.physicsforums.com/showthread.php?t=690871 What is the point of a partial order? http://www.physicsforums.com/showthread.php?t=690816&goto=newpost Thu, 09 May 2013 20:03:36 GMT I have a feeling the question I am about to ask, I won't be able to ask it the way I am trying to...but I will try. I will break it up into two questions.

1) What are its real life applications? Much easier for me to get it when I can see a real life application
2) Why is it the way it is? I get that it is reflexive, transitive, and antisymmetric. Why is that an important combo though? Why did the person who create partial orders decide they should be antisymmetric rather than symmetric? Is there something about the combo of reflexive, transitive, and antisymmetric that is good?

If you are reading this and thinking "What the hell is this guy asking", then I am sorry, lol. ]]> Set Theory, Logic, Probability, Statistics CuriousBanker http://www.physicsforums.com/showthread.php?t=690816