If we tried to define the concept "observed to occur", would your axioms be different?

(What motivates this question is thinking about a sequence of coin flips as a randomly chosen real number in the unit interval. The sequence can be interpreted as a nested sequence of arbitrarily small closed intervals. If the limit lies in an open set then one of the intervals is a subset of it, which is the point at which you observe that the event has occurred. Non-open sets don't have this property. Closed sets can be observed not to occur but not necessarily observed to occur, etc.)

I would drop #4.

And replace it with:

5. if O(A) = 1 and A, B are disjoint then O(B) = 0?

If A and B are disjoint, then by (1), O(A ∩ B) = O(Ø) = 0. So by (3), either O(A) = 0 or O(B) = 0. Hence, (5) is a consequence of (1) and (3).

Looking at it again, though, I would not only drop #4, I would also relax things a bit so that O need not be defined on all of F.

Let me try to rephrase the point of my earlier post as it relates to your question. You asked if events with probability zero can occur. Well, what exactly occurs? Traditionally, we answer this question by looking to the outcome, which is traditionally defined as an element in the sample space.

My suggestion is that we should not be using this definition. The fundamental objects in a probability model are the events and their probabilities. We ought to be able to define an "outcome" intrinsically in terms of the events. Knowing the outcome should mean that we know exactly which events occurred and which did not. This means an "outcome" should be defined as an ultrafilter on the (sigma-)algebra of events, which is what (1)-(4) give us.

In a finite sample space, the traditional definition of "outcome" and my suggested definition are equivalent, since every ultrafilter on a finite sample space is principal. But on an infinite sample space, the definitions are different, and under my suggested definition, it is perfectly consistent to claim that events of probability zero never occur.

If we do not know the outcome, but have made some (partial) observations, then there will be some events we know occurred, some we know did not occur, and some whose occurrence we cannot determine. In that case, what we are looking at is not an ultrafilter of events, but simply a filter.

Thanks, jason1990. Ultrafilter = total information and filter = partial information about the outcome makes a lot of sense.

I don't think I have any more questions, but I did wonder about defining O on all of F. If O is a function into {0, 1} and if O(A) = 1 means that "A has been observed to occur" then presumably O(A) = 0 means that "A has not been observed to occur" rather than "A has been observed not to occur". But that could be changed by making O a function into {0, u, 1}. Alternatively, you could introduce another function N into {0, 1}, where N(A) = 1 means that "A has been observed not to occur".

In the case of a filter, O is not defined on all of F. Strictly speaking, a filter is a collection of events U. The function O is related to U by O(A) = 1 if A is in U, O(A) = 0 if A^c is in U, and O(A) is undefined otherwise. Of course, if you want to, you can define O(A) = u in the third case, instead of leaving it undefined.

For an ultrafilter, given any A, either A or A^c is in U, so O is defined on all of F.

What do you think of Sklansky's opinion that Mathematicians should rule the world and be the primary decision makers in all fields?

jason 1990

Thanks for doing this thread. I hope my historical, entirely non-technical question isn't out of place here, but what can you tell me about the position of probability theory in the history of philosophical logic? Is it strictly a branch of mathematics, the connection to philosophy already entirely severed by the time of de Fermat, Huygens, and Pascal? Or is there a thread to be traced back to classical philosophy? Understanding that logic was closely akin to the study of rhetoric then, I am interested to know how far backward from the mid-17th century it is possible to go--if at all--in strictly history of philosophical logic terms--assuming you know or care anything about such a thing. Which (if any) philosophers interest you? Thanks again for this cool idea for a thread!

I'm learning stochastic calculus, brownian motion, etc and I have a question:

In stochastic calculus we write e.g. dX = dt + dW, and if we call dW^2 = dt, then we get all the right results, like Ito's lemma etc (and of course this is justified more properly). Is there any sense or way in which this can be extended to have an additional term - dZ in dX, s.t. dZ^4 or dZ^3 ~ dt? And if yes, would that add anything interesting (or is Brownian motion basically it for continuous stochastic processes)?

Thanks

Sklansky's ruling class is too broad. Obviously, probabilists are the only mathematicians that are qualified decision makers.

Well, for example, the Dubins-Schwarz theorem says that any continuous martingale is a time-change of Brownian motion. And you can use Girsanov's theorem to e.g. transform a Brownian motion with drift into a Brownian motion without drift by a change of measure.

Ok, I understand the drift part.

Time-change means reparametrizing time? So Dubin-Schwarz covers continuous processes that are non-homogeneous by reparametrization of time such that in the new time it's homogeneous?

Insofar as I understand the question, I think it is interesting, but I do not have an answer. Based on my naive, layman's knowledge of both history and philosophy, it is my understanding that probability (at least the probability calculus) was non-existent before the 17th century. I also think that people were not much concerned with the philosophical interpretation of the probability calculus until much later. But perhaps a professional philosopher/historian will chime in and answer this question.

This is probably a little bit basic but I'm curious to know the answer.

There were 30 in my class in university and 4 different pairs of people shared birthdays. What is the probability of that happening from a random sample of 30 people?

The heuristic expression, (dB)² = dt, is really just shorthand for the fact that the quadratic variation of Brownian motion is the identity function. In order for a process Z to satisfy (dZ)^p = dt, it would have to have t for its p-variation. There are such processes. For example, fractional Brownian motion with Hurst parameter H = 1/p satisfies (dB^H)^p = C_pdt for some constant C_p.

The problem is that one cannot define a stochastic integral for these processes in the usual way. In fact, it can be proven (see this book, for example) that the classical construction of the stochastic integral works only for semimartingales -- and a semimartingale Y will always satisfy (dY)^p = 0 for p > 2.

In order to have an Ito lemma for processes such as fractional Brownian motion, one needs to develop a new definition of the stochastic integral, and as a consequence, a whole new calculus. There are several of these calculi in the literature. A good starting point is this book.

You are probably already aware of the birthday problem, but I am including a link just in case.

By my calculations, the probability of a class of 30 having 26 or fewer distinct birthdays is 0.316527. I will warn you, though, that I spent very little time checking these calculations.

thx!

That wasn't what he was asking. 22 people have different birthdays. Two more share a birthday that is different from those 22. Two more share a birthday that is different from those 23. Two more share a birthday that is different from those 24. Two more share a birthday that is different from those 25.

What he said.

I'd imagine the probability is considerably lower than 0.3.

In the context of the question, it was a reasonable probability to give.

FMP. I forgot to subtract the probability that all 30 birthdays are distinct.

I understood the original question, but the question I answered is a better question. DUCY?

Well, if you absolutely must have the answer to the very specific question you asked, I can say this. The answer is within 1/500 of the answer to this question: "Hey, I just got dealt a J8! What are the odds of that?"

This is probably a basic question, and not even for a math class, but would like to know the answer to this for a paper I'm writing about presidential elections, and can't seem to figure it out.

Question: Say I'm going to spin a roulette wheel 10 times. What are the odds that the red-black split (e.g. the number of times I get red v. number of times I get black), is 5-5 or 6-4 in either direction? What are the odds that I will get 1 color (either red or black) 7 or more times out of 10?

Thank you mucho

i already posted this in the probability section but got zero replies so far, so i hope u can help.

i had a debate at another forum about how to determine variance for tourneys.

1. u take a sample of tourneys and divide it into subsamples (e.g. sessions). for every subsample u calculate roi and compare that with mean roi (the roi for the whole sample). u take the differences from mean roi, square them and add them up.

2. u take the same sample, determine the distribution of ur finishing positions in that sample and add the squared distances of every single finishing position from mean roi.

well mathematically 2. should be the same as 1. with subsamples consisting of single results instead of greater subsamples, so both seem to fit into the variance definition like it is to be read e.g. in wikipedia, but version 2. says nothing about streak lengths and the presence of swings imo. and therefore should not be used to determine variance on a real-life-base. am i wrong?

what do the math cracks out there think? is there any book which describes how to size subsamples to get a decent result? did anyone do this calculation before e.g. comparing turbo and non-turbo tourneys?

thx in advance, franxic

At the risk of getting this wrong and looking like an idiot I'll give this one a go.

It appears to be a binomial distribution. So probability of 5/5 is 0.246

6/4 in one direction is 0.205 so in either direction is 0.41

Red or Black 7 or greater is 1-(0.246+.205+.205) = 0.344

update to the variance question:

Sherman and me have a little debate going on about that topic in the probability-forums. would be great if u could read this since my points and questions get clearer there.

it isn´t thaaaaat long..

thx again, franxic