Well, things are not always as they seem. Upon further consideration, I cannot see any inconsistencies between option (a) and the phenomenon I described. I do not know if (a) is realistic for your situation, but it does appear to be viable.

Incidentally, your ratings are somewhat reminiscent of Elo ratings. Perhaps you might want to look into the theory behind the various systems used to update/estimate Elo ratings, and see if they can be adapted to fit your situation.

Thanks jason and egj

@egj: I don't have historical sales numbers, so I can't do a regression to total sales, just to a binary variable of 'top seller'

@jason: I am familiar with Elo and several other rating systems. My major issue here is how to handle missing information. Thanks for the help

This is just a quick and slightly dumb notational question. I'm goofing around with some calculations and it's getting a little messy, so I'm looking for a way to condense my notation a little bit.

Suppose that f(x) is a probability distribution function and F(x) the cumulative distribution function of f(x). The probability of x being between a and b is expressed as \int_a^b f(x) dx. Is it notationally acceptable to write \int_a^b dF(x) instead?

I recall something like this from measure theory, but since I didn't like real analysis, I've wiped much of that stuff from memory. And just because it's done this way by the analysts does not mean that probabilists would necessarily have adopted the same notation.

Thanks.

Hopefully jason won't mind if I try to save us a little time...

The "dF(x) integrals" are called Stieltjes integrals and yes they are used in the probability and statistics world sometimes -- more commonly in books from the 30s-to-50s that I've seen in libraries than in recent ones. I don't really know if that means they are shunned for formal reasons these days, or if they're simply avoided so that we don't have to explain the notation to a multitude of students who have never seen it.

The purpose in using them is that they allow you to write one formula that covers all distributions, rather than writing an integral-based formulas for continuous distributions, sum-based formulas for discrete distributions, and tearing your hair out if you run into something that's half-and-half. (If F(x) is a step function, then an integral of g(x) dF(x) becomes a weighted sum of g(x) evaluated at the sites of the steps, with the size of the steps as the weights.)

Thanks.

I am doing something where I'm allowing myself to have a bunch of different distributions f(x), which lead to a bunch of different F(x). I am somewhat hoping to keep this general enough so that I can deal with transitioning from a continuous probability distribution function to one that takes on a finite set of discrete values. (I do have Riemann-Stieltjes integrals somewhere in my mind, going back to my undergrad physics days with the formal manipulations of the delta function in various classes.)

(Although, the real reason is that I think I'm making some mistakes by not putting in the weighting factor where I think I should, and I'm looking for ways to keep myself from making mistakes!)

Essentially, it's a game where you get a random number, and then based on that number you make a decision regarding whether you want another random number. The strategy depends on the number you're looking at and the strategy that your opponent is using with his random numbers. This means you've got to compute the probability of winning with two different strategies (take the second number, don't take the second number) and compare them to identify the stronger strategy.

This is leading to some double/triple/quadruple integrals as I'm balancing this outcome against the other. As long as there's nothing wrong with thinking in terms of dF(x) instead of having to always throw in the weighting factor, I think things should work out a little more cleanly.

Here is an example of what is happening early on in the calculation: The probability of the first number being between a and b and the second number between c and d is



(Note: c and d may depend on x_1, which is why it's coming out this way.)

I'd much rather just write



And have this be meaningful and understandable (and of course, mean what I want it to mean). I think it would clean up the presentation quite a bit. But if nobody uses this notation (or it's otherwise wrong), then I'll just have to stick with the other way.

I see nothing wrong with the notation; in measure theory people have "d-mu" all over the place. You just may have to explain to people who haven't seen it before what you're doing, depending who your audience is.

Yes, this is standard notation for the Lebesgue-Stieltjes integral. (Wiki link omitted, since the article is pretty horrible.)

There is a one-to-one correspondence between right-continuous, nondecreasing functions F with F(-∞) = 0 and F(∞) = 1, and Borel probability measures μ on the real line. The correspondence is via the identity μ((a,b]) = F(b) - F(a). The following notations are all equivalent:

If μ has atoms (i.e. if F has jumps), then it is probably best to use one the last three notations, to remind you that a is not included in the interval of integration, but b is.

Your point is well-taken, since probabilists' notation differs in fundamental ways from notation in the rest of analysis. But for the record, probability is a branch of analysis, so probabilists are analysts.

Response to original post. What is the probability that I will post this specific comment?

This question is ill-posed in multiple ways.

(1) The sentence, "I will post this specific comment", is not a proper proposition (or event) because of the presence of indexicals. The sentence must be rephrased to remove "I" and "this". The tense of the sentence also suggests that there is another implied indexical, "now": "I will post this specific comment [later than now]."

For example, if "I" denotes the BrianTheMick, "this specific comment" denotes Post #434 in the thread, and "now" denotes the time of posting, then the sentence becomes, "BrianTheMick will post Post #434 in the 'Ask a probabilist' thread later than 06-17-2010, 02:25 AM GMT." This sentence is trivially false, so the probability is zero.

(2) All probability is conditional, so the question should be, "What is the probability that [...] will post [...], given [...]?" The third ellipsis should be filled in with some specific set of information. If that information includes all information that I presently have, then depending how (1) is resolved, the probability may very well be one.

(3) Once the above two problems are corrected, the question will at least be meaningful. But the problem of determining the probability may still be ill-posed. If the information specified in (2) does not contain enough symmetry relative to the final formulation of the proposition, then it will not be possible to determine a probability.

By the way, for what it's worth, my wife thinks I am wasting my time responding to this particular question.

Dominant analysis itt

How about, "What are the odds that BrianTheMick will make the specific post he did in #434 of the "Ask a probabilist" thread given the exact conditions (all of them) that took place at 06-16-2010 10:25 PM?"?

It was a silly question. Probability calculations are fun.

I have a question regarding moments.
When you have a normal distribution, you know all the moments of the distribution. But what about the inverse problem. If you know several moments can you say it is a normal distribution?

The reason I ask is the following. The Erdos-Kac theorem is a beautiful theorem in probabilistic number theory. It says that asymptotically the number of primes up to x, behaves like a normal distribution with mean loglog x and variance loglog x. The first proof, used sieve theory and the central limit theorem. However, a recent proof was done by estimating the moments. Granville and Soundararajan in a recent paper (last 10 years) proved that all the moments of w(n) (the number of prime divisors of n) asymptotically behave as if it were a normal distribution. It seemed to me that there was a gap in the proof as that doesn't seem to imply that it would be a normal distribution.

Sorry if I wrote in a confusing way.

Yes, this is true for Gaussians (if you know all moments) but not all distributions (e.g. not log-normal RVs). One needs to ensure that the moments do not grow too fast.

So you are interested in P(A | J), where

A = "BrianTheMick posts 'Response to original post. What is the probability that I will post this specific comment?' as Post #434 in the 'Ask a probabilist' thread at 06-17-2010, 02:25 AM GMT."

and

J = "the exact conditions (all of them) that took place at 06-16-2010 10:25 PM"

The string A appears to be a well-defined proposition, so we are fine there. The string J, however, is not even a sentence. We need a well-defined proposition for J.

As blah_blah wrote, if you know all the moments (several are not enough), and they are the moments of a normal distribution, then the distribution is normal.

For more information, see here and here.

"given" doesn't count as part of the phrase? Or is it that my phrase wasn't clear enough?

This is a "run it twice" question, but with no variables. Feel free to rewrite my proposition for education's sake.

Jason, I sincerely apologize if this has been discussed already but I tried to read through the thread and my face almost blew up because I am not at all a person that understand complicated maths. I do plan to try to study the thread a little more this weekend but for now I'm hoping you can just answer this question:

Is there a reason why all probability experts are fantastic LHE players? It seems like the guys who wrote Mathematics of Poker are doing well for themselves and that's probably the most solvable game right now (HU anyway) even though there is still a long way to go.

I've always wondered why you guys don't just quit your jobs and get staked by some super rich dude and take all of the money out of the HU poker world.

Also, before I try to make my way through the thread, any books you'd recommend for someone interested in probability maths but lacks the basic knowledge needed to teach themselves?

Thanks. Back to page 1.

Based on playing poker with many math/probability Ph.Ds, I can confidently say that many of them are huge fish.

Not all people are motivated by money. And if they were, they would almost certainly go into finance instead of poker, which has far higher upside.

In terms of taking all of the money out of the HU poker world, someone stealing the Alberta LHE poker bot is much more dangerous than a couple of probabilists studying LHE for awhile (our advanced training doesn't give us much additional insight into basic algebra, unfortunately).

What careers are there for mathematicians? average income? not sure if this is an appropriate question but i am really interested

a lot of jobs allow engineering/math/physics degrees to apply

median for all engineers is 77k w/ starting ~50k (although in the recent economic decline this is not true) but after 15 years ~150k in a large city and ~100k in a rural area is not unheard of

so an engineering job with a mathematics degree? is that what youre saying?

Your premises here are all based on a combination of confirmation bias and narrative fallacy. Being a math whiz is not in practice particularly beneficial for poker.

I pick 1000 math whizzes and 1000 random people, give them two years to become good enough to beat the Coomerce 15-30 under penalty of death if they fail.

How many in each group do you think will succeed?

What about if the second group was comprised of random 600 math SAT scorers.

Note my qualifier "in practice". But I will concede I phrased my point badly. I should have said something like being a math whiz is not "particularly predictive" of poker skill rather than that it is not "beneficial" for poker skill.


And for the record I greatly admire math whizzes. But it's an empirical fact in my experience that math whizzery and poker skill are not that strongly correlated.

The funny thing is you yourself are actually a very good illustration of my point.