Ok. If you're asking if it would be useful for your job, probably not. You might be better served to take a financial mathematics class that prepares your for the second exam. That material deals with interest theory and would probably be more applicable to your future work. There is a decent amount of calculus involved in the course for continuous annuities and such, but nothing overly complicated (mostly it's just taking intergrals of exponential functions).

As far as finding a job, I don't know what accounting firms are looking for, but I have to imagine it'd help. I think the real value from getting the minor would come from possibly becoming an actuary if you liked the material. As you probably know the money is far better, and there are plenty of actuaries that did not major in actuarial science.

One last thing. It looks like the entire course work for the minor is geared toward passing the first exam, with the exception of the random processes class if it is what I think it is. That prep class will most likely involve just practicing SOA exam specific questions.

Thanks for the advice. I mostly want to do this minor because I feel like I have a math aptitude that accounting doesn't really utilize. I've always been interested in the actuarial science field and of course the money that goes along with it.

If I go through these classes and find out that this is something I'm good at and enjoy what would be my best course of action? I'll most likely be able to get a solid accounting job when I graduate, so would it be more beneficial to start working in a field that doesn't necessarily involve actuarial work or to immediately pursue grad school for an actuarial degree?

If, by "combinatorials," you mean the binomial coefficients, they appear in the binomial distribution.

It depends on the context and on the available information.

I would recommend it because it gives you flexibility. At this stage in your scholastic career, I think you should study what you enjoy and you should strive to build a broad educational base which offers choices. I would even recommend something more general, like a math minor, if you think you would enjoy it and could handle it.

I'll PM you when I get a chance so as not to derail this thread.

I apologize if this has been asked before, but:

Being a probabilist, were you able to do real-time analysis while playing poker? Did you know anyone who did? Supposedly Gus Hansen actually solves fairly elaborate equations in his head while playing.

The mathematics one needs at the table is fairly trivial, so my expertise did not grant me any special advantage in that regard. You might want to read my earlier post in this thread, which is related.

Jason,

Could you try explaining in words why the minimum of exponentials is independent of their ordering? I sort of understand the proof in my text book but I'm having trouble making intuitive sense out of it.

Great thread!

Jason, you seem to have a good understanding of physics too. Do you think that a probabilist could contribute to physics or even convert to being a physicist?

I'm studying financial mathematics and have therefore had my fair share of probability theory. I'm contemplating switching to pure probability theory for my masters in the hope that I could still work with financial mathematics if I choose or maybe explore physics (I have no idea what so ever what I wanna do later in life but I have a lot of flexibility due to poker). Do you think this is possible/a good idea?

Yes and yes. There are many models studied by probabilists which have deep implications for statistical physics, and many probabilists whose work is rooted almost entirely in physics (for example, this is the research page of one of the professors in the probability group at my university).

Perhaps I do not understand your question, since what you seem to be describing has nothing to do with exponentials. The minimum of any iid collection of random variables is independent of their ordering, provided their distribution has no atoms (which is the case if they have a density, for example).

More precisely, let X₁,...,X_n be iid with common cumulative distribution function F(x). Let M = min(X₁,...,X_n) and let σ be the random permutation describing the ordering of the X's. (That is, σ is a random variable taking values in the symmetric group S_n, satisfying X_σ(1) < ... < X_σ(n).) If F is continuous, then M and σ are independent.

In words, I think it is pretty easy to be convinced that this should be true. Just consider two random variables X and Y. Initially, by symmetry, we have

P(X < Y) = P(Y < X) = 1/2.

Now suppose we discover that min(X,Y) = 7, for example. Then again by symmetry, we know that the two possibilities

(1) X < Y and X = 7,
(2) Y < X and Y = 7,

are equally likely. In other words,

P(X < Y | min(X,Y) = 7) = P(Y < X | min(X,Y) = 7) = 1/2.

So learning the value of the minimum tells us nothing about the ordering of the variables.

Sorry I should have been more clear. I was talking about exponential RVs with different rates.

To clarify, here's a specific problem I saw:

X and Y are exponentially distributed with respective rates mu and lambda.

Find E(X|X<Y)

The answer to this just turns out to be E(min(X,Y)).

Great thread btw. Thanks for taking the time to answer all of these questions.

Absolutely. And conversely, among non-probabilists, physicists seem to have the best understanding of probability, in general. I often cringe (facepalm?) when I read statistical analyses written by non-probabilists. But physicists tend to make me cringe the least and in fact often have keen insights that I respect.

Of course you could.

I think it is a great idea. It will provide you with a solid foundation and the flexibility to go in different directions in the future.

A nonnegative random variable X with a density f(t) comes with something called a hazard rate. If F(t) is the distribution function, then the hazard rate is h(t) = f(t)/R(t), where R(t) = 1 - F(t). The interpretation of the hazard rate is the following. Suppose X represents the lifetime of some object. If the object is alive at time t, then the probability that it dies by time t + ε is approximately εh(t). In other words,

P(X ≤ t + ε | X > t) ≈ εh(t).

In fact, to be precise,

P(X ≤ t + ε | X > t) = (∫_t^{t + ε} h(s)R(s) ds)/R(t).

The exponential is the unique distribution that has a constant hazard rate. This is known as the "memoryless property". The hazard rate for an exponential is just the parameter of the exponential.

Suppose X and Y are independent nonnegative random variables, which do not necessarily have the same distribution. The conditional odds ratio for {X < Y}, given min(X,Y) = t, is

P(X = t, t < Y)/P(Y = t, t < X) = (f_X(t)R_Y(t))/(f_Y(t)R_X(t))
= (h_X(t)R_X(t)R_Y(t))/(h_Y(t)R_X(t)R_Y(t))
= h_X(t)/h_Y(t).

For exponentials, this conditional odds ratio does not depend on t. So learning the actual value of the minimum tells us nothing about whether X < Y or Y < X. In other words, it is the memoryless property that is responsible for making the minimum independent of the ordering.

Do you agree with the spirit of this argument?

http://diracseashore.wordpress.com/2...tical-physics/

Ok awesome that makes a lot more sense. I use hazard rates all the time (we call it force of mortality since I'm studying life contingencies but same thing).

Now correct me if I'm wrong, but this means that this property applies to any two distributions whose hazard rates are in constant proportion to each other right?

I do not have the background to understand what this article is about. All I see is someone claiming to apply conditional probability to his personal beliefs. In spirit, I agree with doing this, because I think one's personal beliefs, insofar as they are quantified, should conform to the laws of probability in order to be logically consistent. However, there appears to be a lack of probabilistic content in this article. I see him say that his personal probability used to be X. Then something happened, and now it is Y. Okay, but so what? Will he now use the laws of probability to deduce something about his personal conditional probabilities that led from X to Y? Or did he already have personal conditional probabilities, and he used them to derive Y? It is as though he said, "I used to believe X. Then I applied a logical argument which is difficult for me to describe to you. And now I believe Y."

I also noticed he wrote, "Here I am taking the point of view of Jaynes’ book on Probability theory." I do not think this is accurate. Jaynes' book is not about personal probabilities. Probabilities, according to Jaynes, are meant to be derived objectively from a given set of desiderata. They are not subjective and do not vary from person to person. Berenstein seems to be misunderstanding Jaynes.

Correct.

Hi Jason,

I am interested in a probability problem that I am calling the "conditional deck" problem. Surprisingly, I haven't really ever seen it discussed elsewhere. We are in a 9-handed NLTH poker game. The first 7 players fold. Conditioned on the fact that I hold hand X in the SB AND that these 7 players folded hands within certain assumed folding ranges, what is the conditional probability that the BB holds hand Y? Based on the Monte Carlo simulations that I have run, the BB's range of hands is materially stronger than what the marginal probabilities suggest.

How would you advise that I tackle this problem? Is it even feasible to calculate the precise conditional probabilities (based on my folding range assumptions), or will the combinatorial explosion render this impractical? If precise calculation is feasible, could you roughly outline an approach to solving it? If exact calculation is infeasible, how would you go about determining the best approximation of the probability?

Separately, could you recommend an introductory text on combinatorics for someone with a BS degree that's not in math?

Thanks!

Hi Jason,

I was just wondering if you could give a few examples of the kind of things that professional probabalists do for their day to day work.

My experience with probability has been limited to poker, and to logic puzzles like the Monty Hall and Birthday problems. I have no real concept, outside of these types of things, of what kind of things you work on.

Thanks for your time

I would not do any exact calculations. I would instead design a piece of software that allows the user to input a hand range for each of the 7 players that folded, as well as the hand you hold, and then returns numbers based on simulation.

I do not know any combinatorics textbooks. For that, you should ask a combinatorist.

See here.

I don't know if it was what he was asking, but now I'm curious: What does your typical day look like? Do you meet in an/your office and scribble away or do you meet with colleagues? Or do you work in groups?

I probably should have explained a little more, the main concept was about something going from likely true, to almost certainly true and the rate at which this happens. I can see how this is a personal view but I am ignorant of Jayne's book so I'm not sure how this can be objectively quantified, maybe I'll add it to my reading list. Overall, I agree with Lubos Motl in the comments when he says that it is almost impossible in most cases to know for sure that the new evidence is independent from the old. His example of testing Newton's gravity law for 500 and then 500000 meteors is a good example were your confidence in the law doesn't change at all. Also, it seems David agrees with this so maybe he just didn't explain his view well. Thanks for an interesting thread.