The sequence X_1, ..., X_n, ... is an iid sequence. So each X_j has the same distribution.

Not really, because r depends on the (common) distribution of the X_j's. For example, if they were normal, then r = 3s^4. If they are not normal, then r could be anything. Also, you do not know s. You only know S. I am assuming the whole point of looking at the variance of the sample variance is to get a confidence interval for s. In other words, you want to say something like

P(s in [..., ...]) = 0.95,

where the interval [..., ...] contains expressions involving S. That way, you take what you calculate (S), build an interval, and then claim that the thing you cannot see (s) is likely to be in that interval. If you are going to follow this procedure, then you have to keep s and S separate. You cannot carelessly swap one for the other.

Here is an example. Suppose the X_j's are all normal. Then r = 3s^4. If n is large, then

(r - s^4(n - 3)/(n - 1))/n ~= 2s^4/n.

Hence, S^2 is approximately normal with mean s^2 and variance 2s^4/n. Constructing a 2-standard deviation interval gives

s^2 - 2\sqrt{2/n}s^2 < S^2 < s^2 + 2\sqrt{2/n}s^2,
(1 - 2\sqrt{2/n})s^2 < S^2 < (1 + 2\sqrt{2/n})s^2.

We can rewrite this as

S^2/(1 + 2\sqrt{2/n}) < s^2 < S^2/(1 - 2\sqrt{2/n}).

This gives a 95% confidence interval for s^2 that looks like:

[S^2/(1 + 2\sqrt{2/n}), S^2/(1 - 2\sqrt{2/n})].

ah my bad. I was looking through wolframs sample variance distribution and for some reason thought that m_2 and μ_2 were the same thing.... exactly what you told me not to do already

so you would recommend the bounding of r method, if the sample distribution is far from normal?

If you think you can give a reasonable bound on r or r/s^4, then yes. Note that r/s^4 is the 4th standardized moment, which is related to the kurtosis. There are standard estimators you can use for kurtosis. That is, there are formulas you can apply to your data to get an estimate of r or r/s^4. But then you run into the question, "how accurate is your estimate of the kurtosis?" At some point you have to make some assumptions.

Hi Jason, it's so great you offered to answer some questions. I'm currently being forced to take a Statistics course in order to complete my Sociology major. I'm having some trouble understanding confidence levels and significance.

I figured a CI: (12.996, 13.304) of a sample of alcohol levels in wine and now am being asked "Based on your confidence interval, is the mean alcohol content significantly different at the α = 0.05 level from 12%? From 13%?"

I know it's something like if .05 is lower or higher I think? Actually I'm pretty clueless....Would you be so kind as to enlighten me on this relationship?

I suspect you are making this problem way more complicated than it actually is. The number 13 is inside the confidence interval; the number 12 is not. (At the α = 0.05 level, you are 95% confident the mean alcohol content is inside your confidence interval.)

Gee, it is pretty simple when you think of it like that! I sure wish my book would explain it like that instead of making everything so complicated. Thank you Jason!

thanks again, extremely helpful. It just now hit me how to get a very good estimate for r given my type of data. All set!

I was playing on Pokerstars the other night, and my pocket aces got cracked by 7 2 offsuit. Does this prove online poker is rigged?


Sorry, couldn't resist.

Sorry to clutter your thread, but how tough is it to get into one of these programs?

This clearly signals the beginning of the end of this thread.

It's no clutter, of course, but unfortunately I do not have a great answer for you. It's very tough, as you can imagine. Remember, though, as a grad student, you are not applying to the probability group, but to the department generally. Some of the above are math departments, some stat, some operations research. This is one of the odd things about probability. You have to check each school to see what department the probabilists are in. They are not always in the math department.

Here's an undergrad math problem that I'm struggling with:

evaluate the double integral

1 +2x +42y dA

Over the region R bounded by the curves y^2=x and y^2 =4 -x

I'm not really sure what to put on the limits of integration. I think x varies between 0 and 4 and y varies between 0 and 2. I know what limits I put on depends on which I choose to integrate first, its just the y^2 curves are giving me problems and it all gets a bit messy.

Thanks if anyone can help.

You probably will not be able to do this problem without actually drawing the region R. It is an excellent habit in problems like this to graph the curves and take a look at the region. Once you do, you will see that, yes, x ranges from 0 to 4, but y ranges from -\sqrt{2} to \sqrt{2}. Also, once you look at the picture, it should be clear that you need to integrate dx first (on the inside, that is). If you try to integrate in the order dy dx, you will end up having to break the integral into two separate integrals.

So graph these curves (they are sideways parabolas), identify the region R that is between them, and try using the hints I gave to set up the integral. After you give that a shot, if you still need more help, let me know.

Where does the 1990 come from in your screen name?

Nice one Jason. I've drawn the area. Integrating wrt x first I get:

20 + 168y dy

I had messed up the range of y, forgot it could be below zero, isn't it -2<y<2 ?

Integrating wrt y over this range I get the area as 80, is that right?

Thanks.

His birth year. He's the world's youngest professor.

Well, I had to have something to distinguish me from the other jasons on the internet. And 1990 was a very memorable year. Microsoft released Windows 3.0. Tim Berners-Lee created the World Wide Web. The Communist Party in the Soviet Union surrendered power. And Iraq invaded Kuwait, prompting President Bush to address the nation on September 11th in his speech, "Toward a New World Order."

It was either that or jason1783.

Are there any simple to state problems in probability theory that if solved could win a Field's medal or something similar?

The problem is substantially more difficult than this. The correct setup is

\int_{-\sqrt{2}}^{\sqrt{2}} \int_{y^2}^{4-y^2} (1 + 2x + 42y) dx dy.

It is sometimes hard to tell from a post on a message board, but I suspect you are having significant conceptual problems with this material. My best advice to you is to meet your TA or professor, and have a face-to-face discussion about this problem and others like it. That will be many times more effective than any help you can get online. The concepts and techniques of setting up iterated integral are pretty tricky. Depending on what classes you take in the future, this can really come back to haunt you if you don't get it nailed down.

I couldn't come up with anything flashy off the top of my head. But in my Googling, I found this excellent web page. It's not really an answer to your question, but it's a great advertisement for probability in general, and it does contain some open problems.

Well, folks, it's back to work for this probabilist. I will try to keep an eye on this thread in the future, but if there are big gaps between my posts, or if your question falls through the cracks, then I apologize beforehand.

this is probably the stupidest question you will be asked, but what is the notation you are using traditionally for? is it whatever language matlab uses?

It is LaTeX.

So I take it that you're not too enamored with the idea that every time I flip a coin reality splits into two equally real realities?

Hi Jason:

I thought you might be interested in knowing that I too once worked as a mathematician who specialized in probability theory (although I now have forgotten most everything). Unlike you I was in aerospce working with reliability engineering (which is sort of a blend of engineering and probability theory including a fair amount of mathematical modeling) for the Northrop Corporation. I left my job in 1987 and moved to Nevada.

As a side note, upon leaving I was quite knowledgeable about blowing up the Soviet Union. Of course that has no value now.

best wishes,
Mason

Thanks for doing this. (May be should be in the Probability forum.)
Here's a problem I've seen:

There are two distinct secret numbers: call them X and Y, and assume that X < Y, without loss of generality. You have no clue how I came up with them. They could be anything (positive, negative, rational, irrational, etc). They could come from any probability distribution (discrete or continuous). You have no idea. I flip a fair coin. If the coin shows Heads, I reveal to you the larger number, Y; if it shows Tails, I reveal to you the smaller number, X. You do not get to see the result of the coin flip. Your goal is to guess whether the coin was Heads or Tails, based only on your seeing the one number that I revealed to you. Obviously, if you just decide ``Heads'' is your guess, without taking into account the revealed number at all, then you are correct with probability 0.5. But the goal is to be able to be correct with probability greater than 0.5. Can someone devise a method to do this? Explain.