/m27
There is a trick.

You can use the CDF of the uniform (P(z<Z)) to correspond to some CDF of the radii (P(r>R)). Once you have that, the PDF is the derivative of the CDF.

For problem 4,

I've figured out what the distribution looks like, and I believe I can define it mathematically, but am having trouble doing so. Basically what I would like to do is transform a uniform distribution to the desired one. Here is my logic.

If we look at a cross section of an organelle of radius r, the cross section can be thought of as cutting the organelle at a point x that is generated uniformly from -r..r. This corresponds to the front and back of the sphere relative to the cutting plane.

If we know where the sphere is cut, we can compute the observed radius, Z = sqrt(r^2 - x^2). My questions is how can I use the function for Z and the fact that x is uniform to find the pdf (likelihood function) for Z.

I know this is possible but am having difficulty actually doing it. Thanks.

/m25 answer: Yes, you may assume that the number of cross-sections n is fixed.

/m21 /m24 Thanks. I understand what you mean. One more question. Can we assume that the number of cross-section is always n? If the organelles are distributed randomly, the number of cross-section will actually vary. So, the distribution should involve the probability distribution of n also. But since there is no description about the density of the organelles, I cannot compute the exact distribution. (Also, the problem becomes much more complicated).

/m21 answer: Yes.

/m18 answer: Natural parameters are explained here: http://www-cse.ucsd.edu/users/elkan/291/lect08.pdf

The issue is, when you look for an open rectangle, do you have to look in the space of natural parameters, or can you look in the space of original parameters?

/m19 answer: One lesson to learn from this problem is that widely used terms may not have a universally shared definition. You should make the definition you use clear.

The wikipedia is an excellent reference. Here is a paper that explicitly discusses the confusion:
http://www.hpl.hp.com/research/idl/papers/ranking/ranking.html

For the contemporary importance of power laws see
http://www.sciencemag.org/cgi/content/full/294/5548/1849

\m20 Just as a random variable has a distribution, so does some function of that variable. Since an estimator is just a function of the data, it will some distribution as well. In your example, if x1,...,xn are iid gaussian(\mu,\sigma^2), then avg(x1,...,xn) will follow a gaussian distribution with mean \mu and variance \frac{\sigma^2}{n}

Edited by author 02-11-2005 09:44 PM
In problem 4, I don't get the last sentence. What did you mean by "distribution of estimate"? For example, if avg(x1,..,xn) is the MLE of \mu in Gaussian, what is the distribution of the estimate?

/m17 Thanks for the information. I found dirichlet there, but Zipf distribution is not stated clearly, and there is no power law distribtution. Where can I get these info?

In part 3(c), "Explain carefully whether or not the theorem relies on the exponential family being described using its natural parameters." I don't get that part. What does that mean, and what I am supposed to explain?

from wikipedia.com
Banu

Where can I get the formal definition of Dirichlet distributions, power law distirbutions, and Zipf distributions?

/m10, /m11 answer: Yes, the binomial assumes N is large. You may assume this; I should have mentioned it in the problem statement. No need to do the more difficult hypergeometric calculations.

/m10 answer: I think the reasoning with the binomial is correct, and the hypergeometric is not needed for this problem.

Be careful with this claim: "if the actual N is less than the claimed N, we would expect to see more tagged animals." I'm not saying it's false (or true) just that it requires careful thought to be sure.

/m7 answer: I discussed this with Ryan, and I think he is correct.

This does not mean that the Gaussian-based answer is incorrect, just that it may be unreliable in the real world. It points out the need to investigate which distribution(s) model real-world returns well.