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.

/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.

Thanks, I had figured that out.

Now I'm having trouble finding the MLE, because i'm having trouble breaking up the sum for the total score function.

I know that each x_i ~ p*(x_i,r) = x_i / ( r sqrt(r^2-x^2)

s(x_i,r) = -1/r - r/(r^2-x_i)^2

I need to find s(x,r) = \sum s(x_i,r)

When I do that, I can't seem to separate it into something solveable. Does anyone have any tips on manipulating the sum to give the score function in terms that will put the sum only on the x_i's

Thanks,

Stephen


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/m29 sorry that should read:

s(x_i,r) = -1/r - r/(r^2-x_i^2)

For problem 4, which variable is uniformly distributed? Is the distance of a cross-section or the radius of a sphere itself? How can we decide this? In both cases, we get similar pdf but not the same.

/m31 answer: Your question is perhaps the part of the problem that is conceptually the least clear. I think the answer is that the point at which each organelle is aliced is uniformly distributed, hence the observed radius of the disk obtained from the organelle is not uniformly distributed.

/m29 answer: Remember, you can maximize f(x) by setting its derivative to zero only when the optimal x is in the interior of the allowed range for x.

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