# Likelihood and proportion

I have this thing I need to solve and can't seem to figure out how to do it exactly. I would appreciate some hints as to what exactly it is that I need to do in order to find out a), b) and c).

$$p(k) = 2^{a-1} (a-1) k^{-a}$$

a) Suppose $k_1,\ldots,k_n$ are $n$ samples drawn independently from $p(k)$. Derive an equation for the maximum likelihood estimate $a$. For this one, I've just written that:

The likelihood is the product of $p(k_i)$, with $i$ from $1$ to $n$. Is that enough?

b) Calculate the value of $a$ for the degree data. (I was given a file with a few hundred numbers that represent "degrees" of some nodes).

c) Plot $p(k)$ for $k$ in $\{1,2,3,\ldots, 20\}$ with MLE $=a$, along with the proportion of times each value of $k$ appeared in the degree data. Is it possible to do this one without looking at the values in the file? It would be pretty silly if they required us to look into hundreds of values...

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a) well you can write it out if you want. I would. b) Take the log of the answer in part (a) and find the $a$ that maximizes that (ie, take derivative, set to 0, solve for $a$). c) I think you just have to write a little program to find the frequency of k in your file. –  Apprentice Queue Jan 7 '12 at 15:51
But how can I get a finite answer for a) in order to take the log of it? Can you please explain? –  Sorin Cioban Jan 7 '12 at 16:07
By doing exactly what Apprentice wrote - log the equation and do a standard maximization? –  gnometorule Jan 7 '12 at 16:43
Oh, I think I've got it now :) Thanks. –  Sorin Cioban Jan 7 '12 at 17:20
I don't understand how to take logs and differentiate the product of p(ki), with i from 1 to n. Can anyone help? –  jamess Jan 10 '12 at 12:49