It would help greatly if we had the context for this problem. It looks to me like some sort of least-squares fit (maybe log-likelihood?). Please define every single variable I see. That is, what are each of the following:
$\displaystyle \alpha$
$\displaystyle \beta$
$\displaystyle n$
$\displaystyle LL$
$\displaystyle \mu_{i}?$
Do any of these variables depend on any of the others, aside from $\displaystyle LL$ depending on $\displaystyle \alpha$ and $\displaystyle \beta?$
Thanks for asking!
beta - is a modeling parameter
alfa - is chosen as the maximum likelihood estimate
LL - is the first equation (in square shape)
n - is the index of the numbers in the samle (from i - n)
mu(i) is the value of the element ( for example i = 3; mu(i)=254.2...etc)
This is a sample with some positive numbers. I'm asked to do this: Suppose you are free to choose both parameters alfa as well as beta . Can you derive
the MLE (maximum likelihood estimates) for both parameters simultaneously? I've got the derive fore alfa, but I'm stuck for beta. Is it more clear now? It's a long problem, this was the reason didn't want to post much info, just the equations. Thanks a lot!
Come to think of it, I'm not sure I buy the result you've got there. I'd have thought it would be this:
$\displaystyle \frac{\partial LL}{\partial\beta}=\frac{n}{\beta}+\sum_{i=1}^{n} \ln (\mu_{i})-\alpha\sum_{i=1}^{n}\ln (\mu_{i})\mu_{i}^{\beta}.$
So, more generally, you're asking if
$\displaystyle \sum_{i=1}^{n}a_{i}b_{i}=\left(\sum_{i=1}^{n}a_{i} \right)\left(\sum_{i=1}^{n}b_{i}\right).$
I do not think it very difficult to convince yourself that this is false in general. Let $\displaystyle a_{i}=2,$ a constant, and let $\displaystyle b_{i}=3.$ Then the equation would have us believe that
$\displaystyle \sum_{i=1}^{n}2\cdot 3=\left(\sum_{i=1}^{n}2\right)\left(\sum_{i=1}^{n} 3\right),$ or
$\displaystyle 6n=\left(2n\right)\left(3n\right)=6n^{2},$
which is certainly not true if $\displaystyle n=2,$ say. The result is only true if $\displaystyle n=1,$ in which case your summation really isn't doing anything, as there is only one term!
Does that clear things up?
I tried it with doing some test after I asked, and I concluded that it was a stupid question. I wanted to simplify some things through my equations, but when you are not able to do so, it's better not to invent formulas :PP
Thanks a lot!