How to Find Maximum Likelihood Estimators for Sample Data?

Scootertaj
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problem.jpg

Homework Equations



L(x,p) = \prod_{i=1}^npdf
l= \sum_{i=1}^nlog(pdf)
Then solve \frac{dl}{dp}=0 for p (parameter we are seeking to estimate)

The Attempt at a Solution



I know how to do this when we are given a pdf, but I'm confused how to do this when we have a sample.
 
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Scootertaj said:
problem.jpg



Homework Equations



L(x,p) = \prod_{i=1}^npdf
l= \sum_{i=1}^nlog(pdf)
Then solve \frac{dl}{dp}=0 for p (parameter we are seeking to estimate)

The Attempt at a Solution



I know how to do this when we are given a pdf, but I'm confused how to do this when we have a sample.

Are there no similar examples solved in your textbook or course notes? Could you find nothing at all on-line?

RGV
 
Not so far, though I'll be talking to a graduate TA about it.
 
There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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