MHB How Do You Calculate Estimators and Analyze Complication Rates in Statistics?

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To calculate estimators for the continuous random variable X with the given density function, one must derive both the method of moments estimator and the maximum likelihood estimator for λ, starting with finding the mean. In analyzing complication rates between two surgical procedures, the data shows 35 complications from 150 patients in procedure A and 34 complications from 138 patients in procedure B. A statistical test is needed to determine if there is a significant difference in complication rates at the 1% level, along with calculating the P-value. Providing any progress made on these calculations can help others assist more effectively. Engaging with the community can lead to better guidance on these statistical challenges.
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Really need help with these two questions. I have been working on them for some time and just can't seem to make any significant progress1) X is a continuous random variable with density function

f(x) = (θ + 1)x^θ , 0 < x < 1 and 0 elsewhere

Derive both the method of moments estimator and the maximum likelihood estimator for λ.

*For the MOM estimator you will need to find the mean*2) Two surgical procedures are compares and what is of interest are the complication rates. 150 patients had procedure A and there were 35 complications while procedure B tested 138 patients and there were 34 complications. Does this indicate a difference at a 1% level. What is the P-value?
 
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Hello and welcome to MHB, NYCStats11! (Wave)

Even though you say you have not made significant progress, can you post what progress you have made? This will give our helpers some idea where you are stuck and how to best help you, even if all you can state is what theorems you think may apply.
 

Thread 'Formal derivation of statement from Peano Arithmetic system'

I was reading a Bachelor thesis on Peano Arithmetic (PA). PA has the following axioms (not including the induction schema): $$\begin{align} & (A1) ~~~~ \forall x \neg (x + 1 = 0) \nonumber \\ & (A2) ~~~~ \forall xy (x + 1 =y + 1 \to x = y) \nonumber \\ & (A3) ~~~~ \forall x (x + 0 = x) \nonumber \\ & (A4) ~~~~ \forall xy (x + (y +1) = (x + y ) + 1) \nonumber \\ & (A5) ~~~~ \forall x (x \cdot 0 = 0) \nonumber \\ & (A6) ~~~~ \forall xy (x \cdot (y + 1) = (x \cdot y) + x) \nonumber...
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