MHB What is the Maximum Likelihood Estimator for Uniform Distribution Endpoints?

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The maximum likelihood estimator (MLE) for the endpoints θ1 and θ2 of a uniform distribution based on n independent observations is given by θ1 = min(X) and θ2 = max(X), where X represents the observed data points. The MLE for the mean of the uniform distribution can be calculated as (θ1 + θ2) / 2, which simplifies to (min(X) + max(X)) / 2. This result follows from the properties of the uniform distribution and the definition of the mean. The likelihood function for the uniform distribution is constant within the interval [θ1, θ2] and zero outside, leading to these estimators. Understanding these estimators is crucial for statistical inference in uniform distributions.
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I need help on this problem, anyone know how to do it?

Suppose you have n independent observations from a uniform distribution over the interval [𝜃1, 𝜃2].

a. Find the maximum likelihood estimator for each of the endpoints θ1 and θ2.
b. Based on your result in part (a), what would you expect the maximum likelihood estimator to be for the mean? Explain or prove your result.
 
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The answer will depend on what you know. For example, do you know an expression for the likelihood?
 

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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