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ptolema

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## Homework Statement

The independent random variables [itex]X_1, ..., X_n[/itex] have the common probability density function [itex]f(x|\alpha, \beta)=\frac{\alpha}{\beta^{\alpha}}x^{\alpha-1}[/itex] for [itex]0\leq x\leq \beta[/itex]. Find the maximum likelihood estimators of [itex]\alpha[/itex] and [itex]\beta[/itex].

## Homework Equations

log likelihood (LL) = n ln(α) - nα ln(β) + (α-1) ∑(ln x

_{i})

## The Attempt at a Solution

When I take the partial derivatives of log-likelihood (LL) with respect to α and β then set them equal to zero, I get:

(1) d(LL)/dα = n/α -n ln(β) + ∑(ln x

_{i}) = 0 and

(2) d(LL)/dβ = -nα/β = 0

I am unable to solve for α and β from this point, because I get α=0 from equation (2), but this clearly does not work when you substitute α=0 into equation (1). Can someone please help me figure out what I should be doing?

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