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## Homework Statement

Let [tex]X_{1}, ... , X_{n}[/tex] be a random sample from [tex]f\left(x; \theta\right) = \theta x^{\theta - 1} I_{(0, 1)}\left(X\right)[/tex], where [tex]\theta > 0[/tex].

a. Find the maximum-likelihood estimator of [tex]\theta/\left(1 + \theta\right)[/tex].

b. Is there a function of [tex]\theta[/tex] for which there exists an unbiased estimator whose variance coincides with the Cramer-Rao lower bound?

## The Attempt at a Solution

a.) I understand that in getting the maximum likelihood estimator of [tex]\theta[/tex], we should be finding the value of [tex]\theta[/tex] that will maximize the likelihood function.

We will do this by taking the derivative of the likelihood function with respect to [tex]\theta[/tex] and equate this derivative to zero; or take the derivative of the logarithm of the likelihood function with respect to [tex]\theta[/tex] and equate it to zero.

But I cannot figure out how to find the MLE of [tex]\theta/\left(1 + \theta\right)[/tex].

b.) Please help me also to figure out what to do in solving this problem b.