# Max Likelihood Estimator

I need to calculate the ML estimator of the following probability function and I have doubts about the solution.

$$f(x; \theta) = \theta x^{\theta -1}$$

$$L(\theta) = \theta^n \left(\prod_{i=1}^n x_i\right)^{\theta-1} =$$ $$l(\theta) =\log(L)= n\log(\theta) + (\theta -1) \log \left(\prod_{i=1}^n xi\right) =$$ $$n\log(\theta) + (\theta -1) \sum_{i=1}^n \log(xi)$$

$$l'(\theta) = \cfrac{\partial l}{\partial \theta} = \cfrac{n}{\theta} + \sum_{i=1}^n\log(x_i)$$

$$l'(\theta) = 0 \Leftrightarrow \hat{\theta} = -\cfrac{n}{\sum_{i=1}^n \log(x_i)}$$

Is my result correct? How can it be negative if $$\theta > 0$$ and $$x \in \{0,1\}$$?

Does $$f(x; \theta) = \theta x^{\theta -1}$$ have a specific name? It looks like a Weibull, but it does not have the exponential component.

The description is missing the important part that the support is $$(0,1)$$, i.e. $$f_\theta(x) = \theta x^{\theta-1}\mathbb I_{(0,1)}(x)$$ This is a special case of the Beta distribution, namely the Beta$$(\theta,1)$$ distribution. And, since $$x\in(0,1)$$,$$\log(x)<0$$