I'm looking at this independent, beta distribution and wondering if there is a way to compute the mean and variance according to fθ(x) = θx^θ−1, x ∈ (0, 1), where θ > 0 is a parameter where there is a parametric family of distributions on (0, 1)^n?
Further, is there a way to find the Fisher information of µ by rewriting θ as a function of µ with the corresponding density as fµ(x)? How can I find if the mean achieves equality in the Cramer-rao bound?
Thank you!