So I am given a question whereby I am told that $(1+\theta)Y^\theta$ is my pdf and it has limits of $0\leq y \leq 1$ and I am asked to find the method of moments estimator for $\theta$ and also the MLE which I am pretty sure I can do myself just wanted to get some clarification on the MOM.
So, I know I have to integrate my pdf i.e. $E(x) = \int x\cdot f(x) dx$ with my limits. However, I am just unsure how I am suppose to integrate this out fully with the parameter of $\theta$ attached to $x$.
Any help would be appreciated!
if the random variable $Y$ has pdf $f(y)=(1+\theta)y^\theta$ on $0\leq y \leq 1$ for some parameter $\theta$, then keeping in mind this parameter is fixed,
$$\mathbb{E}(Y)=\int_0^1 y f(y) dy=\int_0^1 (1+\theta)y^{1+\theta} dy$$ Which can be evaluated by the power rule for integrals. Can you take it from here?