Let $X$ be a continuous random variable with pdf $f(x) =\frac{1}{2}(1+ \theta x)$, for $-1 < x < 1$, and $-1 < \theta < 1$
(a) Show that $E(X) = \frac{\theta}{3}$.
(b) Given a random sample of size $n$ from a population with pdf $f(X)$, consider the estimator $\hat{\theta} = 3\bar{X}$. Find the variance, bias, and mean squared error of $\hat{\theta}$.
For (a) I simply integrate $x f(x)$
For (b) I'm not sure it is related to part (a) or the pdf at all. I'm not quiet sure how to proceed.
I know that $\operatorname{Var}(\hat{\theta})=E((\hat{\theta}-E[\theta])^2)$
$\operatorname{Bias}(\hat{\theta})=E(\hat{\theta}-\theta))$, and
$\operatorname{MSE}=\sqrt{\operatorname{Var}(\hat{\theta})}$
Do I just need to manipulate the definitions?