Chapter 6.1, Problem 13E

Consider a random sample X₁,..., X_n from the pdf

$\begin{array}{cccc}f(x;\theta )& =& .5(1\text{ + }\theta x)& -1\le x\le \text{ 1}\end{array}$

where −1 ≤ θ ≤ 1 (this distribution arises in particle physics). Show that $\hat{\theta }\text{ = 3}\bar{X}$ is an unbiased estimator of θ. [Hint: First determine μ = E(X) = E( $\bar{X}$).]