Consider a Random Sample $X _ { 1 } , \ldots , X _ { n }$

Essay

Consider a random sample $X _ { 1 } , \ldots , X _ { n }$ from the pdf $f ( x ; \theta ) = 5 ( 1 + \theta x ) \quad - 1 \leq x \leq 1$ where $- 1 \leq \theta \leq 1$ (this distribution arises in particle physics). Show that $\hat { \theta } = 3 \bar { X }$ is an unbiased estimator of $\theta$ [ Hint: First determine $\mu = E ( X ) = E ( \bar { X } ) . ]$