Let $X$ Denote the Proportion of Allotted Time That a Randomly Selected

Essay

Let $X$ denote the proportion of allotted time that a randomly selected student spends working on a certain aptitude test. Suppose the pdf of $X$ is $f ( x ; \theta ) = \left\{ \begin{array} { l l } ( \theta + 1 ) x ^ { \theta } & 0 \leq x \leq 1 \\0 & \text { otherwise }\end{array} \right.$ where $\theta$ > -1. A random sample of ten students yields data $x _ { 1 } = .92 , x _ { 2 } = .79 , x _ { 3 } = .90 , x _ { 4 } = .65 , x _ { 5 } = .86 , x _ { 6 } = .47 , x _ { 7 } = .73 , x _ { 8 } = .97 , x _ { 9 } = .94 , x _ { 10 } = .77$
a. Use the method of moments to obtain an estimator of $\theta$
and then compute the estimate for this data.
b. Obtain the maximum likelihood estimator of $\theta$
and then compute the estimate for the given data.