Chapter 6.2, Problem 22E

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

$f(x;\theta )\text{ = }\{\begin{array}{l}(\theta \text{ + 1})x^{\theta }\text{ 0}\le \text{ x }\le \text{ 1}\\ 0\text{ otherwise}\end{array}$

where −1 < θ. A random sample of ten students yields data x₁ = .92, x₂ = .79, x₃ = .90, x₄ = .65, x₅ = .86, x₆ = .47, x₇ = .73, x₈ = .97, x₉ = .94, x₁₀ = .77.

1. a. Use the method of moments to obtain an estimator of θ, and then compute the estimate for this data.
2. b. Obtain the maximum likelihood estimator of θ, and then compute the estimate for the given data.