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

question 24

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.

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Definitions:

Utility Function

A mathematical representation in economics to capture the preference of consumers by assigning numerical values to different bundles of goods, indicating their level of satisfaction or happiness.

Rational Consumer

An individual who makes spending decisions to maximize their satisfaction or utility, based on their preferences and budget constraints.

Demand

The desire and ability of consumers to purchase goods and services at different prices, contributing to determining the market demand.

Utility Function

A mathematical representation of how a consumer's preferences over bundles of goods relate to their satisfaction or utility.