Question: To evaluate a consumer electronics product, a rater goes through a checklist of desirable features for the product to have. The rater places a check beside each feature that she thinks the product has. After completing the evaluation, the rater reports the product’s score as x = proportion of features on the checklist that she determined the product to have. Thus, the score x is a number between 0 and 1.

A statistical model is proposed for the evaluation: Let X be a random variable whose outcomes are the rater’s possible scores x. The probability density function for X is proposed as

\[f(x\,;\theta )=\left\{ \begin{array}{*{35}{l}} \theta {{x}^{\theta -1}}\quad \text{if 0}\le \text{x}\le \text{1} \\ 0\quad \text{ otherwise} \\ \end{array} \right.\]

The density function (and thus the statistical model) is incompletely specified, since it depends on the value of a parameter θ, which is known only to be greater than 0.

a) Why does \[f(x\,;\theta )\] qualify as a potential density function for every \[\theta >0\] ?

b) Find the mean and variance of X for any generic \[\theta >0\].