Chapter 8.5, Problem 7E

A spinner from a board game randomly indicates a real number between 0 and 10. The spinner is fair in the sense that it indicates a number in a given interval with the same probability as it indicates a number in any other interval of the same length.

(a) Explain why the function

$f(x)=\{\begin{array}{l}\begin{array}{ll}0.1\hfill & \text{if}\text{0}\le x\le 1\hfill \\ 0\hfill & \text{if}x<0\text{or}x>10\hfill \end{array}\hfill \end{array}$

is a probability density function for the spinner’s values.

(b) What does your intuition tell you about the value of the mean? Check your guess by evaluating an integral.