Chapter 15.10, Problem 3P

Define $s$ by the equation $s^2=(1/n){\displaystyle \underset{i=1}{\overset{n}{\sum }}{\left(x_i-\bar{x}\right)}^2}.$ Show that the expected value of $s^2$ is $[(n-1)/n]{\sigma }^2.$ Hints: Write $={\left(x_i-\mu \right)}^2-2\left(x_i-\mu \right)(\bar{x}-\mu )+{(\bar{x}-\mu )}^2$.

Find the average value of the first term from the definition of ${\sigma }^2$ and the average value of the third term from Problem 2. To find the average value of the middle term write .

Show by Problem 6.14 that , and evaluate (same as the first term). Collect terms to find $E\left(s^2\right)=\frac{n-1}{n}{\sigma }^2$