[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

Let [math]X[/math] be the outcome of a chance experiment with [math]E(X) = \mu[/math] and [math]V(X) = \sigma^2[/math]. When [math]\mu[/math] and [math]\sigma^2[/math] are unknown, the statistician

often estimates them by repeating the experiment [math]n[/math] times with outcomes [math]x_1[/math], [math]x_2[/math], \dots, [math]x_n[/math], estimating [math]\mu[/math] by the sample mean

[[math]] \bar{x} = \frac 1n \sum_{i = 1}^n x_i\ , [[/math]]

and [math]\sigma^2[/math] by the sample variance

[[math]] s^2 = \frac 1n \sum_{i = 1}^n (x_i - \bar x)^2\ . [[/math]]

Then [math]s[/math] is the sample standard deviation. These formulas should remind the reader of the definitions of the theoretical mean and variance. (Many statisticians define the sample variance with the coefficient [math]1/n[/math] replaced by [math]1/(n-1)[/math]. If this alternative definition is used, the expected value of [math]s^2[/math] is equal to [math]\sigma^2[/math]. See Exercise \ref{exer 6.2.19}, part (d).) Write a computer program that will roll a die [math]n[/math] times and compute the sample mean and sample variance. Repeat this experiment several times for [math]n = 10[/math] and [math]n = 1000[/math]. How well do the sample mean and sample variance estimate the true mean 7/2 and variance 35/12?