Chapter 5.2, Problem 5.8LM

Consider the following probability distribution:

x	0	1	4
p(x)	$\frac{1}{3}$	$\frac{1}{3}$	$\frac{1}{3}$

1. a. Find µ and σ².
2. b. Find the sampling distribution of the sample mean $\bar{x}$ for a random sample of n = 2 measurements from this distribution.
3. c. Show that $\bar{x}$ is an unbiased estimator of µ [Hint: Show that $E\left(\bar{x}\right)={\displaystyle \sum \bar{x}p}\left(\bar{x}\right)=\mu$.]
4. d. Find the sampling distribution of the sample variance s² for a random sample of n = 2 measurements from this distribution.
5. e. Show that s² is an unbiased estimator for σ².