The Sample Mean and Variance from a Normal Sample
Recall that our random sample consists of independent, identically distributed random variables
(X1, X2, ..., Xn)
and that the sample mean and variance are defined by
Let us first note a simple but interesting fact.
1. Use basic properties of covariance to show that for each i, following random variables are uncorrelated:
For the remainder of this section, we will derive some special and somewhat surprising properties of the sample mean and variance when we are sampling from a normal distribution. Our analysis hinges on the sample mean and the vector of deviations from the sample mean:
2. Show that
and hence show that the sample variance can be written as a function of D.
3. Show that the sample mean and the vector D have a joint multivariate normal distribution.
4. Use the results of Exercises 1, 2, and 3 to show that the sample mean and the vector D are independent.
5. Use the result of Exercise 3 to show that the sample mean and sample variance are independent.
The next sequence of exercises derives the distribution of a certain multiple of the sample variance.
6. Show that
7. Use the result of Exercise 6 to show that
8. Show that
9. Use the result of Exercise 8 and moment generating functions to show that
has the chi-squared distribution with n - 1 degrees of freedom.