The Sample Mean and Variance from a Normal Sample

Recall that our random sample consists of independent, identically distributed random variables

(X₁, X₂, ..., X_n)

and that the sample mean and variance are defined by

Independence of the sample mean and variance

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 distribution of the sample variance

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

1. The random variable on the left side of the equation in Exercise 7 has the chi-square distribution with n degrees of freedom.
2. The second term on the right in Exercise 6 has the chi-square distribution with 1 degree of freedom.

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.

The Sample Mean