## The Sample Mean and Variance from a Normal Sample |

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

(

X_{1},X_{2}, ...,X_{n})

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

- The random variable on the left side of the equation in
Exercise 7 has the chi-square
distribution with
*n*degrees of freedom. - 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 |