Suppose that is a random sample from the normal distribution with mean and standard deviation . In this section, we will establish several special properties of the sample mean, the sample variance, and some other important statistics. Since the sample (and in particular the sample size ) is fixed, it will be suppressed in the notation.
First recall that the sample mean is
The distribution of follows easily from basic properties of independent normal variables:
Show that is normally distributed with mean and variance .
The standard score of is given as follows. This statistic will appear in several of the derivations in this section.
Show that has the standard normal distribution.
Recall that if is known, a natural estimator of the variance is the statistic
Although the assumption that is known is usually artificial, is very easy to analyze and it will be used in some of the derivations below.
Show that has the chi-square distribution with degrees of freedom.
Use the result of the previous exercise to show that
Of course, these are special cases of the general results obtained in the section on Sample Variance.
Recall that the standard version of the sample variance is the statistic
The sample variance is the usual estimator of when is unknown. The next series of exercises show that the sample mean and the sample variance are independent, a very important and useful property. First we will note a simple but interesting fact that holds for a random sample from any distribution, not just the normal.
Use basic properties of covariance to show that and are uncorrelated for any .
Our analysis hinges on the sample mean and the vector of deviations from the sample mean. Let
Show that can be written as a function of by noting that
Show that the and the vector have a joint multivariate normal distribution.
Use the results of the previous exercises to show that and are independent.
Finally, show that and are independent.
We can now determine the distribution of a simple multiple of the sample variance . Let
Show that where, as usual, is the standard score associated with the sample mean and where is the special version of the sample variance. Hint: In the sum on the left, add and subtract , and expand.
Show that has the chi-square distribution with degrees of freedom. Hint: Use the result of the previous exercise, independence, and moment generating functions.
Use the result of the previous exercise to show that
Of course, these are special cases of the general results obtained in the section on Sample Variance.
In the basic random variable experiment, select the chi-square distribution. Vary the degree of freedom parameter and note the shape and location of the probability density function and the mean, standard deviation bar. For selected values of the parameter, run the experiment 1000 times, updating every 10 runs, and note the apparent convergence of the empirical density and moments to the true distribution density and moments.
The next sequence of exercises will derive the distribution of
Note that is similar to the standard score , but with the sample standard deviation replacing the distribution standard deviation . The statistic plays a critical role in constructing interval estimates for and performing hypothesis tests for .
As usual, let denote the standard score associated with the sample mean and the chi-square statistics associated with the sample variance . Show that
Use previous results to show that has the student distribution with degrees of freedom.
In the basic random variable experiment, select the distribution. Vary the degree of freedom parameter and note the shape and location of the probability density function and the mean, standard deviation bar. For selected values of the parameters, run the experiment 1000 times, updating every 10 runs, and note the apparent convergence of the empirical density and moments to the true distribution density and moments.
The last statistic that we will study arises in the two-sample normal model. Thus, suppose that is a random sample of size from the normal distribution with mean and standard deviation and that is a random sample of size from the normal distribution with mean and standard deviation . Finally, suppose that and are independent. In the following exercises, we will use the basic notation established above, but we will indicate the dependence on the sample.
Show that the random variable given below has the distribution with degrees of freedom in the numerator and degrees of freedom in the denominator:
Show that the random variable given below has the distribution with degrees of freedom in the numerator and degrees of freedom in the denominator:
These variables are useful for estimates and tests of the ratio of the variances . The choice of the variables depends on whether the means and are known or unknown. Usually, of course, the means are unknown and so the statistic in Exercise 18 is used.
In the basic random variable experiment, select the distribution. Vary the degrees of freedom parameters and note the shape and location of the probability density function and the mean, standard deviation bar. For selected values of the parameters, run the experiment 1000 times, updating every 10 runs, and note the apparent convergence of the empirical density and moments to the true distribution density and moments.