The Fisher F Distribution

Java Applet Simulation of the Fisher F experiment

In this section we will study a distribution that has special importance in statistics. In particular, this distribution arises form ratios of sums of squares when sampling from a normal distribution.

The Fisher F Density

1. Suppose that U has the chi-square distribution with m degrees of freedom, V has the chi-square distribution with n degrees of freedom, and that U and V are independent. Show that

has the density function

The distribution defined by the density function in Exercise 1 is known as the F distribution with m degrees of freedom in the numerator and n degrees of freedom in the denominator. The F distribution is named in honor of Sir Ronald Fisher.

Simulation

The applet for this page simulates a random variable with an F distribution. The graph of the density function is shown in blue. The mean and standard deviation are recorded in the first table and shown graphically in terms of the horizontal blue bar, which is centered at the mean and extends one standard deviation on either side. You can vary the parameters m and n with scroll bars. When you run the experiment, independent values from the distribution are generated and recorded in the second table. The empirical density function is graphed in red and the sample mean and variance recorded in the first table and shown graphically in terms of the horizontal red bar.

2. Start the applet and vary the parameters with the scroll bars. Note the shape of the F density function and the location and size of the mean-standard deviation bar.

3. Run the simulation with an update frequency of 10 for several different sets of parameter values. Note the apparent convergence of the empirical density and statistics to the corresponding distribution density and parameters.

Shape

4. Sketch the graph of the F density function given in Exercise 1. In particular, show that

1. f(x) at first increases and then decreases, reaching a maximum at the mode
   x = (m - 2)/(m(n + 2)).
2. f(t) converges to 0 as t approaches infinity.

Thus, the F distribution is unimodal but skewed.

Moments

5. Show that if n > 2, the mean is given by

E(X) = n / (n - 2).

Thus, the mean depends only on the degrees of freedom in the denominator.

6. Show that if n > 4 then the variance is given by

7. Show that if k < n / 2 then

Transformations

8. Suppose that X has the F distribution with m degrees of freedom in the numerator and m degrees of freedom in the denominator. Show that 1/X has the F distribution with n degrees of freedom in the numerator and m degrees of freedom in the denominator.

9. Suppose that T has the t distribution with n degrees of freedom. Show that X = T² has the F distribution with 1 degree of freedom in the numerator and n degrees of freedom in the denominator.

Special Distributions