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6. The F Distribution

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 F Density

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. Let

X = (U / m) / (V / n).

1. Show that X has the density function

f(x) = C_m,n x^{(m - 2) / 2} / [1 + (m / n)x]^{(m + n) / 2} for x > 0,

where the normalizing constant C_m,n is given by

C_m,n = gam[(m + n) / 2] (m / n)^{m / 2} / [gam(m / 2) gam(n / 2)].

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.

2. In the random variable experiment, select the F distribution. Vary the parameters with the scroll bars and note the shape of the density function. With n =3 and m = 2, run the simulation 1000 times, updating every 10 runs, and note the apparent convergence of the empirical density function to the true density function.

3. 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(x) converges to 0 as x approaches infinity.

Thus, the F distribution is unimodal but skewed.

The distribution function and the quantile function do not have simple, closed-form representations. Approximate values of these functions can be obtained from the quantile applet.

4. In the quantile applet, select the F distribution. Vary the parameters and note the shape of the density function and the distribution function. In each of the following cases, find the median, the first and third quartiles, and the interquartile range.

1. m = 5, n = 5
2. m = 5, n = 10
3. m = 10, n = 5
4. m = 10, n = 10

Moments

Suppose that X has the F distribution with m degrees of freedom in the numerator and n degrees of freedom in the denominator. The random variable representation in Exercise 1 can be used to find the mean, variance, and other moments.

5. Show that if n > 2, 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

var(X) = 2 n²(m + n - 2) / [(n - 2)² m (n - 4)].

7. In the simulation of the random variable experiment, select the F distribution. Vary the parameters with the scroll bar and note the size and location of the mean/standard deviation bar. Now with n = 3, m = 5, run the simulation 1000 times, updating every 10 runs, and note the apparent convergence of the empirical moments to the true moments..

8. Show that if k < n / 2 then

E(X^k) = gam[(m + 2k) / 2] gam[(n - 2k) / 2] (n / m)^k / [gam(m / 2) gam(n / 2)].

Transformations

9. Suppose that X has the F distribution with m degrees of freedom in the numerator and n 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.

10. 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.

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