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  1. Virtual Laboratories
  2. 4. Special Distributions
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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 Density Function

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

\[ X = \frac{U / m}{V / n} \]

\(X\) has the probability density function

\[ f(x) = \frac{\Gamma[(m + n) / 2]}{\Gamma(m / 2) \, \Gamma(n / 2)} \left( \frac{m}{n} \right)^{m/2} \frac{x^{(m-2)/2}}{[1 + (m / n)x]^{(m+n)/2}}, \quad x \ge 0 \]

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 was first derived by George Snedecor, and is named in honor of Sir Ronald Fisher.

In the special distribution simulator, select the \(F\) distribution. Vary the parameters with the scroll bars and note the shape of the density function. For selected values of the parameters, run the simulation 1000 times and note the apparent convergence of the empirical density function to the true density function.

The \(F\) probability density function \( f \) satisfies the following properties:

  1. \(f\) at first increases and then decreases, reaching a maximum at the mode \(x = \frac{m-2}{m(n + 2)}\).
  2. \(f(x) \to 0\) as \(x \to \infty\).

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 special distribution calculator and from most mathematical and statistical software packages.

In the special distribution calculator, 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

The random variable representation in Exercise 1 can be used to find the mean, variance, and other moments.

Suppose that \(X\) has the \(F\) distribution with \(m\) degrees of freedom in the numerator and \(n\) degrees of freedom in the denominator. Then

  1. \(\E(X) = \infty\) if \(0 \lt n \le 2\)
  2. \(\E(X) = \frac{n}{n - 2}\) if \(n \gt 2\)

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

Suppose that \(X\) has the \(F\) distribution with \(m\) degrees of freedom in the numerator and \(n\) degrees of freedom in the denominator. Then

  1. \(\var(X)\) is undefined if \(0 \lt n \le 2\)
  2. \(\var(X) = \infty\) if \(2 \lt n \le 4\)
  3. If \(n \gt 4\) then \[ \var(X) = 2 \left(\frac{n}{n - 2} \right)^2 \frac{m + n - 2}{m(n - 4)} \]

In the simulation of the special distribution simulator, select the \(F\) distribution. Vary the parameters with the scroll bar and note the size and location of the mean/standard deviation bar. For selected values of the parameters, run the simulation 1000 times and note the apparent convergence of the empirical moments to the true moments.

Suppose that \(X\) has the \(F\) distribution with \(m\) degrees of freedom in the numerator and \(n\) degrees of freedom in the denominator.

  1. \(\E(X^k) = \infty\) if \(0 \lt n \le 2 \, k\)
  2. If \(n \gt 2 \, k\) then \[ \E(X^k) = \frac{\Gamma[(m + 2\, k)/2] \, \Gamma[(n - 2\, k) / 2]}{\Gamma(m/2) \Gamma(n/2)} \left( \frac{n}{m} \right)^k \]

Transformations

Suppose that \(X\) has the \(F\) distribution with \(m\) degrees of freedom in the numerator and \(n\) degrees of freedom in the denominator. Then \(1 / X\) has the \(F\) distribution with \(n\) degrees of freedom in the numerator and \(m\) degrees of freedom in the denominator.

Suppose that \(T\) has the \(t\) distribution with \(n\) degrees of freedom. Then \(X = T^2\) has the \(F\) distribution with 1 degree of freedom in the numerator and \(n\) degrees of freedom in the denominator.

  1. Virtual Laboratories
  2. 4. Special Distributions
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  18. Answers