The F Distribution

If Z1 and Z2 are independently distributed chi-square variables with k1 and k2 df, respectively, the variable follows (Fisher's) F distribution with k1 and k2 df. An F-distributed variable is denoted by Fk1,k2 where the subscripts indicate the df associated with the two Z variables, k1 being called the numerator df and k2 the denominator df. Geometrically, the F distribution is shown in Figure A.7.

The F distribution has the following properties:

1. Like the chi-square distribution, the F distribution is skewed to the right. But it can be shown that as k1 and k2 become large, the F distribution approaches the normal distribution.

2. The mean value of an F-distributed variable is k2/(k2 — 2), which is defined for k2 > 2, and its variance is which is defined for k2 > 4.

3. The square of a t-distributed random variable with k df has an F distribution with 1 and k df. Symbolically,

Z2/k2

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