F distribution (from http://www.math.wm.edu/˜ leemis/chart/UDR/UDR.html ) The shorthand X ∼ F (n_{1},n_{2}) is used to indicate that the random variable X has the F distribution with parameters n_{1} and n_{2}, which are positive integers known as the degrees of freedom for the numerator and the degrees of freedom for the denominator. The F distribution is also known as the variance ratio distribution and the Fisher–Snedecor distribution. An F random variable X with n_{1} and n_{2} degrees of freedom has probability density function

f(x) = \frac{Γ((n_{1}+ n_{2})/2)(n_{1}/n_{2})^{n_{1}/2}x^{n_{1}/2−1}}{Γ(n_{1}/2)Γ(n_{2}/2)[(n_{1}/n_{2})x + 1]^{(n_{1}+n_{2})/2}} x > 0,

for n_{1} = 1,2,... and n_{2} = 1,2,.... The F distribution is used for statistical inference concerning ratios of variances of two normal populations. The F distribution is used for statistical inference concerning ratios of rates of two exponential populations. The probability density function is plotted below for three parameter combinations: n_{1} = 1 and n_{2} = 1; n_{1} = 5 and n_{2} = 1; n_{1} = 75 and n_{2} = 75.

PSfrag replacements

x
f(x)
n_{1} = 1, n_{2} = 1
n_{1} = 5, n_{2} = 1

n_{1} = 75, n_{2} = 75

The cumulative distribution function, survivor function, hazard function, cumulative hazard function, inverse distribution function, median, moment generating function, and characteristic function can’t be written as closed-form expressions. The mode of X is

\frac{n_{2}(n_{1}− 2)}{n_{1}(n_{2}+ 2)} n_{1} > 2.

The population mean, variance, skewness, and kurtosis of X are

E[X] = \frac{n_{2}}{n_{2}− 2} n_{2} > 2

V [X] = \frac{2n_{2}^{2}(n_{1}+ n_{2}− 2)}{n_{1}(n_{2}− 2)^{2}(n_{2}− 4)} n_{2} > 4

E [( \frac{X − μ}{σ} ) ^{3}] = \frac{(2n_{1}+ n_{2}− 2)\sqrt{8(n_{2}− 4)}}{(n_{2}− 6)\sqrt{n_{1}(n_{1}+ n_{2}− 2)}} n_{2} > 6

E [( \frac{X − μ}{σ} ) ^{4}] = \frac{3(n_{2}− 4) (10n_{1}^{2}+ n_{1}^{2}n_{2}− 20n_{1}+ 8n_{1}n_{2}+ n_{1}n_{2}^{2}+ 16 − 16n_{2}+ 4n_{2}^{2})}{n_{1}(n_{1}+ n_{2}− 2) (n_{2}− 6) (n_{2}− 8)} n_{2} > 8.

APPL verification: The APPL statements

X := FRV(n1, n2); Mean(X); Variance(X); Skewness(X); Kurtosis(X);

verify the population mean, variance, skewness, and kurtosis.