### 7.4.8 The Fisher-Snédécor distribution

#### The probability density function for the Fisher-Snédécor distribution: fisher fisherd snedecor snedecord

The Fisher-Snédécor distribution (also called the F-distribution) with n1 and n2 degrees of freedom has density function given by for x ≥ 0,

fisher(n1,n2,x) =

(n1/n2)n1/2Γ((n1+n2)/2)
Γ(n1/2)Γ(n2/2)

x(n1−2)/2
(1+(n1/n2)x)(n1+n2)/2

(The snecedor command is the same as the fisher command.) If you enter

fisher(5,3,2.5)

you will get

0.10131184472

#### The cumulative distribution function for the Fisher-Snédécor distribution: fisher_cdf snedecor_cdf

The cumulative distribution function for the Fisher-Snédécor distribution with n1 and n2 degrees of freedom at a value x is fisher_cdf(n1,n2,x) = snedecor(n1,n2,x) = Prob(X ≤ x); if you enter

fisher_cdf(5,3,9)

you will get

Beta(5/2,3/2,15/16,1)

which can be numerically approximated with

evalf(fisher_cdf(5,3,9,10))

which is

0.949898927032

#### The inverse distribution function for the Fisher-Snédécor distribution: fisher_icdf snedecor_icdf

The inverse distribution function for the Fisher-Snédécor distribution with n1 and n2 degrees of freedom is computed with fisher_icdf(n1,n2,h); recall that this will return the value x with fisher_cdf(n1,n2,x) = h. If you enter

fisher_icdf(5,3,0.95)

you will get

9.01345516752

#### The upper tail cumulative function for the Fisher-Snédécor distribution: UTPF

The UTPF (the Upper Tail Probability - Fisher-Snédécor distribution) will compute Prob(X > x). If you enter

UTPF(5,3,9)

you will get

0.050101072968