# The Fisher-Snédécor distribution

### 8.4.10 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

(7)

The fisher command computes this density function.
fisherd, snedecor and snedecord are synonyms for fisher.

- fisher takes three arguments:
  - n1 and n2, integers (the degrees of freedom).
  - x, a non-negative real number.
- fisher(n1,n2,x) returns the value of the Fisher-Snédécor density function with n1 and n2 degrees of freedom, given in (7).

#### Example.

Input:

fisher(5,3,2.5)

Output:

0.10131184472

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

The fisher_cdf (or snedecor_cdf) command computes the cumulative distribution function for the Fisher-Snédécor distribution.

- fisher_cdf takes three mandatory arguments and one optional argument:
  - n1 and n2, integers (the degrees of freedom).
  - x, a real number.
  - Optionally, y, a real number.
- fisher_cdf(n1,n2,x) returns Prob(X ≤ x) for the Fisher-Snédécor distribution with n1 and n2 degrees of freedom
- fisher_cdf(n1,n2,x,y) returns Prob(x ≤ X < y).

#### Examples.

- Input:
  fisher_cdf(5,3,9)
  Output:

  β

  ⎛
  ⎜
  ⎜
  ⎝

  5
  2

  ,

  3
  2

  ,

  15
  16

  ,1

  ⎞
  ⎟
  ⎟
  ⎠

  (See Section 5.8.16.)
  This can be numerically approximated with:
  Input:
  evalf(fisher_cdf(5,3,9))
  Output:

  0.949898927032

- Input:
  evalf(fisher_cdf(5,3,9,10))
  Output:

  0.0066824173023

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

The fisher_icdf (or snedecor_icdf) command computes the inverse distribution for the Fisher-Snédécor distribution.

- fisher_icdf takes three arguments:
  - n1 and n2, integers (the degrees of freedom).
  - h, a real number between 0 and 1.
- fisher_icdf(n1,n2,h) returns the inverse distribution for the Fisher-Snédécor distribution with n1 and n2 degrees of freedom; namely, the value of x for which Prob(X ≤ x) = h.

#### Example.

Input:

fisher_icdf(5,3,0.95)

Output:

9.01345516752

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

The UTPF (the Upper Tail Probability - Fisher-Snédécor distribution) computes Prob(X > x) for the Fisher-Snédécor distribution.

- UTPF takes three arguments:
  - n1 and n2, integers (the degrees of freedom).
  - x, a real number.
- UTPF(n1,n2,x) returns Prob(X > x) for the Fisher-Snédécor distribution with n1 and n2 degrees of freedom.

#### Example.

Input:

UTPF(5,3,9)

Output:

0.050101072968