9.4.10  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,

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:

Output:

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 6.8.16.)
  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:

Output:

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:

Output: