20.4.10 Fisher-Snédécor distribution
The probability density function for the Fisher-Snédécor distribution.
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
or fisherd
or snedecor
command computes this density function.
-
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
The cumulative distribution function for the Fisher-Snédécor distribution.
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
(See Section 7.3.16.) This can be numerically approximated with:
| evalf(fisher_cdf(5,3,9,10)) |
The inverse distribution function for the Fisher-Snédécor distribution.
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
The upper tail cumulative function for the Fisher-Snédécor distribution.
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