### 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

fisher(5,3,2.5)

0.10131184472 |  |  |  |  |  |  |  |  |  |

##### 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

fisher_cdf(5,3,9)

β

⎛
⎜
⎜
⎝

5
2

,

3
2

,

15
16

,1

⎞
⎟
⎟
⎠

(See Section 7.3.16.) This can be numerically approximated with:

evalf(fisher_cdf(5,3,9))

0.949898927032 |  |  |  |  |  |  |  |  |  |

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

0.0066824173023 |  |  |  |  |  |  |  |  |  |

##### 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

fisher_icdf(5,3,0.95)

9.01345516752 |  |  |  |  |  |  |  |  |  |

##### 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

UTPF(5,3,9)

0.050101072968 |  |  |  |  |  |  |  |  |  |