fCdf¶

Evaluates the F distribution function.

Synopsis¶

fCdf (f, dfDenominator, dfNumerator)

Required Arguments¶

float f (Input): Point at which the F distribution function is to be evaluated.
float dfNumerator (Input): The numerator degrees of freedom. The argument dfNumerator must be positive.
float dfDenominator (Input): The denominator degrees of freedom. The argument dfDenominator must be positive.

Return Value¶

The probability that an F random variable takes a value less than or equal to the input point, f.

Description¶

The function fCdf evaluates the distribution function of a Snedecor’s F random variable with dfNumerator and dfDenominator. The function is evaluated by making a transformation to a beta random variable and then by evaluating the incomplete beta function. If X is an F variate with \(\nu_1\) and \(\nu_2\) degrees of freedom and \(Y=(\nu_1X)/(\nu_2+ \nu_1 X)\), then Y is a beta variate with parameters \(p=\nu_1/2\) and \(q=\nu_2/2\).

The function fCdf also uses a relationship between F random variables that can be expressed as follows:

\(F_F(f,\nu_1,\nu_2)=1-F_F(1/f,\nu_2,\nu_1)\) where \(F_F\) is the distribution function for an F random variable.

Figure 9.22 — Plot of \(F_F(f,1.0,1.0)\)

Example¶

This example finds the probability that an F random variable with one numerator and one denominator degree of freedom is greater than 648.

from __future__ import print_function
from numpy import *
from pyimsl.math.fCdf import fCdf

f = 648.0
df_numerator = 1.0
df_denominator = 1.0

p = 1.0 - fCdf(f, df_numerator, df_denominator)
print("The probability that an F(1,1) variate ", end=' ')
print("is greater than 648 is %6.4f" % (p))

Output¶

The probability that an F(1,1) variate  is greater than 648 is 0.0250