chiSquaredCdf¶

Evaluates the chi‑squared cumulative distribution function (CDF).

Synopsis¶

chiSquaredCdf (chiSquared, df)

Required Arguments¶

float chiSquared (Input): Argument for which the chi‑squared distribution function is to be evaluated.
float df (Input): Number of degrees of freedom of the chi‑squared distribution. Argument df must be greater than 0.

Return Value¶

The probability p that a chi‑squared random variable takes a value less than or equal to chiSquared.

Description¶

Function chiSquaredCdf evaluates the distribution function, \(F(x,ν)\), of a chi‑squared random variable x = chiSquared with ν = df degrees of freedom, where:

\[F(x,v) = \frac{1}{s^{v/2} \mathit{\Gamma}(v/2)} \int_0^x e^{-t/2} t^{v/2-1} dt\]

and \(\Gamma(\cdot)\) is the gamma function. The value of the distribution function at the point x is the probability that the random variable takes a value less than or equal to x.

For \(v>\nu_{max}=1.e7\), chiSquaredCdf uses the Wilson-Hilferty approximation (Abramowitz and Stegun [A&S] 1964, Equation 26.4.17) for p in terms of the normal CDF, which is evaluated using function normalCdf.

For \(v\leq\nu _{max}\), chiSquaredCdf uses series expansions to evaluate p: for \(x<\nu\), chiSquaredCdf calculates p using A&S series 6.5.29, and for \(x\geq\nu\), chiSquaredCdf calculates p using the continued fraction expansion of the incomplete gamma function given in A&S equation 6.5.31.

Figure 9.21 — Plot of Fx (x, df)

Example¶

Suppose X is a chi‑squared random variable with two degrees of freedom. In this example, we find the probability that X is less than 0.15 and the probability that X is greater than 3.0.

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

chi_squared = 0.15
df = 2.0
p = chiSquaredCdf(chi_squared, df)
print("The probability that chi-squared ", end=' ')
print("with 2 df is less than 0.15 is %6.4f" % (p))

chi_squared = 3.0
p = 1.0 - chiSquaredCdf(chi_squared, df)
print("The probability that chi-squared ", end=' ')
print("with 2 df is less than 3.0 is %6.4f" % (p))

Output¶

The probability that chi-squared  with 2 df is less than 0.15 is 0.0723
The probability that chi-squared  with 2 df is less than 3.0 is 0.2231

Informational Errors¶

IMSL_ARG_LESS_THAN_ZERO Since “chiSquared” = #is less than zero, the distribution function is zero at “chiSquared.”

Alert Errors¶

IMSL_NORMAL_UNDERFLOW Using the normal distribution for large degrees of freedom, underflow would have occurred.