9.4.9  The χ² distribution

The probability density function for the χ² distribution: chisquare

The χ² distribution with n degrees of freedom has density function given by

The chisquare command computes this density function.

• chisquare takes two arguments:
  • n, an integer (the degrees of freedom).
  • x, a real number.
• chisquare(n,x) returns the value of the χ² density function with n degrees of freedom, given in (6).

Example.
Input:

Output:

which can be numerically approximated by:
Input:

Output:

The cumulative distribution function for the χ² distribution: chisquare_cdf

The chisquare_cdf command computes the cumulative distribution function for the χ² distribution.

• chisquare_cdf takes two mandatory arguments and one optional argument:
  • n, an integer (the degrees of freedom).
  • x, a real number.
  • Optionally, y, a real number.
• chisquare_cdf(n,x) returns Prob(X ≤ x) for the χ² distribution with n degrees of freedom.
• chisquare_cdf(n,x,y) returns Prob(x ≤ X < y).

Examples.

• Input:
  chisquare_cdf(5,11)
  Output:

  0.948620016517

• Input:
  chisquare_cdf(3,1,2)
  Output:

  0.22884525243

The inverse distribution function for the χ² distribution: chisquare_icdf

The chisquare_icdf command computes the inverse distribution for the χ² distribution.

• chisquare_icdf takes two arguments:
  • n, an integer (the degrees of freedom).
  • h, a real number between 0 and 1.
• chisquare_icdf(n, h) returns the inverse distribution for the χ² distribution with n degrees of freedom; namely, the value of x for which Prob(X ≤ x) = h.

Example.
Input:

Output:

The upper tail cumulative function for the χ² distribution: UTPC

The UTPC (the Upper Tail Probability - Chi-square distribution) computes Prob(X > x) for the χ² distribution.

• UTPC takes two arguments:
  • n, an integer (the degrees of freedom).
  • x, a real number.
• UTPC(n,x) returns Prob(X > x) for the χ² distribution with n degrees of freedom.

Example.
Input:

Output: