Maxima distrib核心库cdf_binomial函数疑似Bug问题求助

Root Cause #

The cdf_binomial(S, n, p) function in Maxima's distrib library uses the incomplete beta function beta_incomplete in its general formula. When S = n (e.g., S=4, n=4), this formula produces beta_incomplete(0, 5, 1-p)—an undefined expression. Direct substitution doesn't trigger the function's built-in edge case checks, but calling cdf_binomial(4,4,p) directly works because the function explicitly returns 1 when S equals n.

Fixes for Substitution Issues #

• Direct Call (Simplest): Instead of substituting into the general form, call cdf_binomial with the specific value:

  cdf_binomial(4,4,p); /* Returns 1 without errors */
  

• Custom Piecewise Function: If you need to work with symbolic S, define a wrapper that handles edge cases:

  my_binom_cdf(S, n, p) := 
    if S >= n then 1 
    else if S < 0 then 0 
    else cdf_binomial(S, n, p);
  

  Now substituting S=4 will work correctly:

  subst([S=4], my_binom_cdf(S,4,p)); /* Returns 1 */
  

Fixes for Integration Issues #

If you need to integrate the CDF over p up to 1:

• Explicit Edge Case Handling: When S=4, the integrand is 1, so the integral from a to 1 is 1 - a. For S < 4, use the standard integral of cdf_binomial(S,4,p).
• Use the Custom Piecewise Function: Integrate the wrapped function to automatically handle all cases:

  integrate(my_binom_cdf(S,4,p), p, 0, 1);
  

  This returns 1 when S >=4, and computes the correct integral for S <4.
• Symbolic Piecewise Definition: For fully symbolic work, use Maxima's piecewise function:

  symbolic_binom_cdf(S, n, p) := piecewise(
    [S >= n, 1],
    [S < 0, 0],
    [true, cdf_binomial(S, n, p)]
  );
  

  Integrals or substitutions with this function will resolve correctly for all S values.

内容的提问来源于stack exchange，提问作者ivo Welch