t-Test Probability Formulas

Below you will find descriptions and details for the 4 formulas that are used to compute probabilities (p-values) for a t-test.

Beta function:

Cumulative distribution function (CDF) for the t-distribution:

Variable definitions:
v = degrees of freedom
t = upper limit of integration
I = regularized lower incomplete beta function

Lower incomplete beta function:

Regularized lower incomplete beta function:

Variable definitions:
the numerator = lower incomplete beta function
the denominator = beta function