Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications

B.17 BETA FUNCTIONS

Definition: Beta Function The beta function is

(B.94)

for a, b > 0.

The beta function is symmetric with respect to its parameters: B(a, b) = B(b, a), as illustrated in Figure B.14. The incomplete beta function is obtained by replacing the definite integral with an indefinite integral as follows:

Figure B.14 Beta function B(a, b).

(B.95)

Obviously, B1(a, b) = B(a, b). Sometimes it is notationally convenient to use the regularized incomplete beta function:

(B.96)

which is a normalized form (this notation should not be confused with the indicator function).