Consider the c.d.f. of the beta distribution. Several numerical methods have been developed specifically for its evaluation (see, for example, Kennedy and Gentle, 1980; and Press, Teukolsky, Vetterling and Flannery, 1989), but here we consider two ‘direct’ ways: (i) direct numeric integration, and (ii) use of the identity

(this is proved in Volume II in the context of order statistics; see also Example 1.28). Construct a program which compares the accuracy and computation time of the above two methods, along with the pre-programmed method betainc in Matlab (which uses one of the specific numeric algorithms). For the numerical integration, use the quadl function and, for the combinatoric exact solution, use the binocdf function, which evaluates the c.d.f. of the binomial distribution.

= k=j n! (j-1)! (nj)! Jo x-(1-x)"- dx