564 Fishery Bulletin 90(3). 1992 



/ r(o+/3) \ „ , 



\ r(a) r{p) j 



where a and p are positive constants and r(-) is the gamma function, which, except for r(l)= 1, can be described 

 in terms of the recursion formula 



r(a) = (a-l)r(a-l). (10) 



By Equations (9) and (10), then, the integral in Equation (8) can be evaluated as follows: 



■ip(q), / r(a + /3) \ r' w, .«_, . / r(a + /3) Wr(a)r(/3-l)\ 



r ^dq=P^^ f q-(l-q)^-dq 



J ^, 1-q \r(a)r(/?)|J„ 



r(a)r(/j)/\ r(a+/3-i) 



fr(/?-l)W r(a + P) \ a + /J-l 



(11) 



\ r(/?) /\r(a + /5-l)/ /3-1 



Substituting Equation (11) into Equation (8) then gives 



[2 1n(l + F') - ln(l + K" + F')](a + /?-l) 



Ei{L[z(F, q)]} = - In(F') + ^ ^^^ -. (12) 



^-1 



Differentiating Equation (12) with respect to F' and setting the resulting expression equal to zero yields the 

 quadratic expression 



aF'2 + [K"(2a + /3-l) + a - /? + 1] F' - (/3-l)(K"+l) = 0. (13) 



Before solving Equation (13), it would be helpful to cast the solution in terms of parameters which are more 

 intuitive than a and ft, for example the mean and variance of P(q). The beta distribution has mean m and variance 

 V as follows: 



a a/? 



m = and v = . (14) and (15) 



a + p {a + P)~{a + p+l) 



Conversely, Equations (14) and (15) can be solved simultaneously to describe a and /? in terms of m and v: 



/m(l-m) \ /m(l-m) \ 

 a = [^ ^-1 m and P = \— ^-1 (1-m). (16) and (17) 



Unlike the normal distribution, the variance of the beta distribution exhibits a maximum possible value for a 

 given mean. Remembering that a and p are constrained to be positive, the maximum possible value of v can be 

 derived from either Equation (16) or Equation (17) by setting the left-hand side equal to and solving for v. This 

 exercise results in a maximum v equal to m(l - m). Thus, a and p can be written in terms of the mean and a scaled 



variance v | = 1 as follows: 



m(l-m) 



