Expected Value of Yield 



Most users of Griffin's equation (Griffin et al. 

 1976; Louisiana State University (footnote 3) es- 

 timated yield by using the mean (or expected) 

 value of the independent variables, discharge and 

 effort. Yet it can easily be shown that, for a general 

 two- variable function, ifx,y are random variables 

 and g an arbitrary twice differentiable function of 

 X, y such that: 



z=gix,y) (2) 



then E[z] =E\g{x,y)] ^ g(E[xl Ely]) (3) 



or if E[x] = iq^ and E\y] 7]y 



then E[z]-^g(r]^,r]y) 



(4) 



where Ei ) denotes expectation of random vari- 

 able. 



Hence, yield estimates obtained using mean 

 values as in Equation (4) are generally not accu- 

 rate. It may be shown (Papoulis 1965) that Equa- 

 tion (4) may be correctly approximated as: 



To compute the estimated yield and its variance 

 we required the first, second, and cross partial 

 derivatives of the yield equation. The derivation 

 was tedious and hence not reproduced here. By the 

 necessary partial differentiation of Equation (1) 

 we could write: 



ay 



15 ' "^''"^ 



32-1 



(1-^0 



dD 



02-2 



dY 

 dE 



bDdE 



-doD^- i3f log J, 



-/3o£>^^/3f (log./?,)' 



-U^D^'-'' |3f log^/3, 



(9) 



2 = ^2^2-1) l^oD'''^ (l-i3f ) (10) 



(11) 



(12) 

 (13) 



Griffin et al. (1976) have estimated the equation 

 parameters to be: Po = 6593, A = 0.995701, and fi^ 

 = -0.60134. 



^l^] =^(r?.,r?j4(B ol .0 ol .|^ cov(.,y)} 



2 ^dx 



dy 



bxdy 



(5) 



where the o^'s are the variances of variables x,y. The variance of the estimate is as follows: 



o: 



^dx^ 



^"^^(^ 



^dx by' 



cov(x,y) + 



(6) 



Thus, Equation (4) is only a first approximation, 

 with Equation (5) providing the second term. Ad- 

 ditional terms may be obtained by continuing 

 Taylor's series expansion ofg{x,y) around^(T7^, 17^). 

 For the purpose of the test, however, the second 

 term was sufficient. 



For Griffin's equation the independent (random) 

 variables were river discharge D and vessel effort 

 E. So, the expected value of the dependent (ran- 

 dom) variable yield Y could be expressed as: 



Expected yield of shrimp can be determined by 

 using the means, variances, and covariances of 

 river discharge and effort. Following the approach 

 of one user of Griffin's equation (Christmas and 

 Etzold 1977), yield was estimated by using mean 

 values of variables for the years 1970-74. A listing 

 of the data and numerical values of means and 

 covariances are given in Table 1. 



E[Y] = y(r7^,r7^)4{0 



^D 



2 9'r 



Ve 



2 o ^^Y 



^D'^E 



COX {D,E)). (7) 



Similarly, the variance of the estimate was given by: 





o 



^D 



D 



^dE' 



Ve 



a| + 



^dD dE' 



rjij ,Ve 



co\{D,E) 



(8) 



974 



