A Simple Model of Technological Research 145 



and the density function is 



/z^(z) = «F"-i(z)/(z). (8.2) 



The analysis will be illustrated in terms of the exponential dis- 

 tribution: 



f(x) =\e-^^''-^\d<x (8.3) 



F(x)= 1 -^-^(^-e) (8.4) 



E(x) = d + \/\ (8.5) 



Var(x)= 1/X2. (8.6) 



The cumulative distribution of the largest values is, employing 

 8.1, 



and the probability density function is 



h^(z) = \n [1 _e"^(z-0)]'?-i g-x(2-e) 



The function/z„(z)is illustrated in figure 8.2 for \ = \,d =0,n = 

 1,2,... 5. 

 The expected value and the variance of z are (Gumble 1958) 



Var„(z) = ^t I, «« 



Two versions of the model are shown here; in both cases it is 

 assumed that a learning process takes place in the research 

 system. Thus the experimenter can confine his search to samples 

 that are all better than the available technology. Formally, assume 



x>y, (8.9) 



and that y = 0. 



Version 1: In this version the scientist picks the largest observa- 

 tion in a set of trials and the technology associated with this ob- 

 servation replaces the currently practiced technology. 



