PART II: MODEL DEFINITIONS 



^asic Statistical Notation 



10. The models used here are all defined* basically as probability 

 density functions p of specified crest- to- trough wave height H in the form 

 p(H) , having dimensions of inverse length. When multiplied by an incremental 

 range of height dH , the resulting expression p(H) dH is dimensionless and 

 gives the probability that a randomly chosen height A lies in the range 

 between H and H + dH . This relationship is written as 



p(H) dH = Prob[H < fi < H + dH] (1) 



where Prob means the probability that. 



11. If Equation 1 is integrated over all increments of height between 

 zero and some specified height H , one obtains the cumulative distribution 

 function (cdf) denoted as P(H) , which is the probability that a randomly 

 chosen height A is less than a specified height H . This expression takes 

 the form 



'"' ' II 



P(H) = I p(x) dx 



= Prob[fl < H] (2) 



where x is simply the dummy integration variable. In Equation 2, it can be 

 seen that F(a>) = 1 because all wave heights are less than infinitely high. 

 One can take the complement of the cumulative probability to obtain the 

 exceedence probability function Q(H) , which is the probability that a 

 randomly chosen height fl is greater than a specified height H . Formally, 

 Q(H) is found from the probability density function by 



Q(H) = r p(x) dx 



= Prob [ft > H] (3) 



For convenience, symbols and abbreviations are listed in the Notation 

 (Appendix A) . 



