P(x) = / p(.x) dx 



where 



p{x) is the probabiHty density of x 



X is the variate 



^ is the mean value of the variate 



0-2 is the variance of the population 

 Thus the two parameters ^ and o- define the dis- 

 tribution completely. 



The logarithmically normal or log-normal dis- 

 tribution is a normal distribution of the logarithm 

 of X, and 



I -(\o%x - »)' 



^^ ^ xff V27r 



where u is the mean value of log x and a is the 

 standard deviation of log x. 



The Rayleigh distribution is defined as follows 



p{x) = -^e-^'/'E, X > 



where £ is the mean value of the squares of x; thus 



Table 2 USCGC Unimak — Computation of Confi- 

 dence Limits for Data Shown in Figs. 2 and 3. 



(Pitch angle, sea state 5, head seas, ship speed 7V2 knots.) 



Variation in 



Pilch AnglE 



d!8 



Piobability Density 

 (per ilegree) 



P 



Probability 



Slandaril Beviation 

 at Fraclile 



Upper Limit 

 oti 



Lower Limit 

 oil 



"^V „ 



67% Confiden 



oe Liniits, det 



O.S 



0.0210 



0.050 



0.2166 



0.72 



0.28 



1 



0.0113 



0.0206 



0.2239 



1.22 



0.78 



2 



0.0775 



0.0809 



0.2290 



2.23 



1.77 



3 



0.1046 



0.1729 



0.2353 



3.24 



2.76 



4 



0.1193 



0J923 



0.2482 



4.25 



3.75 



5 



0.1244 



0.4097 



0.2573 



5.26 



4.74 



e 



0.1190 



0.5294 



0.2730 



6J7 



5.73 



7 



O.lOSl 



0.6441 



0.2965 



7J0 



S.70 



8 



0.0875 



0.7407 



0.3260 



8.33 



7.67 



9 



0.0G8B 



0.8188 



0JG44 



9.36 



1.63 



10 



0.0511 



0.8786 



0.4160 



10.42 



9.58 



11 



0.03 62 



0.9221 



0.4820 



11.48 



10.52 



i; 



0.0243 



0.9519 



0.5732 



12.57 



11.53 



14 



0.0118 



0.9784 



0.8019 



14.80 



13.20 



IE 



0.0052 



0.9911 



1.1756 



17.18 



14.82 



Th= 67 fc 



cat conlldcuc ll,nlL. •,<: cWc. b) 



.♦irilheOOpercen! 



byx ti.6S<7-. 



he 9S l-reenl 



by 1 + 1.96 cr 



• la] tbe 99 perceal by c + 2.S8 o. 









The valis 



r the standard deviation cnmpuled 



e ite table aasomee 



hat there are 



etTc»a mtro- 



doced In (Iw 





the Utter orroea 



l.rou,l,l,<j' 



= 0.30. Thua the conlldence llnut 



a would be broader tb 



«n elven In the table if an | 



■"""""" 



'"""""""" ''"'— 



-— "^^ 



♦?' 





Tesl Conditions; Sea State 5, Head Seas, Stiip Speed 7 !z knots 

 Significant Wave Heigtit21 ft. 



Variation in Pitcti Angle - degrees 



Fig. 2 USCGC Unimak, Distribution of Variation in Pitch 

 Angle — Sample 1 



(This sample is representative of 129 samples that were analyzed.) 



the single parameter E defines the distribution. 

 It is possible to represent the distributions cor- 

 responding to different values of £ by a single 

 distribution, employing a change of variables. 

 Thus let 



Ve 



then -^/e p{x) = 2ve- 



Integrating to obtain the cumulative probability 

 VE X'' p{x) dx= y/E P{Xi) = V£ [1 - e-"'] 



P{xi) = [1 - e-^'] 



and 



