Characters of Log-normal Distribution and Probability Density Function whose Spectrum has Many Peaks
Daikaku Manabe · Journal of the Society of Naval Architects of Japan · 1961
According to the time-series theory, the probability function can be expressed as followsp (x) =1/√2π {εex2/2ε2+√1-ε2xe-x2/2∫-∞√1-ε2/εxe-u2/2du} (Theory of maxima, Cartwright and Longuet-Higgins) xe-x2+a2/2Io (xa) (Theory of envelope, Rice) Both extremity of ε=1 and a= ∞, correspond to Gauss-Laplace's distribution, namelyP (x)=1/√2πe-x2/2;1/√2πe- (x-a) 2/2Now. if we change independent variable from x to Z, whereZ= log x, so probability function becomes : P (Z) =2/√2πexp {Z-e2z/2} _??_2/√2πee-2Z2/e;1/√2πexp {Z- (eZ-a) 2/2} _??_a/√2πe-a2/2 (Z-loga) 2, which are summarized macroscopically as follows : P (Z) =1/√2πσexp- (Z-Z) 2/2σ2, where σ is dispersion and Z is total mean of Z.This is log-normal distribution, and we find that this annexes both theory of maxima and envelope.Spectrum has sometimes many peaks. Consequently in the frequency distribution of oscillation appear also many peaks. If we express irregular motion asξ=∑Nn=1Cncos (ωnt+ψn) and letting R be the amplitude of envelope, then probability density isP (R) = R∫o∞r ΠNn=1Jo (Cnr) Jo (Rr) drwhose number of peaks is about equal to N.