対数正規分布の特性

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According to the time-series theory, the probability function can be expressed as follows・[numerical formula]Both extremity of ε=1 and a=∞, correspond to Gauss-Laplace's distribution, namely [numerical formula]Now. if we change independent variable from x to Z, where Z=logx, so probability function becomes : [numerical formula]which are summarized macroscopically as follows : P(Z)=1/(√<2πσ>)exp-((Z-Z^^-)^2)/(2σ^2), 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 as ξ=Σ^^N__<n=1> C_ncos(ω_nt+ψ_n) and letting R be the amplitude of envelope, then probability density is P(R)=R∫^^∞__0rΠ^^N__<n=1>J_0(C_nr) J_0 (Rr) dr whose number of peaks is about equal to N.
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