excess length distribution of log normal

lognormal.jl

using Distributions
using SpecialFunctions
using QuadGK
using Plots

function eqcdf(d::LogNormal, x)
    mu, sigma = params(d)
    return 0.5*(1+x*erfc((log(x) - mu)/(sigma*sqrt(2)))*exp(-(mu + (sigma^2)/2)) - erf((mu + sigma^2 - log(x))/(sigma*sqrt(2))))
end

function rmst(d::UnivariateDistribution, x::Real)
    return mean(d)*eqcdf(d,x)
end


function genericeqcdf(y, d::UnivariateDistribution)
    v,_ = quadgk(x -> ccdf(d,x),0,y)
    return v/mean(d)
end

plot(x-> eqcdf(LogNormal(2.0,0.2),x),0,100,legend=false)
plot!(x-> genericeqcdf(x, LogNormal(2.0,0.2)),0,100,legend=false)


plot(x-> eqcdf(LogNormal(0,0.5),x),0,10,legend=false)
plot!(x-> genericeqcdf(x, LogNormal(0,0.5)),0,10,legend=false)


plot(x-> eqcdf(LogNormal(-1,1.5),x),0,100,legend=false)
plot!(x-> genericeqcdf(x, LogNormal(-1,1.5)),0,100,legend=false)