Lukas Huppertz asked . 2022-07-08

How to integrate a shifted lognormal distributed random variable

% Generating random variable W

W = lognrnd(0, 0.05, 1, 2000);

% Shifting W to have lognormal random variable w with zero mean and
% has a support of [-1, Inf)

w = W - 1;

% Transforming omega to s_w = a*w with a = 9.0083

s_w = 9.0083*w;

% My aim is now to integrate s_w from -1 to 0 and from 0 to Inf
% Since I can only integrate a continous random variable numerically I
% tried to hard code the density and use the integral function

%mu = mean(s_w);
%sigmasq = var(s_w);

%s_w = @(x) exp(-(log(x) - mu).^2./(2.*sigmasq)./(x.*sqrt(2.*pi.*sigmasq)));

%S_neg = integral(@(x)s_w(x), -1, 0);
%S_pos = integral(@(x)s_w(x), 0, Inf);

%d = -S_neg/S_pos;

% This does obviously not work because of the support of log yet I actually
% need to integrate

% Even my approach of tranforming W to s(W) doesn't work since both values
% are close to zero which they shouldn't be according to a density plot

s_W = 9.0083*W;

mu = mean(s_W);
sigmasq = var(s_W);

s_W = @(x) exp(-(log(x) - mu).^2./(2.*sigmasq)./(x.*sqrt(2.*pi.*sigmasq)));

S_neg = integral(@(x)s_W(x), (-1+mu), mu);
S_pos = integral(@(x)s_W(x), mu, Inf);

d = -S_neg/S_pos;

I hope I gave all the necessary information.

integration , integral , lognormal distribu... , AI, Data Science, and Statistics