How to integrate a shifted lognormal distributed random variable
Dear all, I'm quite new to Matlab and struggeling to integrate a continuous random variable by two parts. In the beginning I assume W=1+w is lognormally distributed with mean = 1 and standard deviation of 0.05. The mean is specifically chosen such that w has a zero mean and has a support of [-1, Inf). I tried to integrate a hard coded version of a lognormal density. This doesn't give me a useful result since log is not defined for -1 yet I tried the same procedure with W the result still doesn't make sense. % 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.
Kshitij Singh answered .
2025-11-20
For the edited version of your post,
a=-1+mu; b=mu; sigma=sqrt(sigmasq); Sneg = logncdf(b,mu,sigma) - logncdf(a,mu,sigma); Spos = logncdf(b,mu,sigma,'upper');