Hi Raj,
MODIFIED
Since the pearson type iii is a gamma distribution with a shifted and scaled variable, that means you can use the Matlab gamrnd function to produce random draws and proceed accordingly.
The pearson type iii distribution depends on three parameters. This answer assumes that the task is to create random draws from a distribution with three known parameters. For parameters a,b,p and for the log distribution, the pdf is 'pdfa' below. If the task is fitting the parameters from a known data set, then this answer does not apply.
Instead of gamrnd, you can use the inverse cdf method as you mentioned, which is shown in fig. 2 below. As is almost always the case with that method, there are some problems producing exteme outliers when the cdf is close to 1. But it works reasonably well.
% pearson type iii in log variable:
% pdf = (1/b)*logxs.^(p-1).*exp(-logxs)/gamma(p)
% where
% logxs = (log(x)-a)/b is a shifted and scaled log(x),
% with log(x) >= a
n = 1e5; % number of random draws
p = 3;
a = 4.7;
b = 2.4;
% constuct pdfa, the analytic pdf
logx = a:.01:50;
logxs = (logx-a)/b; % shifted and scaled log(x) [1]
pdfa = (1/b)*gampdf(logxs,p,1);
% random draws using gamrnd
randlogxs = gamrnd(p,1,1,n); % shifted and scaled log(x)
randlogx = randlogxs*b +a; % undo [1] for pearson type iii
figure(1)
histogram(randlogx,'Normalization','pdf'); grid on
hold on
plot(logx,pdfa)
hold off
% random draws using inverse of cdf
cdfa = cumtrapz(logx,pdfa);
r = rand(1,n);
r(r>max(cdfa)) = [];
randlogx = spline(cdfa,logx,r); % or randlogx = interp1(cdfa,logx,r,'spline');
figure(2)
histogram(randlogx,'Normalization','pdf'); grid on
hold on
plot(logx,pdfa)
hold off
