Drawing random samples from a given Probability Distribution is excellently explained by Carson Chow at https://sciencehouse.wordpress.com/2015/06/20/sampling-from-a-probability-distribution/.
You will need the inverse of the Cumulative Distribution Function. The Lomax CDF is given by Wikipedia as
The inverse function gives the x value corresponding to a given cumulative probability r as
The code below shows how to draw samples from the Lomax PDF. The resulting distribution is compared to the analytical PDF for verification.
% Lomax PDF parameters:
m = 1;
a = 2;
% Draw random samples from uniform distribution in range 0 to 1:
n_samples = 100000;
r = rand(n_samples,1);
% Find the CDF values corresponding to the samples
x = m*((1 - r).^(-1/a)-1); % Inverse Lomax CDF
% Calculate histogram with bin width 0.1:
binwidth = 0.1;
bins = 0:0.1:5;
N = histcounts(x,bins); % Number of x values in each bin
f = N/n_samples/binwidth; % Observed frequency per x unit
bin_centres = (bins(1:end-1)+bins(2:end))/2;''
figure;
bar(bin_centres,f)
% Compare with analytic pdf
x = 0.05:0.1:4.95;
p = a/m*(1+x/m).^-(a+1); %Lomax PDF
hold on;
h = plot(x,p,'ok'); % Plot the PDF using circles
set(h,'MarkerFaceColor','w')
hold off
str = sprintf('Lomax PDF, m = %3.1f, a = %3.1f',m,a);
title(str)
legend('Sampled','Analytical')
