Question

Generate 500 samples of 10 numbers according to the cdf -exp(-25x/1132) using matlab, please include the code.

Answer 1

We can generate random numbers given as CDF using Inverse Transform Samping:

The inverse transform sampling method works as follows:

1. Generate a random number ${\displaystyle u}$ from the standard uniform distribution in the interval ${\displaystyle [0,1]}$ , e.g. from ${\displaystyle U\sim \mathrm {Unif} [0,1].}$
2. Find the inverse of the desired CDF, ${\displaystyle F_{X}^{-1}(x)}$ e.g. .
3. Compute ${\displaystyle X=F_{X}^{-1}(u)}$ . The computed random variable ${\displaystyle X}$ has distribution ${\displaystyle F_{X}(x)}$ .

Please find required MATLAB code along with necessary details in comments below:

clear all, close all, clc

% Plot the cumulative distribution function over the range 0-20.
syms x
f(x)=-exp(-25.*x./1132);

x1 = 0:20;
distCDF = f(x1);

subplot(2,1,1);
plot(x1, distCDF, 'LineWidth', 3);
title('True CDF');

sample=500; % number of samples to be generated
num_random=10; % number of points generated in each sample

% If you want to see graphically that given CDF of generated random numbers
% actually approximate true CDF, genrate larger number of random number in
% each sample by uncommenting lines below
% sample=10;
% num_random=500;

for i=1:sample
% Generate numberOfRandoms uniformly distributed random numbers.
unif_rn = rand(num_random, 1);
% Invert the CDF of the given CDF to get a function that can
% generate random numbers drawn from a required distribution,
% given numbers drawn from a uniform distribution.
inv_f = finverse(f); % calculate F^-1
X=round(double(inv_f(unif_rn))); % calculate F^-1(u)
Rand_num(:,i)=X; % store random number
  
% plot the ith sample cdf
subplot(2,1,2);
hold on
pause(0.01)
cdfplot(X)
xlim([0 20])
title('Numerical CDF')
end

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To plot the numerical CDF of a data, I have used cdfplot() available at mathworks, which I have included below for your ease of reference:

function cdfplot(X)
% cdfplot(X)
% displays a plot of the Empirical Cumulative Distribution Function
% (CDF) of the input array X in the current figure. The empirical
% CDF y=F(x) is defined as the proportion of X values less than or equal to x.
% If input X is a matrix, then cdfplot(X) parses it to the vector and
% displays CDF of all values.
%
% EXAMPLE:
% figure;
% cdfplot(randn(1,100));
% hold on;
% cdfplot(-log(1-rand(1,100)));
% cdfplot(sqrt(randn(1,100).^2 + randn(1,100).^2))
% legend('Normal(0,1) CDF', 'Exponential(1) CDF', 'Rayleigh(1) CDF', 4)
% Version 1.0
% Alex Podgaetsky, September 2003
% [email protected]
%
% Revisions:
% Version 1.0 - initial version
tmp = sort(reshape(X,prod(size(X)),1));
Xplot = reshape([tmp tmp].',2*length(tmp),1);
tmp = [1:length(X)].'/length(X);
Yplot = reshape([tmp tmp].',2*length(tmp),1);
Yplot = [0; Yplot(1:(end-1))];
figure(gcf);
hp = plot(Xplot, Yplot);
ColOrd = get(gca, 'ColorOrder');
ord = mod(length(get(gca,'Children')), size(ColOrd,1));
set(hp, 'Color', ColOrd((ord==0) + (ord>0)*ord, :));
if ~ishold
xlabel('X', 'FontWeight','b','FontSize',12);
ylabel('F(X)', 'FontWeight','b','FontSize',12);
title('Empirical CDF', 'FontWeight','b','FontSize',12);
grid on;
end

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You can verify the nature of random numbers by plotting the numerical cdf after generating a lot of random numbers per samples. For 500 points generated in each of the 10 samples, we obtain two cdf as :

---------------------------------------- SCREENSHOT OF CODE

The matrix Rand_num contains the generated 500 samples of 10 random numbers each.