Simulating the Central Limit Theorem with Non Uniform Distribution in MATLAB

How can I simulate the Central Limit Theorem with a Non-Uniform Distribution function in MATLAB, including the "rand" function?

Answer 1

[Alfonso Nieto-Castanon]

 x = mean(rand(100,1000));
 hist(x);

Answer 2

[Image Analyst]

rand() generates numbers from a uniform distribution function, non "Non Uniform", though of course any given array generated from rand() won't have all the same values -- so not sure what you mean there.

Anyway, here's some code:

% Startup code.
clc; % Clear command window.
clearvars; % Get rid of variables from prior run of this m-file.
workspace; % Make sure the workspace panel with all the variables is showing.
close all;  % Close all imtool figures.
format long g;
format compact;
fontSize = 20;
% Define initial data
% m = ones(1, 10)  % Pick uniform, or random data to start with.
m = rand(1, 10)
iteration = 0;
again = 1;
while again == 1 && iteration < 12  
  % Plot it:
  plot(m, 'b.-', 'LineWidth', 2, 'MarkerSize', 30);
  grid on;
  caption = sprintf('This was after %d convolutions.', iteration);
  title(caption, 'FontSize', fontSize);
  
  % Set up figure properties:
  % Enlarge figure to full screen.
  set(gcf, 'Units', 'Normalized', 'OuterPosition', [0 0 1 1]);
  % Get rid of tool bar and pulldown menus that are along top of figure.
  set(gcf, 'Toolbar', 'none', 'Menu', 'none');
  % Give a name to the title bar.
  set(gcf, 'Name', 'Demo by ImageAnalyst', 'NumberTitle', 'Off')
  
  % Do the convolution for this iteration:
  m = conv(m,m, 'full');
  % Normalize it so we don't overflow.
  m = m / max(m);

  % See if user wants to do another iteration.
  promptMessage = sprintf('This was after %d convolutions.\nDo you want to do another iteration,\nor Cancel to abort processing?', iteration);
  titleBarCaption = 'Continue?';
  button = questdlg(promptMessage, titleBarCaption, 'Continue', 'Cancel', 'Continue');
  if strcmpi(button, 'Cancel')
    again = false; % Bail out
    break;
  end  
  iteration = iteration + 1;
end

[Debajyoti Datta]

Some corrections in the logic :

% Startup code.

clc; % Clear command window.

clearvars; % Get rid of variables from prior run of this m-file.

workspace; % Make sure the workspace panel with all the variables is showing.

close all; % Close all imtool figures.

format long g;

format compact;

fontSize = 20;

% Define initial data

m = 0.5*ones(1, 2); % Picking uniform pmf,

x = 0.5*ones(1, 2);

iteration = 1;

again = 1;

while again == 1 && iteration < 40

% Plot it:

plot(m);

grid on;

caption = sprintf('This was after %d tosses.', iteration);

title(caption, 'FontSize', fontSize);

% Set up figure properties:

% Enlarge figure to full screen.

set(gcf, 'Units', 'Normalized', 'OuterPosition', [0 0 1 1]);

% Get rid of tool bar and pulldown menus that are along top of figure.

set(gcf, 'Toolbar', 'none', 'Menu', 'none');

% Give a name to the title bar.

set(gcf, 'Name', 'Demo by ImageAnalyst', 'NumberTitle', 'Off')

% Do the convolution for this iteration:

m = conv(m,x); % normalization not needed as we had choosen a pmf initially

% See if user wants to do another iteration.

promptMessage = sprintf('This was after %d tosses.\n Do you want to do another iteration,\n or Cancel to abort processing?', iteration);

titleBarCaption = 'Continue?';

button = questdlg(promptMessage, titleBarCaption, 'Continue', 'Cancel', 'Continue');

if strcmpi(button, 'Cancel')

again = false; % Bail out

break;

end

iteration = iteration + 1;

end

[Image Analyst]

I would not call those "corrections". You're doing a different scenario. From Wikipedia Central Limit Theorem: "In probability theory, the central limit theorem (CLT) establishes that, in some situations, when independent random variables are added, their properly normalized sum tends toward a normal distribution (informally a "bell curve") even if the original variables themselves are not normally distributed."

Note that your code does not use random numbers at all. It does not even call rand() at all like the original poster asked you: "How can I simulate the Central Limit Theorem with a Non-Uniform Distribution function in MATLAB, including the "rand" function?"

[Pooja Jangra]

Debajyoti Datta thank u so much