Trying to graph a cumulative distributive function with erf function, lost.

%Sn = X1 + X2 ... + Xn

%Xi's are independent random variables 

%uniform on the interval [-a,a]

a1 = 1;
a2 = 5;
a3 = 10;

%σ^2 = o^2 and is the variance of Xi

%μ = u which is expectation

n = 4;

x = -a1:0.1:a1;

for i = -a1:a1
    o = (i - mean(x)) / (n-1);
    i = i + 1;
end

y = (1/2)*(1+erf((x-n*mean(x))/sqrt(n*o^2)));
plot(x,y)
grid on
title('CDF of normal distribution with \mu = 0 and \sigma = 1')
xlabel('x')
ylabel('CDF')

Trying to make a graph that looks something like the one below, although I am lost. Solving for the expectation and variance is difficult since it is supposed to be graphed and it isn't a simple math problem. I tried using the mean function and using a for loop to calculate variance with each x value. However I keep running into roadblocks.

Answer 1

[Sudarsanan A K]

Hello Kylenino,

I understand that you are looking for plotting the Cumulative Distribution Function (CDF) of a random variable , that is formed by taking the sum of n Independent and Identically Distributed (IID) random variables which are uniformly distributed in the interval .

I also note that you want to consider different values of n (say n = [4, 20, 50]), and plot the resulting CDF in the same figure.

Here is an example how you could achieve this:

% Parameters
mu = 0;
sigma = 1;
a = 2;

% Values of n
n_values = [4, 20, 50];

% Generate x values for plotting
x = linspace(-a, a, 5000); 

% Plotting
figure('Position', [100, 100, 1000, 400]); % Adjust the figure size here
hold on;

% Define line styles, colors, and marker styles
line_styles = {'--', ':', '-'};
colors = {'b', 'r', 'g'};
%marker_styles = {'o', 's', '*'};

for idx = 1:numel(n_values)
    n = n_values(idx);

    % Generate n uniform random variables
    X = 2 * a * rand(n, 1) - a;

    % Calculate the sum of the uniform random variables
    S_n = cumsum(X);

    % Calculate the mean of the uniform random variables
    mean_X = mean(X);

    % Calculate the random variable Z_n
    Z_n = (S_n - n * mean_X) / (sigma * sqrt(n));

    % Calculate the CDF for Z_n
    cdf = zeros(size(x));
    for i = 1:length(x)
        cdf(i) = sum(Z_n <= x(i)) / length(Z_n);
    end

    % Plot the CDF with customized line styles, colors, and marker styles
    plot(x, cdf, line_styles{idx}, 'Color', colors{idx}, 'LineWidth', 1.5);
end

% Add labels and title
xlabel('$x$', 'Interpreter', 'latex');
ylabel('$F_{Z_n}(z)$', 'Interpreter', 'latex');
title('CDF of $Z_n$', 'Interpreter', 'latex');

% Add a legend
legend('n = 4', 'n = 20', 'n = 50');

% Adjust the axis limits
xlim([-3, 3]); % Modify the x-axis limits here
ylim([0, 1]);

% Set grid lines
grid on;

% Set background color
set(gca, 'Color', [0.95, 0.95, 0.95]);

% Set font size
set(gca, 'FontSize', 12);

% Set line width of legend
set(findobj(gca, 'Type', 'Line'), 'LineWidth', 2);

% Hold off to end plotting
hold off;

Note that you can use the "erf()" function to get a continuous CDF plot (removing the staircase nature) by replacing the code in the calculation of CDF of "Z_n" with the following command:

% Calculate the CDF for Z_n
cdf = 0.5 * (1 + erf((x - mean(Z_n)) / std(Z_n)));

To know the "erf()" function better, you can additionally refer to the MathWorks documentation in the following link:

https://mathworks.com/help/symbolic/erf.html

I hope this helps!