How to generate a 1D random walk of 100 sample paths, along with sample mean and variance?

So I need to generate a figure that represents 100 random walks (1D) and the mean, square root of variance with it. It should look like the example above, with much more lines to represent the walks. How can I code this problem?

I've tried the following, but it only produced the mean and variance. I'm not very good at MATLAB, so any kind of specific help would be welcome!

close all;

clc;

n = 100; % number of steps

nwalk = 10000; % number of walks

X = zeros(nwalk,n);

X2avg = zeros(n);

% r = rand();

for j = 1:nwalk

X(j,1) = 0; % initial position

r = rand(); % getting random number to decide step left or step right

for i = 2:n

if r < 0.5

X(j,i) = X(j,i-1)+1; % step to the right

elseif r > 0.5

X(j,i) = X(j,i-1)-1; % step to the left

end

X2avg(i) = X2avg(i) + X(j,i)^.2; % accumulating squared displacement

end

end

for i = 2:n % normalizing the squared displacement

X2avg(i) = X2avg(i)/nwalk;

end

plot(1:n,X2avg);

xlabel('Step Number (Time)');

ylabel('<x^2>');

title('1-Dimensional Random Walk');

Answer 1

[Benjamin Thompson]

Try this to take advantage of vectorization in MATLAB. I may be confused about how you define "walks" and "steps", and whether you wanted the mean at each step or at the end. And if you want the first row of X to be the initial value zero that is a pretty small change.

close all;

clc;

n = 100; % number of walks

nwalk = 1000; % number of steps

X2 = rand(nwalk,n);

X3 = zeros(nwalk, n);

% Use index matrices to populate X3 with +1 and -1 as needed

Iright = X2 < 0.5;

Ileft = X2 > 0.5;

X3(Iright) = 1;

X3(Ileft) = -1;

% Sum along each column to create X

X = cumsum(X3, 1);

% Get the average of each row

Xavg = mean(X, 2);

% Get the variance of each row

Xvar = var(X, [], 2);

figure, plot(1:nwalk, X);

hold on;

plot(1:nwalk, Xavg, 'r--', 'LineWidth', 5)

plot(1:nwalk, Xvar, 'b--', 'LineWidth', 5)

hold off;

[jae-hun]

it was a lot of help, thanks!

Answer 2

[Image Analyst]

I've done lots of random walk demos. You might want to run the attached demos and see if any of them can be adapted or used.