How would I start this problem?

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Gaëtan Poirier 2017-10-8

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https://ww2.mathworks.cn/matlabcentral/answers/360274-how-would-i-start-this-problem

评论： Image Analyst 2017-10-9

We consider again the one-dimensional version of the drunkard problem. This time he starts out at position x (e.g., x=10), and takes unit steps left or right with equal probability. Suppose at 0 and a (a>x>0, e.g., a=30) are two bars. When he reaches either one, he is trapped forever.

We want to determine the average time t (the mean number of steps) before the drunkard is trapped. This is a diffusion problem and the 'time' t is called the first passage time. Do a simulation to compute t. (Again note that only the average of t has a meaning and it is that average that we want to compute.)

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采纳的回答

Image Analyst 2017-10-8

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https://ww2.mathworks.cn/matlabcentral/answers/360274-how-would-i-start-this-problem#answer_284803

See my attached collection of random walk programs/demos I've written. Adapt as needed.

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Image Analyst 2017-10-8

在 MATLAB Online 中打开

Where exactly does it ask the user for the number of steps? It doesn't. stepNumber is a loop iterator. maxSteps in a parameter - you can set it very high to make sure that you never run out of steps before you reach the boundary. I've done that for you. See if you can finish it yourself. I mean, you got to do something for your homework, right?

% Monte Carlo experiment for random walk
clc;    % Clear the command window.
close all;  % Close all figures (except those of imtool.)
clear;  % Erase all existing variables. Or clearvars if you want.
workspace;  % Make sure the workspace panel is showing.
format long g;
format compact;
fontSize = 20;
% maxStepLength = max length of one step, from minus this to + this.
maxStepLength = 1;
% Define Monte Carlo parameters
numExperiments = 3000;
maxSteps = 10000;
% Initialize data.
x = zeros(maxSteps, numExperiments);
% Since there is only 1 row it should only
%record the fist time it crosses the boundary
out_of_bounds=zeros(1, numExperiments);
% Define a and b.
a = 0;
B = 30; % Boundary is 30 = max allowed distance before we go to next particle
% Better is to use inputdlg() to ask the user for these numbers.
for experiment = 1 : numExperiments
  for stepNumber = 2: maxSteps
    % Find the location of the next point
    deltaX = maxStepLength * (2 * rand(1) - 1);
    x(stepNumber, experiment) = x(stepNumber-1, experiment) + deltaX;
    if x(stepNumber, experiment) < a || x(stepNumber, experiment) >= B
      % Then the particle is out of bounds.
      % Record the number of steps 
      % it took to get out of bounds.
      out_of_bounds(experiment) = stepNumber;
      break; % Quit tracking this particle.
    end  
  end
end
maxNumberOfSteps = max(out_of_bounds);
% DONE!  Now make fancy plots.
% Plot the number of steps to get out of bounds.
subplot(1,2,1);
plot(out_of_bounds, 'b-', 'LineWidth', 2);
xlim([0, numExperiments]);
caption = sprintf('Steps until out of bounds for %d experiments', numExperiments);
title(caption, 'FontSize', fontSize);
xlabel('Experiment Number', 'FontSize', fontSize);
ylabel('Number of Steps Until Out of Bounds', 'FontSize', fontSize);
grid on;
subplot(1,2,2);
histogram(out_of_bounds);
grid on;
caption = sprintf('Distribution of the Number of Steps until out of bounds');
title(caption, 'FontSize', fontSize);
xlabel('Number of Steps', 'FontSize', fontSize);
ylabel('Count of Number of Experiments', 'FontSize', fontSize);
% Enlarge figure to full screen.
set(gcf, 'Units', 'Normalized', 'OuterPosition', [0 0 1 1]);
% Give a name to the title bar.
set(gcf, 'Name', 'Demo by ImageAnalyst', 'NumberTitle', 'Off')

Gaëtan Poirier 2017-10-9

在 MATLAB Online 中打开

So here it is so far:

for N=1:10000 %number of "walkers"
    x(1)=10; %starting position
out_of_bounds = stepNumber;
for stepNumber=20;  %number of steps limited to each "walker"
    g(n)=rand(); %random number
    if g(n)<0.5
        x(n+1)=x(n)-1; %if random number is less than, go left
    else x(n+1)=x(n)+1; %else go right
    end
    if x(n+1)>29||x(n+1)<1 %second bar––if "walker" reaches 30, the loop terminates
        break
    else continue % else it continues 
    end
end 
end
histogram(out_of_bounds);