Question

python / visual studio

Problem 1: Random Walk A random walk is a stochastic process. A stochastic process is a series of values that are not determined functionally, but probabilistically. The random walk is supposed to describe an inebriated person who, starting from the bar, intends to walk home, but because of intoxication instead randomly takes single steps either forward or backward, left or right. The person has no memory of any steps taken, so theoretically, the person shouldn't move too far from where he or she starts. Random walks are used to model many phenomena, like the size of the web or changes in financial instruments. We will model a 2D random walk with two arrays x and y where x represents moving left or right and y represents forward or backward. The index i will represent the step and xy[ will represent the location at step i. So, for i-0, we have x[0],y[o] (starting place). Using random we choose from the list [1,2,3,4]. If the value is one then we move right from the previous x position 1x[i-xi-11 If the value is two, then we move left from the previous x position: 1 x[i] x[i-1] -1 = If the value is three, we move up from the previous y position: 1 x[i]-x[i-1] 2 y[i]-yi-11 And when the value is four, we move down from the previous y position Here is another way to describe this step(0)0 eft step(n 1) step/(n)right step/n -1) up step(n -1) down step(n 1) Stepl Start Step1 Step 3 Step 2 Step 4 Step 5 Figure 1: The first few steps of a random walk 1 import numpy as np 2 import matplotlib.pyplot as plt 3 import random as rn 5 6 #translates a random int into a step along the random walk 7 #parameters: int i for the step index, numpy array x for tracking the left. /right location at index i, 8 #numpy array y for tracking the forward/backward location at index i 9 #return: none 10 def step(x.y,i) direction rn.randint(1,4) #70 D0 : implement this function 12 13 15 def graphit(x,y,n) 16 plt . title( "Random {0} Walk1nLast Location {1},{2)".format(n, int(x(n- 17 18 19 1]),int(yn))) plt.plot(x.y) plt.plot([x[1],x y11-10,y[1+10], "b-" plt.plot([x[1]-10,x[1]+10], [y1].y "b-") Page 4 Assignment No10 Maps and Matrices and Correlation plt.plot([x[n-1]-10,x[n-11+10], [y[n-1.y[n-1]], "r-") plt.plot([x[n-1].x[n-1],[y[n-11-10,y[n-1]+10], "r- plt.savefig("rand_walk"+str(n)+" .png", bbox_inches-"tight" ,dpi-600) plt.show() 20 21 23 25 26 nint (input("Number of steps:")) 27 28 xnp.zeros(n) 29 y - np.zeros (n) 30 31 for i in range(1,n) 32 step(x,y,i) 34 graphit(x,y,n) Session Output Number of Steps: 100000 Random 100000 Walk Last Location 236,27 150 100 50 -100 100-50 50 00 150 200 250 Figure 2: The blue cross is where we started and the red cross is where we ended Programming Problem 1: Random Walk Complete the step function. Experiment with different numbers of steps. Do you see any number of steps that seems to always move significantly away from the start? Put your code into a new module named randomwalk.py

Answer 1

On experimenting over different number of steps, it is being observed that even for a large number of steps, the maximum distance in x-direction is +/- 11 and same goes for y direction over multiple runs. This may vary but not much.

The code with step function completed is present below

import numpy as np
import random as rn
import matplotlib.pyplot as plt

def step (x,y,i):
    direction = rn.randint(1,4)
    if (direction == 1) :
        x[i] = x[i-1] + 1
        y[i] = y[i-1]
    elif (direction == 2) :
        x[i] = x[i-1] - 1
        y[i] = y[i-1]
    elif (direction == 2) :
        x[i] = x[i-1]
        y[i] = y[i-1] + 1
    elif (direction == 3) :
        x[i] = x[i-1]
        y[i] = y[i-1] - 1

def graphit(x,y,n):
    plt.title("Random {0} Walk\nLast Location {1},{2}".format(n,
        int(x[n-1]),int(y[n-1])) )
    plt.plot(x,y)
    plt.plot([x[1],x[1]], [y[1]-10, y[1]+10], "b-")
    plt.plot([x[1]-10, x[1]+10], [y[1], y[1]], "b-")
    plt.plot([x[n-1]-10,x[n-1]+10], [y[n-1], y[n-1]], "r-")
    plt.plot([x[n-1], x[n-1]], [y[n-1]-10, y[n-1]+10], "r-")
    plt.savefig("rand_walk"+ str(n)+".png", bbox_inches="tight",dpi=600)
    plt.show()

n = int(input("Number of steps"))

x = np.zeros(n)
y = np.zeros(n)

for i in range(1,n):
    step(x,y,i)

graphit(x,y,n)