Question

This problem is also a Monte Carlo simulation, but this time in the continuous domain: must use the following fact: a circle inscribed in a unit square

has as radius of 0.5 and an area of ?∗(0.52)=?4.π∗(0.52)=π4.

Therefore, if you generate num_trials random points in the unit square, and count how many land inside the circle, you can calculate an approximation of ?

For this problem, you must create code in python

(A) Draw the diagram of the unit square with inscribed circle and 500 random points, and calculate the value of ?

Answer 1

SOLUTION:

Given That data

==>a Monte Carlo simulation, but this time in the continuous domain: must use the following fact: a circle inscribed in a unit square has as radius of 0.5.

==>an area of ?∗(0.52)=?4.π∗(0.52)=π4

Therefore, if you generate number of_trials random points in the unit square, and count how many land inside the circle, you can calculate an approximation of ? For this problem, you must create code in python.

So

A)

The R code for drawing the diagram and calculating the value of PI is given below. (500 random points are generated).

from numpy import random
import math
import matplotlib.pyplot as plt

x =-0.5+random.random_sample(500)
y = -0.5+random.random_sample(500)
x1 = x[x*x+y*y<0.25]
y1 = y[x*x+y*y<0.25]

circle = plt.Circle((0, 0),fill=None, radius=0.5)
points = [[-0.5, -0.5], [0.5, -0.5], [0.5, 0.5], [-0.5, 0.5],[-0.5,-0.5]]
line = plt.Polygon(points, fill=None, edgecolor='r')
plt.gca().add_patch(circle)
plt.gca().add_patch(line)
plt.scatter(x1,y1)
plt.axis('scaled')
pi = 4*len(x1)/500
plt.title("n=500, $\pi$=%0.4f" % pi)
plt.show()

The diagram given below.

The value of PI is==> .