Estimating the value of π. The following figure suggests how to estimate the value of π with a simulation. In the figure, a circle with area equal toπ/4 is inscribed in a square whose area is equal to 1. One hundred points have been randomly chosen from within the square. The probability that a point is inside the circle is equal to the fraction of the area of the square that is taken up by the circle, which is π/4. We can therefore estimate the value of π/4 by counting the number of points inside the circle, which is 79, and dividing by the total number of points, which is 100, to obtain the estimate π/4 ≈ 0.79. From this we conclude that π ≈ 4(0.79) = 3.16. This exercise presents a simulation experiment that is designed to estimate the value of π by generating 1000 points in the unit square.

 

a. Generate 1000 x coordinates X∗1, . . . , X∗1000. Use the uniform distribution with minimum value 0 and maximum value 1.

b. Generate 1000 y coordinates Y∗1 , . . . , Y∗1000, again using the uniform distribution with minimum value 0 and maximum value 1.

c. Each point (X∗i , Y∗i ) is inside the circle if its distance from the center (0.5, 0.5) is less than 0.5. For each pair (X∗i , Y∗i ), determine whether its distance from the center is less than 0.5. This can be done by computing the value (X∗i −0.5)2+(Y∗i −0.5)2, which is the squared distance, and determining whether it is less than 0.25.

d. How  many of the points are inside the circle? What is your estimate of π? (Note:With only 1000 points, it is not unlikely for your estimate to be off by as much as 0.05 or more. A simulation with

10,000 or 100,000 points is much m