Estimating the value of ?. The following figure suggests how to estimate the value of ? with a simulation. In the figure, a circle with area equal to ?/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 ?/4. We can therefore estimate the value of ?/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 ?/4 ? 0.79. From this we conclude that ? ? 4(0.79) = 3.16. This exercise presents a simulation experiment that is designed to estimate the value of ? by generating 1000 points

Question

in the unit square.





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

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

c. Each point is inside the circle if its distance from the center (0.5, 0.5) is less than 0.5. For each pair , determine whether its distance from the center is less than 0.5. This can be done by computing the value , 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 ??

Answer 1

(a-d) Answers will vary.