Suppose that 2000 points are selected independently and at random from the unit square {(x, y) : 0 ≤ x < 1, 0 ≤ y < 1}. Let W equal the number of points that fall into A = {(x, y) : x2 + y2 < 1}.

(a) How is W distributed?

(b) Give the mean, variance, and standard deviation ofW.

(c) What is the expected value of W/500?

(d) Use the computer to select 2000 pairs of random numbers. Determine the value of W and use that value to find an estimate for π. (Of course, we know the real value of π, and more will be said about estimation later in this text.)

(e) How could you extend part (d) to estimate the volume V = (4/3)π of a ball of radius 1 in 3-space?

(f) How could you extend these techniques to estimate the “volume” of a ball of radius 1 in n-space?

Answer

Step 1 of 8

The condition is x2+y2<1

P is the point lie in the shaded region

p=