Chapter 2.4, Problem 7E

Suppose that 2000 points are selected independently and at random from the unit square $\left\{\left(x,y\right):0\le x<1,0\le y<1\right\}$.

Let W equal the number of points that fall into $A=\left\{\left(xy\right):x^2+y^2<1\right\}$.

(a) How is W distributed?

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

(c) What is the expected value of $\frac{w}{500}$?

(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 n. (Of course, we know the real value of r, and more will be said about estimation later on in this text.)

(e) How could you extend part (d) to estimate the volume $V=\left(\frac{4}{3}\right)\pi$ of a ball of radius 1 in 3-space?

(f) How could you extend these techniques to estimate the b6volumet of a ball of radius 1 in n-space? NOTE: The answer given in terms of the $\tau$ function (see Equation 3.2-2) is $V_n=\frac{{\pi }^{\frac{n}{2}}}{\tau \left(\frac{n}{2}+1\right)}$.