The volume of a region in n-dimensional Euclidean space R n is the integral of 1 over that region. The unit ball in R n is {(x1, . . . , xn) : x 2 1 + + x 2 n 1}, the ball of radius 1 centered at 0. As mentioned in Section A.7 of the math appendix, the volume of the unit ball in n dimensions is vn = n/2 (n/2 + 1), where is the gamma function, a very famous function which is defined by (a) = Z 0 x a e x dx x for all a > 0, and which will play an important role in the next chapter. A few useful facts about the gamma function (which you can assume) are that (a+1) = a(a) for any a > 0, and that (1) = 1 and ( 1 2 ) = . Using these facts, it follows that (n) = (n 1)! for n a positive integer, and we can also find (n + 1 2 ) when n is a nonnegative integer. For practice, please verify that v2 = (the area of the unit disk in 2 dimensions) and v3 = 4 3 (the volume of the unit ball in 3 dimensions). Let U1, U2, . . . , Un Unif(1, 1) be i.i.d. (a) Find the probability that (U1, U2, . . . , Un) is in the unit ball in R n . (b) Evaluate the result from (a) numerically for n = 1, 2, . . . , 10, and plot the results (using a computer unless you are extremely good at making hand-drawn graphs). (c) Let c be a constant with 0 < c < 1, and let Xn count how many of the Uj satisfy |Uj | > c. What is the distribution of Xn? (d) For c = 1/ 2, use the result of Part (c) to give a simple, short derivation of what happens to the probability from (a) as n