Question is in the attachment:

The volume of a region in n-dimensional Euclidean space Rn is the integral of 1 over that region. The unit ball in Rn is {(21, ...,",) : x; + ... +. 0. A few useful facts about the gamma function (which you can assume) are that I(a + 1) = al(a) for any a > 0, and that D(1) = 1 and D(1) = VT. Using these facts, it follows that I(n) = (n - 1)! for n a positive integer, and we can also find D'(n + =) when n is a nonnegative integer. For practice, verify that v2 = (the area of the unit disk in 2 dimensions) and v3 = 2a (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 Rn. (b) Evaluate the result from (a) numerically for n = 1, 2,..., 10, and plot the results with R. The facts above about the gamma function are sufficient so that you can do this without doing any integrals, but you can also use the command gamma in R to compute the gamma function. (c Let c be a constant with 0 c. What is the distribution of X,? Write the probability that a point drawn uniformly is inside the unit ball using this new r.v. Xn. (d) Explain what all this means geometrically