# Need help with problem number 3 ( Thevolume of a region in

Need help with problem number 3 Thevolume of a region in n-dimensional Euclidean space Rn is the integral of1 over that region. The unit ball in Rn is {(x1, . . . , xn) : x21 + · ·· + x2n ≤ 1}, the ball of radius 1 centered at 0. As mentioned inSection A.7 of the math appendix, the volume of the unit ball in ndimensions is vn =πn/2Γ(n/2 + 1),where Γ is the gamma function, a veryfamous function which is defined byΓ(a) = Z ∞0xae−xdxx for all a > 0,and which will play an important role in the next chapter. A few usefulfacts about the gamma function (which you can assume) are that Γ(a + 1) =aΓ(a) for any a > 0, and that Γ(1) = 1 and Γ( 12) = √π. Using thesefacts, it follows that Γ(n) = (n−1)!for n a positive integer, and we canalso find Γ(n +12) when n is a non-negative integer.For practice, pleaseverify that v2 = π (the area of the unit disk in 2 dimensions) andv3=43π (the volume of the unit ball in 3 dimensions). Let U1, U2, . . . ,Un ∼ Un if(−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(using acomputer unless you are extremely good at making hand-drawn graphs).The facts above about the gamma function are sufficient so that you cando this without doing any integrals, but you can also use the commandgamma in R to compute the gamma function.(c) Let c be a constant with 0c. What is the distribution of Xn?(d) For c = 1/√2, use the result ofPart (c) to give a simple, short derivation of what happens to theprobability from (a) as n → ∞ hw7.pdf