Probability and Statistics

timer Asked: Nov 6th, 2015

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Need help with problem number 3 :( 

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 {(x1, . . . , xn) : x21 + · · · + x2n ≤ 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 =πn/2Γ(n/2 + 1),where Γ is the gamma function, a very famous 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 useful facts 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 these facts, it follows that Γ(n) = (n−1)!for n a positive integer, and we can also find Γ(n +12) when n is a non-negative integer.For practice, please verify 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 a computer unless you are extremely good at making hand-drawn graphs). 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 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 → ∞


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