6. (25 points) Let U₁, U₂,... are an i.i.d. sequence of uniform random variables on (0, 1), iid U₁, U2,... Uniform(0, 1)

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6 25 Points Let U U Are An I I D Sequence Of Uniform Random Variables On 0 1 Iid U U2 Uniform 0 1 1 (74.16 KiB) Viewed 34 times

6. (25 points) Let U₁, U₂,... are an i.i.d. sequence of uniform random variables on (0, 1), iid U₁, U2,... Uniform(0, 1). (a) (5 points) Define Sn = U₁. Prove that the p.d.f. of Sn is given by vi=1 12 -1, x <k, 1 fn(x) = (−1)* (") (x−k)"-¹'sgn(x−k), sgn(x–k) : = 0, x = k, 2(n-1)! k=0 1, x>k, If it is not a correct p.d.f., then find a correct p.d.f. fn(x), the p.d.f. of Sn. (b) (5 points) Draw fn, the p.d.f. of Sn for n = 2, 3, 4, 5 and discuss fn approaches the p.d.f. of a normal distribution as n increases. (c) (10 points) Let N be the minimum value of n for which Sn> 1. Find the p.m.f. of N. You may use the fact, P(N = n) = P(Sn-1 < 1) - P(S₂ < 1). (d) (5 points) Using a Monte Carlo simulation, obtain the p.m.f. of N numerically and compare your result with the exact solutions in (c).