Question

1. Let X be an iid sample of size n from a continuous distribution with mean /i, variance a2 and such that Xi e [0, 1] for all i e {1,...,n}. Let X = average. For a E (0,1), we wish to obtain a number q > 0 such that: (1/n) Xi be the sample Р(X € |и — 9. и + q) predict with probability approximately In other words, we wish to sample of size n, the average X will land in an interval around /u of length 2q. a that if we take an iid (a) Use the following tools to find formulas for q: i. The Central Limit Theorem (CLT): Vn(X P noo n=2° ¢(u) < u for u E R, and where P(.) is the cdf of the standard Normal ii. Chebyschev's inequality: P(Xn-k)S with an appropriate k > 0. ii. Hoeffding's inequality: P([X, - /| > t) < 2e-2nt2 with an appropriate t > 0. (b) Given the pdf [0, 1] - { av те f (x) = 0.w i. Find a such that f(-) is a proper pdf. ii. Find E[X] and Var(X) iii. For n 20 and a = 0.99 compute numerical values for L = /u - q and U = /i + q using the three methods above

Answer 1

Central Limit Theorem :- Also s P - u (u) ) j.e = P P ST dニ md As normal distribution So (4)-u) u)-1+ (u)= 2(u)- Accor dong t the problem 2(u) = « 1- 2. 2. u percentile f N(os) dsbribubion upper PE Thus compartng with

Chebysheu's inequoli by P(1- P1n K2 -k < X-k P(-k m+k) > P k2 Compa Cn iPxe [-, ineualiby Hoeff ding's '(1->t) < 2e-2nt2 tu7- Pt<R t) >1 - 1e-2nt2 Ptn t1-2e-2t2 ef-3, Comparing with p - 2nt2 1-2e ) 2n 2n

xED, 1. (x) dherwise /2 2S/2 Sl2 72 Var(b) - E(x*) - Elo»>] = 즉 -응) : 2. 12 n20 Con fidence in terval :-- 0.0052-S76 Thus volue s obtained fron the standora normal table 2-576 o 151 LA= -o 1S1O 449 1S1 0-7S! Cheby shev's mequatty - 2-6186 L -2-6186 = -2-0186 2686 = 3.2186 U ut Hoeff ding's aneguetity -0o- o-364 36 .964 U