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§ : •.fs. iJt 1t 2016. 10. 21 1. (10%) Let P(Ai) = 1 'Vi. Show that P(n~I Ai) = 1.
2. (15%) Let X andY be mutually independent continuous random variables with the cumu- lative distributions Fx(x) and Fy(y), respectively. Derive the distribution of X conditioning on {Z = 0}, where Z = I(X ~ Y).
3. (7%) (8%) LetT= L:~I Xi and E[N2] < oo. Conditioning on N = n, XI,···, Xn are further assumed to be independent and identically distributed with mean p, and variance CI2• Derive the mean and variance ofT.
4. (15%) Let XI, ... , Xn i.f_vd. N(p., CI2) with unknown Jl· and CI2• Derive the distribution of the random quantity .fii(Xn- p,)/ Sn, where Xn = L:~=I Xi/nand S~ = L:~=I (Xi- Xn? /(n -1).
5. (15%) Let XI, ... , Xn be a random sample from a geometric distribution P(X = x) = p(1 - p)x-I J{l,2 , ... }(x) and p have a uniform prior distribution on [0, 1]. Find the Bayes estimator of p based on the loss function L(p, 8(XI, ... , Xn)) = I8(XI, ... , Xn)-
PI·
6. (15%) Let XI, ... , Xn+I be a random sample from Bernoulli(1r) and h(1r) = P(L:~=I Xi>
Xn+Ii7r). Find the maximum likelihood estimator of h(1r).
7. (15%) Let XI, ... , Xn be a random sample from Beta(O, 1). Find a (1- a) confidence interval by inverting the likelihood ratio test for Ho :
