97年 - 97 國立高雄大學_博士班招生考試試題_統計學研究所：數理統計#147980

1. Suppose that nX is a random sample with variance 2 . LetX 21 ,, ni iXnU 21 121 andni iXnV 21 21 . Consider Chebyshev inequality and Center Limit Theorem, respectively, tosuggest a value of n which such that 95.0)|(| VUP . Which method gives a smallersuggested value of n ? Why? (40%)

(a) Find the value of k .

(b) Find ).| (25%).10|,|),( 22 yxxykyxf( 12 yYeXE YY

3. Find the value of ))|( in terms of ),( YX . (15%),2( XYEXCov Cov

4. Let nX be i.i.d. random variables which has distribution ),( 2 . Prove that thesample meanX ,,1 Nni iXnX 1 and sample variance1 ni iX1 (nS 211 X 2) are independentto each other. (20%)