Example Class2 - THE UNIVERSITY OF HONG KONG DEPARTMENT OF...

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THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT 1302 PROBABILITY AND STATISTICS II (2010-11) EXAMPLE CLASS 2 1. Let X 1 ,X 2 ,...,X n be iid N ( μ,σ 2 ). Show by the Central Limit Theorem, n ( ¯ X n - μ ) D -→ N (0 2 ), as n → ∞ . 2. Let X 1 ,X 2 ,...,X n be iid Poisson( λ ). Show by the Central Limit Theorem, n ( ¯ X n - λ ) converges in distribution to N (0 ), as n → ∞ . 3. Let X 1 ,X 2 ,...,X n and Y 1 ,Y 2 ,...,Y n be the random numbers of two inde- pendent random samples, each of size n , from the distributions that have the respective means μ 1 and μ 2 , and the common variance σ 2 . Find the limiting distribution of ( ¯ X n - ¯ Y n ) - ( μ 1 - μ 2 ) σ p 2 /n , where ¯ X n and ¯ Y n are the respectively means of the samples. 4. In an experiment of tossing fair coins, n is the number of all tosses in the experiment. How large, at least, is the n such that the proportion of heads [0 . 4 , 0 . 6] is greater than or equal to 0.9? Use CLT to determine it.
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Example Class2 - THE UNIVERSITY OF HONG KONG DEPARTMENT OF...

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