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Unformatted text preview: THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT 1302 PROBABILITY AND STATISTICS II (2010-11) EXAMPLE CLASS 3 1. Assume the height of a student is normally distributed with mean μ and variance σ 2 , the heights of 5 randomly selected students were measured to be 1.76, 1.68, 1.82, 1.75 and 1.78. Write down the likelihood function. 2. A sample of size 10 is drawn at random without replacement from a population of size N = 1000, in which there are m males and N- m females and m is not known. It is found that there are two males in the selected sample. Write down the likelihood function. 3. If X 1 ,X 2 ,...,X n constitute a random sample of size n from a Bernoulli population with parameter θ , show that T = ˆ θ = X 1 + X 2 + ... + X n n is sufficient for θ . 4. Let X 1 ,X 2 ,...,X n be a random sample of size n from a geometric distribution with p.m.f. f ( x | θ ) = (1- θ ) x θ , for x = 0 , 1 ,... . Find the sufficient statistic for θ . 5. Let X 1 ,X 2 ,...,X n be a random sample from the normal distribution N ( θ,σ 2 ),-∞ < θ < ∞ , where the variance σ 2 is known. Show by the factorization criterion that ¯ x is a sufficient statistic for the mean...
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This note was uploaded on 11/17/2011 for the course MUSIC 1001 taught by Professor Dr.yun during the Spring '11 term at Princess Sumaya University for Technology.
- Spring '11