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hw9 - Guidance Use the fact that χ 2 is a special case of...

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STAT 225 - Homework 9 Due Thursday 6/25 Do the following problems : 1. Let the random variable X have the following density function: f X ( x ) = αθ α x α +1 x > θ α > 0 , θ > 0 0 otherwise Find the distribution of Y = ln X θ 2. Let X exp( β ). Find the distribution of Y = e - X/β and identify the disttribution you got. 3. Page 318, problem 6.34 ( Hint: For (b) E ( Y ) = E ( U 1 2 ) = . . . and E ( Y 2 ) = E ( U ) = . . . 4. Let X N ( μ , σ 2 ). The random variable Y = e X is said to have a log - normal distribution with parameters μ and σ . (a) Find the density function of Y . (b) Find E ( Y ) and V ( Y ) ( Hint: do not integrate. Use the relationship between X and Y and the moment generation function of X ) 5. Page 324, problem 6.49 6. page 324, problem 6.57 7. page 325, problem 6.59
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Unformatted text preview: Guidance: Use the fact that χ 2 is a special case of Gamma, and the result you proved in the previous problem. 8. Page 323, problem 6.40 Guidance: Use a result from class about the distributions of Y 2 1 and Y 2 2 , and the result you proved in the previous problem. 1 9. Page 323, problem 6.41 10. Page 375, problem 7.49 11. Page 388, problem 7.96 12. Let X 1 ,X 2 ,...,X n be independent random variables, all having a Beta( α , 1) distri-bution. (a) Prove that X ( n ) also has a beta distribution and identify the parameters. (b) Based on part (a), what is E ( X ( n ) )? 2...
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