stat 330 problems - STATISTICS 330 PROBLEMS 1. Consider the...

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STATISTICS 330 PROBLEMS 1. Consider the following functions: a. f x kx 0.3 x , x 1,2, b. f x kx 2 , x c. f x k 1 x 2 1 , − x d. f x ke | x | , x e. f x k 1 x 5 ,0 x 1 f. f x kx 2 e x , x 0, 0 g. f x kx 1 , x 0, 0 h. f x ke x / 1 e x / 2 , x , 0 i. f x kx 3 e 1/ x , x 0, 0 In each case: Determine k so that f x is a p.f./p.d.f. and sketch f x . Let X be a random variable with p.f./p.d.f. f x . Find the c.d.f of X . Find E X and Var X . Find P 0.5 X 2 and P X 0.5| X 2 . To obtain numerical answers for f i use 1for f , g , 2for h and i . 2. For the p.d.f.’s in f i of Problem 1 determine if the parameter is a location or scale parameter for the distribution. 3. Suppose X GEO p . a. Show that P X k j | X k P X j where k and j are nonnegative integers. Explain why this is called the memoryless property. b. The only other distribution with this property is the exponential distribution. Show that Y EXP satisfies the memoryless property. 4. a. If X GAM , then find the p.d.f. of Y e X . b. If X GAM , then show Y 1/ X IG , . c. If X N , 2 then find the p.d.f. of Y e X . d. If X N , 2 then find the p.d.f. of Y X 1 . e. If X UNIF 2 , 2 then show that Y tan X CAU 1,0 . f. If X PAR , then show that Y log X / EXP 1 . g. If X DE then find the p.d.f. of Y X 2 . h. If X t k then show that Y X 2 F 1, k . 5. Suppose that f 1 x , f 2 x , , f k x are p.d.f.’s with supports A 1 , A 2 , , A k , means 1 , 2 , , k , and finite variances 1 2 , 2 2 , , k 2 respectively. Suppose also that 0 p 1 , p 2 , , p k 1and i 1 k p i 1. Show that g x i 1 k p i f i x is a p.d.f. . Let X be a random variable with p.d.f. g x . Find the support of X , the mean of X and the variance of X . 6. Suppose T t n . a. Show that E T 0if n 1.
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b. Show that Var T n / n 2 if n 2. 7. If E | X | k exists for some k Z then show that E | X | j exists for j 1, , k 1.
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This note was uploaded on 10/21/2010 for the course MATHEMATIC stats taught by Professor Various during the Spring '10 term at Waterloo.

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stat 330 problems - STATISTICS 330 PROBLEMS 1. Consider the...

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