Quiz1Bans - & ² ³ ≥< ≤< 5 1 5 2 21 4 2 2 x x x...

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STAT 410 Fall 2008 Version B Name ANSWERS . Quiz 1 (10 points) Be sure to show all your work, your partial credit might depend on it. No credit will be given without supporting work. 1. Consider a continuous random variable X with p.d.f. f X ( x ) = ± ² ³ < < otherwise 0 5 2 21 2 x x a) (4) Find the moment-generating function M X ( t ). M X ( t ) = E ( e t X ) = ´ 5 2 21 2 x d x e x t = ´ 5 2 21 2 x d e x x t = 2 5 2 1 1 21 2 µ µ · ¸ ¸ ¹ º - x t x t e t e x t = t t t t e t e t e t e t 2 2 2 5 2 5 21 2 21 4 21 2 21 10 + - - , t 0. M X ( 0 ) = 1.
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b) (6) Let Y = 2 X 1 . Find the probability distribution of Y. ( That is, either find the c.d.f. of Y, F Y ( y ) OR find the p.d.f. of Y, f Y ( y ). ) f X ( x ) = ± ² ³ < < otherwise 0 5 2 21 2 x x F X ( x ) = ±
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Unformatted text preview: & ² ³ ≥ < ≤-< 5 1 5 2 21 4 2 2 x x x x Y = 2 X 1 . 2 < x < 5 ´ 25 1 < y < 4 1 . g ( x ) = 2 1 x g – 1 ( y ) = y 1 = y – 1 / 2 y x d d = – 2 1 y – 3 / 2 f Y ( y ) = f X ( g – 1 ( y ) ) y x d d = ( 21 2 y – 1 / 2 ) ( 2 1 y – 3 / 2 ) = 21 1 y – 2 , 25 1 < y < 4 1 . OR F Y ( y ) = P ( Y ≤ y ) = P ( 2 X 1 ≤ y ) = P ( X ≥ y 1 ) = 1 – P ( X < y 1 ) = 1 – F X ( y 1 ). = 1 – 21 4 1--y = 21 25 1--y , 25 1 < y < 4 1 . F Y ( y ) = & & & & ± & & & & ² ³ ≥ < ≤-<-4 1 1 4 1 25 1 21 25 25 1 1 y y y y f Y ( y ) = & & ± & & ² ³ < <-otherwise 4 1 25 1 21 1 2 x y...
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This note was uploaded on 10/22/2009 for the course STAT 410 taught by Professor Alexeistepanov during the Fall '08 term at University of Illinois at Urbana–Champaign.

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Quiz1Bans - & ² ³ ≥< ≤< 5 1 5 2 21 4 2 2 x x x...

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