Quiz1Aans - 9 4 65 2 x x F X ( x ) = &...

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STAT 410 Spring 2008 Version A 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 9 4 65 2 x x a) (4) Find the moment-generating function M X ( t ). M X ( t ) = E ( e t X ) = ´ 9 4 65 2 x d x e x t = ´ 9 4 65 2 x d e x x t = 4 9 2 1 1 65 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 4 2 4 9 2 9 65 2 65 8 65 2 65 18 + - - , t 0. M X ( 0 ) = 1.
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b) (6) Let Y = X 1 . Find the probability distribution of Y. ( That is, either find the c.d.f. of Y, F Y ( y ), – < y < , OR find the p.d.f. of Y, f Y ( y ), – < y < . ) f X ( x ) = ± ² ³ < < otherwise
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Unformatted text preview: 9 4 65 2 x x F X ( x ) = &amp; &amp; &amp; &amp; &amp; &amp; &lt; -&lt; 9 1 9 4 65 16 4 2 x x x x Y = X 1 . 4 &lt; x &lt; 9 3 1 &lt; y &lt; 2 1 . g ( x ) = x 1 g 1 ( y ) = 2 1 y = y 2 y x d d = 2 y 3 f Y ( y ) = f X ( g 1 ( y ) ) y x d d = ( 65 2 y 2 ) ( 2 y 3 ) = 65 4 y 5 , 3 1 &lt; y &lt; 2 1 . OR F Y ( y ) = P ( Y y ) = P ( X 1 y ) = P ( X 2 1 y ) = 1 P ( X &lt; 2 1 y ) = 1 F X ( 2 1 y ). = 1 65 16 4--y = 65 81 4--y , 3 1 &lt; y &lt; 2 1 . F Y ( y ) = &amp; &amp; &amp; &amp; &amp; &amp; &amp; &amp; &lt; -&lt;-2 1 1 2 1 3 1 65 81 3 1 4 y y y y f Y ( y ) = &amp; &amp; &amp; &amp; &lt; &lt;-otherwise 2 1 3 1 65 4 5 x y...
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Quiz1Aans - 9 4 65 2 x x F X ( x ) = &amp;amp;amp;...

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