Quiz1Bans - ( x ) = & &...

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STAT 410 Spring 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 ), – < y < , OR find the p.d.f. of Y, f Y ( y ), – < y < . ) f X ( x ) = ± ² ³ < < otherwise 0 5 2 21 2 x x F X
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Unformatted text preview: ( x ) = &amp; &amp; &amp; &amp; &amp; &amp; &lt; -&lt; 5 1 5 2 21 4 2 2 x x x x Y = 2 X 1 . 2 &lt; x &lt; 5 25 1 &lt; y &lt; 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 &lt; y &lt; 4 1 . OR F Y ( y ) = P ( Y y ) = P ( 2 X 1 y ) = P ( X y 1 ) = 1 P ( X &lt; y 1 ) = 1 F X ( y 1 ). = 1 21 4 1--y = 21 25 1--y , 25 1 &lt; y &lt; 4 1 . F Y ( y ) = &amp; &amp; &amp; &amp; &amp; &amp; &amp; &amp; &lt; -&lt;-4 1 1 4 1 25 1 21 25 25 1 1 y y y y f Y ( y ) = &amp; &amp; &amp; &amp; &lt; &lt;-otherwise 4 1 25 1 21 1 2 x y...
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Quiz1Bans - ( x ) = &amp;amp;amp; &amp;amp;amp;...

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