8solution - ECEN 303: Assignment 8 Problems: 1. If X is a...

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Unformatted text preview: ECEN 303: Assignment 8 Problems: 1. If X is a random variable that is uniformly distributed between- 1 and 1, fnd the PDF of radicalbig | X | and the PDF of- ln | X | . Let Y = radicalbig | X | . We have, for 0 y 1, F Y ( y ) = Pr( Y y ) = Pr parenleftBig radicalbig | X | y parenrightBig = Pr(- y 2 X y 2 ) = y 2 , and therefore by differentiation, f Y ( y ) = 2 y, for 0 y 1 . Let Y =- ln | X | . We have, for y 0, F Y ( y ) = Pr( Y y ) = Pr(ln | X | - y ) = Pr( X e- y ) + Pr( X - e- y ) = 1- e- y , and therefore by differentiation f Y ( y ) = e- y , for y . 2. Find the PDF of e X in terms of the PDF of X . Specialize the answer to the case where X is uniformly distributed between 0 and 1. Let Y = e X . We first find the CDF of Y, and then take the derivative to find its PDF. We have Pr( Y y ) = Pr ( e X y ) = braceleftBigg Pr( X ln y ) , if y > , , otherwise . Therefore f Y ( y ) = braceleftBigg d dx F X (ln y ) , if y > , , otherwise , = braceleftBigg 1 y f X (ln y ) , if y > , , otherwise . When X is unifrom on [0 , 1], the answer simplifies to f Y ( y ) = braceleftBigg 1 y , if 1 < y e, , otherwise , 1 3. Find the PDF of | X | 1 / 3 and | X | 1 / 4 in terms of the PDF of X . Let Y = | X | 1 / 3 . We have F Y ( y ) = Pr( Y y ) = Pr parenleftBig | X | 1 / 3 y parenrightBig = Pr (- y 3 X y 3 ) = F X ( y 3 )- F X (- y 3 ) , and therefore, by differentiating, f Y ( y ) = 3 y 2 f X ( y 3 ) + 3 y 2 f X (- y 3 ) , for y > . Let Y = | X | 1 / 4 . We have F Y ( y ) = Pr( Y y ) = Pr parenleftBig | X | 1 / 4 y parenrightBig = Pr (- y 4 X y 4 ) = F X ( y 4 )- F X (- y 4 ) , and therefore, by differentiating, f Y ( y ) = 4 y 3 f X ( y 4 ) + 4 y 3 f X (- y 4 ) , for y > ....
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This note was uploaded on 02/11/2010 for the course ECEN 303 taught by Professor Chamberlain during the Fall '07 term at Texas A&M.

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8solution - ECEN 303: Assignment 8 Problems: 1. If X is a...

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