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

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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 0 . 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 > 0 , 0 , otherwise . Therefore f Y ( y ) = braceleftBigg d dx F X (ln y ) , if y > 0 , 0 , otherwise , = braceleftBigg 1 y f X (ln y ) , if y > 0 , 0 , otherwise . When X is unifrom on [0 , 1], the answer simplifies to f Y ( y ) = braceleftBigg 1 y , if 1 < y e, 0 , otherwise , 1
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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 > 0 . 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 > 0 . 4. Let X and Y be independent random variables, uniformly distributed in the interval [0 , 1]. Find the CDF and the PDF of | X - Y | . Let Z = | X - Y | . We have F Z ( z ) = Pr( | X - Y | ≤ z ) = 1 - (1 - z ) 2 . (To see this, draw the event of interest as a subset of the unit square and calculate its area.) Taking derivatives, the desired PF is f Z ( z ) = braceleftBigg 2(1 - z ) , if 0 z 1 , 0 otherwise .
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