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Unformatted text preview: 180A HW 5 Solutions 1) Suppose X has the uniform distribution on (0 , 1). Compute the probability density function and expected value of: ( a ) X α , where α > 0 ( b ) log( X ); ( c ) exp ( X ); ( d ) sin(2 πX ) answer Let Y = X α . Then, clearly: F Y ( y ) = P ( X α ≤ y ) = P ( X ≤ y 1 /α ) = y 1 /α so f Y ( y ) = 1 α y 1 α α for 0 ≤ y ≤ 1, 0 otherwise and E [ Y ] = integraldisplay 1 1 α x 1 α dx = 1 α · α 1 + α x (1+ α ) /α vextendsingle vextendsingle vextendsingle vextendsingle 1 = 1 α + 1 Let Y 2 = log( X ). Then, for y < 0, (as log x < 0 for x < 1), we have F Y 2 ( y ) = P (log( X ) < y ) = P ( X ≤ e y ) = e y Thus f Y 2 ( y ) = e y for y < 0, 0 otherwise. We note that via the change of variables x = y and last weeks HW. E [ Y 2 ] = integraldisplay −∞ ye y dy = integraldisplay ∞ xe − x dx = 1 Let Y 3 = exp( X ). Then for 1 < y < e (as if 0 < x < 1, we have 1 < e x < 1) F Y 3 ( y ) = P (exp( X ) < y ) = P ( X < log( y )) = log( y ) so f Y ( y ) = 1 y for 1 < y < e , and 0 otherwise. Then E [ Y 3 ] = integraldisplay e 1 y (1 /y ) dy = e 1 . Let Y 4 = sin(2 πX ). This is slightly trickier, as sin(2 πx ) is not monotonic on (0 , 1). We can easily see that E [ Y 4 ] = integraltext 1 x sin(2 πx ) dx = 0 (from our work last week). To get f Y 4 ( y ) we note that for 0 < y < 1 F Y 4 ( y ) = P (sin(2 πX ) < y ) = P (0 ≤ 2 πX < arcsin( y ) or π arcsin( y ) < 2 πX ≤ 2 π ) = P (0 ≤ X < arcsin( y ) / (2 π )) + P (1 /...
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This homework help was uploaded on 02/03/2008 for the course MATH MATH 180A taught by Professor Castro during the Fall '08 term at UCSD.
 Fall '08
 Castro
 Math, Probability

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