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Unformatted text preview: . 4.5: If X has density function f ( x ) = 4 x 3 for 0 < x < 1 and Y =2ln X 4 then the distribution function for Y is F ( y ) = P ( Y ≤ y ) = P (2ln X 4 ≤ y ) = P (4ln X =y/ 2) = P ( X > ey/ 8 ) which reduces to Z 1 e (y/ 8) 4 x 3 dx = 1ey/ 2 so f ( y ) = F ( y ) = ey/ 2 2 , which is the density function for the random variable χ 2 (2). 4.11: If X 1 and X 2 are random variables from the distribution N (0 , 1) and Y = X 2 1 + X 2 2 then F ( y ) = P ( Y ≤ y ) = P ( X 2 1 + X 2 2 ≤ y ) Z Z x 2 1 + x 2 2 ≤ y 1 2 π ex 2 1 2x 2 2 2 dx 1 dx 2 . 1 Changing to polar coordinates reduces this integral to 1 2 * π Z 2 π Z √ y er 2 / 2 rdrdθ = 1ey/ 2 so that f ( y ) = F ( y ) = 1 2 ey/ 2 which is the density function for a random variable with χ 2 (2) distribution. 2...
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This note was uploaded on 09/10/2009 for the course STATS 517 taught by Professor Song during the Fall '07 term at Purdue.
 Fall '07
 Song
 Probability

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