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Problem 8.48 If X has a density fX(x), find the density of Y where a) Y = for constants a and b. baX + b) Y = 2X . c) Y = X , assuming X is a non-negative random variable. Solution (a) If Y , using the results of Section 8.3 for Y = g(X) baX += ( ) ( ) aa byf dy ydgygfyf X XY 1 )()( 1 1 \u239f\u23a0 \u239e\u239c\u239d \u239b \u2212= = \u2212 \u2212 (b) If 2XY = , then ( ) ( ) ( )( ) \u239f\u239f\u23a0 \u239e \u239c\u239c\u239d \u239b++\u2212= y yfyfyf XXY 2 1 (c) If XY = , then we must assume X is positive valued so, this is a one-to-one mapping and ( ) ( ) yyfyf XY 22 \u22c5= Excerpts from this work may be reproduced by instructors for distribution on a not-for-profit basis for testing or instructional purposes only to students enrolled in courses for which the textbook has been adopted. Any other reproduction or translation of this work beyond that permitted by Sections 107 or 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful. page\u20268-59