sm8_48
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sm8_48


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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= 
 
 
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