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

⎟⎠
⎞⎜⎝

⎛ −=

=
−

−

(b) If 2XY = , then

( ) ( ) ( )( ) ⎟⎟⎠
⎞

⎜⎜⎝
⎛++−=

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 ⋅=

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