Let the random variable
𝑋
have the following pdf:
𝑓
(
𝑥
)
=
3
(
1
−
𝑥
)
2
where
0

Question

Let the random variable 
𝑋
 have the following pdf:  
𝑓
(
𝑥
)
=
3
(
1
−
𝑥
)
2
 where 
0
<
𝑥
<
1
.

Find 
𝑔
(
𝑦
)
, the pdf of 
𝑌
=
(
2
−
2
𝑋
)
3
, which is a decreasing function of 
𝑋
. Also, give the support of 
𝑔
(
𝑦
)
.

BlackTom AI · Accepted Answer

For a monotonic transformation of a random variable, the pdf of Y = h(X) is g(y) = f(h^{-1}(y)) · |d/dy[h^{-1}(y)]|. Use this change-of-variables formula with absolute derivative because h is monotone (here decreasing).

1) Relevant knowledge points and concepts
- Change-of-variables formula for continuous random variables.
- Inverse function and derivative; use absolute value of dx/dy even when h is decreasing.
- Mapping of supports: y takes values obtained from x ∈ (0,1).

2) Step-by-step reasoning and inversion
Given Y = (2 − 2X)^3. Factor and simplify:
Y = [2(1 − X)]^3 = 8(1 − X)^3.
Invert to express X in terms of Y:
(1 − X)^3 = Y/8 ⇒ 1 − X = (Y/8)^{1/3} ⇒ X = 1 − (Y/8)^{1/3}.

3) Derivative and intermediate steps
Compute dx/dy:
( Y/8 )^{1/3} = Y^{1/3}/2, so X = 1 − Y^{1/3}/2.
dX/dY = −(1/3) Y^{-2/3}/2 = −(1/6) Y^{-2/3}.
Take absolute value: |dX/dY| = (1/6) Y^{-2/3}.

Evaluate f at x = h^{-1}(y). Given f(x) = 3(1 − x)^2 and 1 − x = (Y/8)^{1/3} = Y^{1/3}/2:
f(h^{-1}(y)) = 3[(Y^{1/3}/2)^2] = 3·Y^{2/3}/4.

Multiply to get g(y):
g(y) = f(h^{-1}(y)) · |dX/dY|
= (3·Y^{2/3}/4) · (1/6) Y^{-2/3}
= (3/4)·(1/6) · (Y^{2/3}·Y^{-2/3})
= 3/24 = 1/8.

4) Support determination
As x runs over (0,1), 1 − x runs over (0,1), so Y = 8(1 − X)^3 runs over (0,8). Because the original interval is open, support is 0 < y < 8.

5) Why other alternatives would be incorrect
Common mistakes: forgetting absolute value (yielding a negative density), incorrect inversion of the cube (wrong power or factor 2), arithmetic errors in derivatives, or incorrect support endpoints (e.g., [0,8] instead of (0,8)). Any such mistakes produce densities that do not integrate to 1 or are negative.

6) Conclusion
Therefore g(y) = 1/8 for 0 < y < 8, which matches the provided answer.

Let the random variable 𝑋 have the following pdf: 𝑓 ( 𝑥 ) = 3 ( 1 − 𝑥 ) 2 where 0 < 𝑥 < 1 . Find 𝑔 ( 𝑦 ) , the pdf of 𝑌 = ( 2 − 2 𝑋 ) 3 , which is a decreasing function of 𝑋 . Also, give the support of 𝑔 ( 𝑦 ) .单项选择题

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