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✵✳✶✳ ◆♦t ❖♥❡✲❚♦✲❖♥❡ ❚r❛♥s❢♦r♠❛t✐♦♥

❙✉♣♣♦s❡ t❤❛t t❤❡ ❢✉♥❝t✐♦♥ ♦❢

{x | fX (x) > 0}✳

t❤❛t ✐s ♦♥❡✲t♦✲♦♥❡ ♦✈❡r ❡❛❝❤

y = g (x)

u (x)✐s

Aj ✳

❤❛s ❛ ✉♥✐q✉❡ s♦❧✉t✐♦♥

A=
A1 , A2 , ... s✉❝❤
g (x)✱ t❤❡ ❡q✉❛t✐♦♥

♥♦t ♦♥❡✲t♦✲♦♥❡ tr❛♥s❢♦r♠❛t✐♦♥ ♦✈❡r

■t ✐s ♣♦ss✐❜❧❡ t♦ ♣❛rt✐t✐♦♥

A

❚❤❡♥ ❢♦r ❡❛❝❤

xj = hj (y)

✐♥t♦ ❞✐s❥♦✐♥t s✉❜s❡ts

y

✐♥ t❤❡ r❛♥❣❡ ♦❢

♦✈❡r t❤❡ s❡t

Aj ✳

■t ❢♦❧❧♦✇s t❤❛t ❚❤❡♦✲

r❡♠ ❄❄ ❛♥❞ ❄❄ ❝❛♥ ❜❡ ❡①t❡♥❞❡❞ t♦ ❢✉♥❝t✐♦♥s t❤❛t ❛r❡ ♥♦t ♦♥❡✲t♦✲♦♥❡ ❜② r❡♣❧❛❝✐♥❣
❡q✉❛t✐♦♥s ❄❄ ❛♥❞ ❄❄ ✇✐t❤

fY (y) =

✭✵✳✶✳✶✮

X

fX (hj (y))

j

❢♦r t❤❡ ❞✐s❝r❡t❡ ❝❛s❡ ❛♥❞✱

fY (y) =

✭✵✳✶✳✷✮

X
j

❢♦r t❤❡ ❝♦♥t✐♥✉♦✉s ❝❛s❡✳
✶✳ ▲❡t

❊①❛♠♣❧❡

fX (x) =

4
31



 dhj (y) 


fX (hj (y)) 
dy 

1 x
2

, x = −2, −1, 0, 1, 2

❙♦❧✉t✐♦♥✳ ❈❧❡❛r❧②✱ B = {0, 1, 2}❛♥❞
4
fY (0) = fX (0) = 31
2
8
+ 31
=
fY (1) = fX (−1) + fX (1) = 31
16
1
fY (2) = fX (−2) + fX (2) = 31 + 31
=

❛♥❞ ❝♦♥s✐❞❡r

Y = |X|✳

10
31
17
31

❆♥♦t❤❡r ✇❛② t♦ ❡①♣r❡ss t❤✐s ✐s

fY (0) =
fY (1) =

4
31 ,hy = 0


4
1 −y
31

2

+

✷✳ ❙✉♣♣♦s❡

❊①❛♠♣❧❡



1 y
2

i

, y = 1, 2

X ∼ U N IF (−1, 1)

❛♥❞

Y = X 2✳

❉❡t❡r♠✐♥❡ ♣❞❢ ♦❢

Y✳

A = (−1, 1) ✐♥t♦ ❞✐s❥♦✐♥t s✉❜s❡ts A1 = (−1, 0) ❛♥❞
x = 0 ❝❛♥ ❜❡ ♥❡❣❧❡❝t❡❞✳ ❚❤❡♥ ❢♦r ❡❛❝❤ y ✐♥
√
t❤❡ r❛♥❣❡ ♦❢ g (x)✱ t❤❡ ❡q✉❛t✐♦♥ y = g (x) ❤❛s ❛ ✉♥✐q✉❡ s♦❧✉t✐♦♥ x1 = h1 (y) = − y
√
♦✈❡r t❤❡ s❡t A1 ❛♥❞ x2 = h2 (y) = y ♦✈❡r t❤❡ s❡t A2 ✳ ❚❤✉s t❤❡ ♣❞❢ ♦❢ Y ✐s




√
 1 
√


1

√  + fX
fY (y) = fX − y  2−1
y  2√ y  = 2√
y
y , y ∈ (0, 1)
■t ✐s ♣♦ss✐❜❧❡ t♦ ♣❛rt✐t✐♦♥

A2 = (0, 1)✳

❙✐♥❝❡

✐s ❝♦♥t✐♥✉♦✉s t❤❡♥

✸✳ ▲❡t

❊①❛♠♣❧❡

♣❞❢ ♦❢

A

Y✳

fX (x) =

x2
3 , −1