✵✳✶✳ ◆♦t ❖♥❡✲❚♦✲❖♥❡ ❚r❛♥s❢♦r♠❛t✐♦♥
❙✉♣♣♦s❡ t❤❛t t❤❡ ❢✉♥❝t✐♦♥ ♦❢ u(x)✐s ♥♦t ♦♥❡✲t♦✲♦♥❡ tr❛♥s❢♦r♠❛t✐♦♥ ♦✈❡rA =
{x|fX(x)>0}✳ ■t ✐s ♣♦ss✐❜❧❡ t♦ ♣❛rt✐t✐♦♥ A ✐♥t♦ ❞✐s❥♦✐♥t s✉❜s❡ts A1, A2, ... s✉❝❤
t❤❛t ✐s ♦♥❡✲t♦✲♦♥❡ ♦✈❡r ❡❛❝❤Aj✳ ❚❤❡♥ ❢♦r ❡❛❝❤y✐♥ t❤❡ r❛♥❣❡ ♦❢g(x)✱ t❤❡ ❡q✉❛t✐♦♥
y =g(x)❤❛s ❛ ✉♥✐q✉❡ s♦❧✉t✐♦♥ xj =hj(y)♦✈❡r t❤❡ s❡tAj✳ ■t ❢♦❧❧♦✇s t❤❛t ❚❤❡♦✲
r❡♠ ❄❄ ❛♥❞ ❄❄ ❝❛♥ ❜❡ ❡①t❡♥❞❡❞ t♦ ❢✉♥❝t✐♦♥s t❤❛t ❛r❡ ♥♦t ♦♥❡✲t♦✲♦♥❡ ❜② r❡♣❧❛❝✐♥❣ ❡q✉❛t✐♦♥s ❄❄ ❛♥❞ ❄❄ ✇✐t❤
✭✵✳✶✳✶✮ fY(y) =
X
j
fX(hj(y))
❢♦r t❤❡ ❞✐s❝r❡t❡ ❝❛s❡ ❛♥❞✱
✭✵✳✶✳✷✮ fY(y) =
X
j
fX(hj(y))
dhj(y)
dy
❢♦r t❤❡ ❝♦♥t✐♥✉♦✉s ❝❛s❡✳ ❊①❛♠♣❧❡ ✶✳ ▲❡tfX(x) =
4 31
1 2
x
, x=−2,−1,0,1,2❛♥❞ ❝♦♥s✐❞❡rY =|X|✳
❙♦❧✉t✐♦♥✳ ❈❧❡❛r❧②✱B ={0,1,2}❛♥❞
fY(0) =fX(0) =
4 31
fY(1) =fX(−1) +fX(1) =
8 31+
2 31 =
10 31 fY(2) =fX(−2) +fX(2) =
16 31+
1 31 =
17 31 ❆♥♦t❤❡r ✇❛② t♦ ❡①♣r❡ss t❤✐s ✐s
fY(0) =
4 31, y= 0 fY(1) =
4 31
h
1 2
−y+ 1 2
yi
, y= 1,2
❊①❛♠♣❧❡ ✷✳ ❙✉♣♣♦s❡X∼U N IF(−1,1)❛♥❞Y =X2
✳ ❉❡t❡r♠✐♥❡ ♣❞❢ ♦❢Y✳ ■t ✐s ♣♦ss✐❜❧❡ t♦ ♣❛rt✐t✐♦♥A = (−1,1) ✐♥t♦ ❞✐s❥♦✐♥t s✉❜s❡ts A1 = (−1,0) ❛♥❞ A2= (0,1)✳ ❙✐♥❝❡A✐s ❝♦♥t✐♥✉♦✉s t❤❡♥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❡tA1 ❛♥❞x2=h2(y) =√y ♦✈❡r t❤❡ s❡tA2✳ ❚❤✉s t❤❡ ♣❞❢ ♦❢Y ✐s
fY(y) =fX −√y
−1 2√y
+fX
√y
1 2√y
=
1
2√y , y∈(0,1) ❊①❛♠♣❧❡ ✸✳ ▲❡t fX(x) = x
2
3,−1 < x <2 ❛♥❞ ❝♦♥s✐❞❡r Y =X
2✳ ❉❡t❡r♠✐♥❡ ♣❞❢ ♦❢Y✳
❚❤❡r❡ ❛r❡ t✇♦ ✐♥✈❡rs❡ tr❛♥s❢♦r♠❛t✐♦♥✱ x1 = h1(y) = −√y ❢♦r x < 0 ❛♥❞ x2=h2(y) =√y ❢♦rx >0✳ ❚❤✉s t❤❡ ♣❞❢ ♦❢Y ✐s
fY(y) =
fX −√y
−1 2√y
+fX
√y
1 2√y
=
1 2√y
(−√y)2
3 + (√y)2
3
,0< y <1
fX √y
−1 2√y
=
1 2√y
(−√y)2
3
,1< y <4
