✵✳✶✳ ❏❖■◆❚ ❚❘❆◆❙❋❖❘▼❆❚■❖◆ ✷
❊①❛♠♣❧❡ ✸✳ ▲❡tX1 ❛♥❞X2 ❜❡ ✐♥❞❡♣❡♥❞❡♥t ❛♥❞ ❡①♣♦♥❡♥t✐❛❧✱X ∼EXP(1)✳
❚❤✉s✱ t❤❡ ❥♦✐♥t ♣❞❢ ✐s
fX1,X2(x1, x2) = exp (−(x1+x2)),(x1, x2)∈A
✇❤❡r❡A={(x1, x2)|x1>0, x2>0}✳
❈♦♥s✐❞❡r t❤❡ r❛♥❞♦♠ ✈❛r✐❛❜❧❡sY1 =X1 ❛♥❞Y2=X1+X2✳ ❚❤✐s ❝♦rr❡s♣♦♥❞s t♦ t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ y1 = x1 ❛♥❞ y2 = x1 +x2✱ ✇❤✐❝❤ ❤❛s ❛ ✉♥✐q✉❡ s♦❧✉t✐♦♥✱
x1=y1❛♥❞x2=y2−y1✳ ❚❤❡ ❏❛❝♦❜✐❛♥ ✐s|J|=
1 0
−1 1
= 1❛♥❞ t❤✉s
fY1,Y2(y1, y2) =fX1,X2(y1, y2−y1)|J|= exp (−y2),(y1, y2)∈B
❛♥❞ ③❡r♦ ♦t❤❡r✇✐s❡✳ ❚❤❡ s❡t ♦❢B ✐s ♦❜t❛✐♥❡❞ ❜② tr❛♥s❢♦r♠✐♥❣ t❤❡ s❡tA✱ ❛♥ t❤✐s ❝♦r✲ r❡s♣♦♥❞s t♦y1=x1>0❛♥❞y2−y1=x2>0✳ ❚❤✉sB={(y1, y2)| ∞> y1> y2>0}✳
❚❤❡ ♠❛r❣✐♥❛❧ ♣❞❢ ♦❢Y1❛♥❞Y2❛r❡ ❣✐✈❡♥ ❛s ❢♦❧❧♦✇s✿ fY1(y1) =...
fY2(y2) =...
❊①❛♠♣❧❡ ✹✳ ❙✉♣♣♦s❡ t❤❛t✱ ✐♥st❡❛❞ ♦❢ t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ t❤❡ ♣r❡✈✐♦✉s ❡①✲ ❛♠♣❧❡✱ ❛ ❞✐✛❡r❡♥t tr❛♥s❢♦r♠❛t✐♦♥✱y1=x1−x2❛♥❞y2=x1+x2 ✐s ❝♦♥s✐❞❡r❡❞✳
❚❤❡ s♦❧✉t✐♦♥ ✐sx1=...❛♥❞x2=...✳ ❚❤❡ ❏❛❝♦❜✐❛♥ ✐s|J|=...✳ ❚❤❡ ❥♦✐♥t ♣❞❢ ✐s ❣✐✈❡♥ ❜②
fY1,Y2(y1, y2) =fX1,X2(..., ...)|...|=...∈B
✇❤❡r❡B={(y1, y2)|...}✳ ❚❤❡ ♠❛r❣✐♥❛❧ ♣❞❢ ♦❢ Y1❛♥❞Y2 ❛r❡ ❣✐✈❡♥ ❛s ❢♦❧❧♦✇s✿ fY1(y1) =...
