❈❍❆P❚❊❘ ✶
❱❛r✐❛❜❧❡ ❘❛♥❞♦♠ ❚r❛♥s❢♦r♠❛t✐♦♥
❙✉♣♣♦s❡X ✐s ❛ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ❛♥❞Y =g(X)❢♦r s♦♠❡ ❢✉♥❝t✐♦♥g✳ ●✐✈❡♥ t❤❡
❞✐str✐❜✉t✐♦♥ ♦❢X✱ ✇❤❛t ✐s t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢Y❄
✶✳✶✳ ❖♥❡✲❚♦✲❖♥❡ ❚r❛♥s❢♦r♠❛t✐♦♥
▲❡tg(x)❜❡ ❛ r❡❛❧✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥ ♦❢ ❛ r❡❛❧ ✈❛r✐❛❜❧❡x✳ ■❢ t❤❡ ❡q✉❛t✐♦♥y=g(x)
❝❛♥ ❜❡ s♦❧✈❡❞ ✉♥✐q✉❡❧②✱ s❛②x=h(y)t❤❡♥ t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ ✐s ♦♥❡✲t♦✲♦♥❡✳
❚❤❡♦r❡♠ ✶✳ ❙✉♣♣♦s❡ t❤❛tX ✐s ❛ ❞✐s❝r❡t❡ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ✇✐t❤ ♣❞❢fX(x)❛♥❞
t❤❛t Y =g(X) ❞❡✜♥❡s ❛ ♦♥❡✲t♦✲♦♥❡ tr❛♥s❢♦r♠❛t✐♦♥✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ❡q✉❛t✐♦♥
y=g(x)❝❛♥ ❜❡ s♦❧✈❡❞ ✉♥✐q✉❡❧②✱ s❛②x=h(y)✳ ❚❤❡♥ t❤❡ ♣❞❢ ♦❢Y ✐s
✭✶✳✶✳✶✮ fY(y) =fX(h(y)), y∈B ✇❤❡r❡B={y|fY(y)>0}✳
Pr♦♦❢✳ ❚❤✐s ❢♦❧❧♦✇ ❜❡❝❛✉s❡
fY(y) =P[Y =y] =P[g(X) =y] =P[X =h(y)] =fX(h(y))
❊①❛♠♣❧❡ ✷✳ ▲❡t X ∼GEO(p)✱ s♦ t❤❛t fX(x) =pqx−1, x = 1,2,3, . . .✳ ▲❡t
Y =X−1✱ t❤❡♥g(x) =x−1✱h(y) =y+ 1✱ ❛♥❞fY(y) =pq(y+1)−1, y= 0,1,2, . . .✳
❚❤❡♦r❡♠ ✸✳ ❙✉♣♣♦s❡ t❤❛t X ✐s ❛ ❝♦♥t✐♥✉♦✉s r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ✇✐t❤ ♣❞❢fX(x)
❛♥❞ ❛ss✉♠❡ t❤❛tY =g(X)❞❡✜♥❡s ❛ ♦♥❡✲t♦✲♦♥❡ tr❛♥s❢♦r♠❛t✐♦♥ ❢r♦♠A={x|fX(x)>0}
♦♥ t♦ B={y|fY(y)>0} ✇✐t❤ ✐♥✈❡rs❡ tr❛♥s❢♦r♠❛t✐♦♥x=h(y)✳ ■❢ t❤❡ ❞❡r✐✈❛t✐✈❡
dh(y)
dy ✐s ❝♦♥t✐♥✉♦✉s ❛♥❞ ♥♦♥ ③❡r♦ ♦♥B✱ t❤❡♥ t❤❡ ♣❞❢ ♦❢Y ✐s
✭✶✳✶✳✷✮ fY(y) =fX(h(y))
dh(y)
dy
, y∈B
Pr♦♦❢✳ ■❢y =g(x) ✐s ♦♥❡✲t♦✲♦♥❡ tr❛♥s❢♦r♠❛t✐♦♥✱ t❤❡♥ ✐t ✐s ❡✐t❤❡r ♠♦♥♦t♦♥✐❝
✐♥❝r❡❛s✐♥❣ ❛♥❞ ♠♦♥♦t♦♥✐❝ ❞❡❝r❡❛s✐♥❣✳ ■❢ ✐t ✐s ❛ss✉♠❡❞ t❤❛t ✐t ✐s ✐♥❝r❡❛s✐♥❣✱ t❤❡♥ g(x)≤y ✐❢ ❛♥❞ ♦♥❧② ✐❢x≤h(y)✱ ❛♥❞ t❤✉s
FY(y) =P[Y ≤y] =P[g(X)≤y] =P[X ≤h(y)] =FX(h(y))
❛♥❞ ❝♦♥s❡q✉❡♥t❧②✱
fY(y) =dFX(h(y))
dy =
dFX(h(y))
dh(y)
dh(y)
dy =fX(h(y))
dh(y)
dy
❜❡❝❛✉s❡g′(x)>0✇❤❡♥g(x)✐s ✐♥❝r❡❛s✐♥❣ t❤❡♥ dh(y)
✶✳✶✳ ❖◆❊✲❚❖✲❖◆❊ ❚❘❆◆❙❋❖❘▼❆❚■❖◆ ✸
✭✹✮ ▲❡t X ❤❛✈❡ fX(x) =x2
24,−2< x <4✱ ❞❡t❡r♠✐♥❡ t❤❡ ♣❞❢ ♦❢Y =X 2
❚❤❡♦r❡♠ ✽✳ ■❢X ✐s ❝♦♥t✐♥✉♦✉s ✇✐t❤ ❈❉❋F(X)✱ t❤❡♥U =F(X)∼U N IF(0,1)✳
Pr♦♦❢✳ ■t ✐s ❛ss✉♠❡❞ t❤❛tF(x)✐s ♦♥❡✲t♦✲♦♥❡ tr❛♥s❢♦r♠❛t✐♦♥ t❤❡♥F−1(u)✐s
❡①✐st❡❞✳
FU(u) =P[U ≤u] =P[F(X)≤u] =P
X ≤F−1(
u)
=F F−1(
u)
=u ❙✐♥❝❡0≤F(x)≤1✱ t❤❡♥FU(u) = 0✐❢u≤0❛♥❞FU(u) = 1✐❢u≥1✳ ❚❤❡r❡❢♦r❡✱
FU(u) =
0, u,
1,
0≥u
0< u <1
u≥1
