✵✳✶✳ ❙✉♠s ♦❢ ❘❛♥❞♦♠ ❱❛r✐❛❜❧❡s
❙✉♠s ♦❢ ✐♥❞❡♣❡♥❞❡♥t r❛♥❞♦♠ ✈❛r✐❛❜❧❡s ♦❢t❡♥ ❛r✐s❡ ✐♥ ♣r❛❝t✐❝❡✳ ■♥ ❡①❛♠♣❧❡s✱ ❞❡t❡r♠✐♥❡ t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ ❛ s✉♠ ♦❢S =X1+X2✇❤❡r❡X1❛♥❞X2❛r❡ ❝♦♥t✐♥✉♦✉s
r❛♥❞♦♠ ✈❛r✐❛❜❧❡s✳
❊①❛♠♣❧❡ ✶✳ ▲❡tX1 ❛♥❞X2❜❡ ✐♥❞❡♣❡♥❞❡♥t ❛♥❞ ✉♥✐❢♦r♠✱ Xi∼U N IF(0,1)✳
❈♦♥s✐❞❡r t❤❡ tr❛♥s❢♦r♠❛t✐♦♥T =X1 ❛♥❞S=X1+X2✳ ❋✐♥❞ t❤❡ ♣❞❢ ♦❢S✳
❙♦❧✉t✐♦♥✳ ❚❤❡ ♣❞❢ ♦❢X ✐s
fS(s) =
(
s ,0< s <1 2−s ,1≤x <2
❊①❛♠♣❧❡ ✷✳ ▲❡tfU(u) =e−u, u >0 ❛♥❞ fV (v) = 2v,0 < v <1 ✳ U ❛♥❞ V
❛r❡ ✐♥❞❡♣❡♥❞❡♥t✳ ■❢X =U+V✱ t❤❡♥ t❤❡ ♣❞❢ ♦❢X ✐s❄ ❙♦❧✉t✐♦♥✳
✭✶✮ ❈♦♥s✐❞❡r t❤❡ tr❛♥s❢♦r♠❛t✐♦♥X =U +V ❛♥❞ Y =V✳ ❚❤❡ ♣❞❢ ♦❢ X ✐s
fX(x) =
(
0 ´x
2ye−(x−y)dy=... ,0< x <1
0 ´1
2ye−(x−y)dy=... ,1≤x <∞
✭✷✮ ❈♦♥s✐❞❡r t❤❡ tr❛♥s❢♦r♠❛t✐♦♥X =U +V ❛♥❞ Y =U✳ ❚❤❡ ♣❞❢ ♦❢X ✐s
fX(x) =
(
0´x2 (x−y)e−ydy=... ,0< x <1
x−1
´x
2 (x−y)e−ydy=... ,1≤x <∞
❚❤❡ ♣❞❢ ♦❢X ✐s
fX(x) =
(
2x+ 2e−x−2 ,0< x <1
2e−x , x≥1
❊①❛♠♣❧❡ ✸✳ ▲❡tX1❛♥❞X2 ❜❡ ✐♥❞❡♣❡♥❞❡♥t ❛♥❞ ✉♥✐❢♦r♠✱Xi∼U N IF(0, 1)✳
❈♦♥s✐❞❡r t❤❡ tr❛♥s❢♦r♠❛t✐♦♥Y1=XX12 ❛♥❞Y2=X1.X2✳ ❋✐♥❞ t❤❡ ♣❞❢ ♦❢Y1 ❛♥❞Y2✳ ❙♦❧✉t✐♦♥✳
❊①❛♠♣❧❡ ✹✳ ▲❡tX1 ❛♥❞ X2 ❛r❡ ✐♥❞❡♣❡♥❞❡♥t ❣❛♠♠❛ ✈❛r✐❛❜❧❡s✱ f(x1, x2) = 1
Γ(α)Γ(β)x α−1
1 x
β−1
2 e−x1−x2,0< xi<∞✳ ❈♦♥s✐❞❡r t❤❡ tr❛♥s❢♦r♠❛t✐♦♥Y1=X1+X2
❛♥❞Y2=X1X+X1 2✳
❚❤❡ ❥♦✐♥t ♣❞❢ ♦❢Y1 ❛♥❞Y2 ✐s
fY1,Y2(y1, y2) =...,(y1, y2)∈...
❆♥❞ t❤❡ ♣❞❢ ♦❢Y1 ✐s
❊①❛♠♣❧❡ ✺✳ ▲❡t X1✱ X2 ❛♥❞ X3 ❛r❡ ✐♥❞❡♣❡♥❞❡♥t ❣❛♠♠❛ ✈❛r✐❛❜❧❡s✱ Xi ∼
GAM(1, αi), i = 1,2,3✳ ❈♦♥s✐❞❡r t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ Yi = P3Xi
j=1
Xj
, i = 1,2 ❛♥❞
Y3= 3
P
j=1
Xj✳
✵✳✶✳ ❙❯▼❙ ❖❋ ❘❆◆❉❖▼ ❱❆❘■❆❇▲❊❙ ✷
❚❤❡ ❥♦✐♥t ♣❞❢ ♦❢Y1✱Y2✱ ❛♥❞Y3 ✐s
fY1,Y2,Y3(y1, y2, y3) =... , (y1, y2, y3)∈... ❆♥❞ t❤❡ ♣❞❢ ♦❢Y3 ✐s
❆ t❡❝❤♥✐q✉❡ ❜❛s❡❞ ♦♥ ♠♦♠❡♥t ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥s ✉s✉❛❧❧② ✐s ♠✉❝❤ ♠♦r❡ ❝♦♥✲ ✈❡♥✐❡♥t t❤❛♥ ✉s✐♥❣ tr❛♥s❢♦r♠❛t✐♦♥s ❢♦r ❞❡t❡r♠✐♥✐♥❣ t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ s✉♠s ♦❢ ✐♥❞❡♣❡♥❞❡♥t r❛♥❞♦♠ ✈❛r✐❛❜❧❡s✳
❚❤❡♦r❡♠ ✻✳ ■❢ X1, X2, ..., Xn ❛r❡ ✐♥❞❡♣❡♥❞❡♥t r❛♥❞♦♠ ✈❛r✐❛❜❧❡s ✇✐t❤ ▼●❋s
MXi(t) t❤❡♥ t❤❡ ▼●❋ ♦❢Y =
n
P
i=1
Xi ✐s
MY (t) =MX1(t)...MXn(t)
Pr♦♦❢✳ ◆♦t✐❝❡ t❤❛tetY =et(X1+...+Xn)=etX1...etX1 s♦ MY (t) =E etY
=E et(X1+...+Xn)=E etX1...E etX1=M
X1(t)...MXn(t)
❊①❛♠♣❧❡ ✼✳ ▲❡tX1, X2, ..., Xk❜❡ ✐♥❞❡♣❡♥❞❡♥t ❜✐♥♦♠✐❛❧ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s ✇✐t❤
r❡s♣❡❝t✐✈❡ ♣❛r❛♠❡t❡rsni ❛♥❞p✱Xi∼BIN(ni, p)✱ ❛♥❞ ❧❡tY = k
P
i=1
Xi✳
■t ❢♦❧❧♦✇s t❤❛tMY (t) =...
❚❤✉s✱Y ∼...
❊①❛♠♣❧❡ ✽✳ ▲❡t X1, X2, ..., Xn ❜❡ ✐♥❞❡♣❡♥❞❡♥t P♦✐ss♦♥✲❞✐str✐❜✉t❡❞ r❛♥❞♦♠
✈❛r✐❛❜❧❡s ✇✐t❤ r❡s♣❡❝t✐✈❡ ♣❛r❛♠❡t❡rsni ❛♥❞p✱Xi∼P OI(µi)✱ ❛♥❞ ❧❡tY = n
P
i=1
Xi✳
■t ❢♦❧❧♦✇s t❤❛tMY (t) =...
❚❤✉s✱Y ∼...
❊①❛♠♣❧❡ ✾✳ ▲❡t X1, X2, ..., Xn ❜❡ ✐♥❞❡♣❡♥❞❡♥t ●❛♠♠❛✲❞✐str✐❜✉t❡❞ ✇✐t❤ r❡✲
s♣❡❝t✐✈❡ s❤❛♣❡ ♣❛r❛♠❡t❡rκ1, κ2, ..., κn❛♥❞ ❝♦♠♠♦♥ s❝❛❧❡ ♣❛r❛♠❡t❡rθ✱Xi∼GAM(θ, κi)
❢♦ri= 1,2, ..., n✱ ❛♥❞ ❧❡tY = Pn
i=1
Xi✳
■t ❢♦❧❧♦✇s t❤❛tMY (t) =...
❚❤✉s✱X ∼...
❊①❛♠♣❧❡ ✶✵✳ ▲❡tX1, X2, ..., Xn ❜❡ ✐♥❞❡♣❡♥❞❡♥t ♥♦r♠❛❧❧② ❞✐str✐❜✉t❡❞ r❛♥❞♦♠
✈❛r✐❛❜❧❡s✱Xi ∼N µi, σ2i
✱ ❛♥❞ ❧❡tY =
n
P
i=1
Xi✳
■t ❢♦❧❧♦✇s t❤❛tMY (t) =...
❚❤✉s✱Y ∼...
❚❤✐s ✐♥❝❧✉❞❡s t❤❡ s♣❡❝✐❛❧ ❝❛s❡ ♦❢ ❛ r❛♥❞♦♠ s❛♠♣❧❡X1, X2, ..., Xn ❢r♦♠ ❛ ♥♦r✲
♠❛❧❧② ❞✐str✐❜✉t❡❞ ♣♦♣✉❧❛t✐♦♥✱ s❛② Xi ∼ N µ, σ2
✳ ■♥ t❤✐s ❝❛s❡✱ µ = µi ❛♥❞
σ2 =σ2
i ❢♦r ❛❧❧ i= 1,2, ..., n✱ ❛♥❞ ❝♦♥s❡q✉❡♥t❧② n
P
i=1
Xi ∼N nµ, nσ2
✳ ■t ❛❧s♦ ❢♦❧✲
❧♦✇s r❡❛❞✐❧② ✐♥ t❤❡ ❝❛s❡ t❤❛t t❤❡ s❛♠♣❧❡ ♠❡❛♥✱X¯ =
n
P
i=1
Xi
n ✐s ♥♦r♠❛❧❧② ❞✐str✐❜✉t❡❞✱
¯
X ∼Nµ,σ2
n
