❈❍❆P❚❊❘ ✶
❘❛♥❞♦♠ ❱❛r✐❛❜❧❡
❈♦♥s✐❞❡r ❛♥ ❡①♣❡r✐♠❡♥t ✇❤❡r❡ t❤r❡❡ ❝♦✐♥s ✐s t♦ss❡❞✳ ❋♦r s✉❝❤ ❡①♣❡r✐♠❡♥ts✱ t❤❡r❡ ❛r❡ ♠❛♥② ❡✈❡♥t t❤❛t ❝❛♥ ❜❡ ❞❡✜♥❡❞ ♦♥ Ω✱ ❜✉t ♦♥❧② t❤♦s❡ ❡✈❡♥ts ✐♥✈♦❧✈✐♥❣ ✐s ❝♦♥s✐❞❡r❡❞✱ ❢♦r ❡①❛♠♣❧❡✱ t❤❡ ♥✉♠❜❡r ♦❢ ❤❡❛❞s t❤❛t ❛♣♣❡❛r✳ ■♥ t❤✐s ❝❛s❡✱ t❤❡ s❡t ♦❢ ❛❧❧ ♣♦ss✐❜❧❡ ✈❛❧✉❡s ♦❢ t❤❡ ♥✉♠❜❡r ♦❢ ❤❡❛❞s ❝❛♥ ❜❡ ❞❡✜♥❡❞ t❛❦✐♥❣ ♦♥ ♦♥❡ ♦❢ t❤❡ ✈❛❧✉❡s ✵✱ ✶✱ ✷✱ ❛♥❞ ✸ ✇✐t❤ r❡s♣❡❝t✐✈❡ ♣r♦❜❛❜✐❧✐t✐❡s
P{Y = 0}=P{(T, T, T)}= 1 8
P{Y = 1}=P{(H, T, T),(T, H, T),(T, T, H)}= 3 8
P{Y = 2}=P{(H, H, T),(H, T, H),(T, H, H)}=3 8
P{Y = 3}=P{(H, H, H)}=1 8
❈♦♥s✐❞❡r ❛♥♦t❤❡r ❡①♣❡r✐♠❡♥t ♦❢ s❡❧❡❝t✐♥❣ t❤r❡❡ ❜❛❧❧s r❛♥❞♦♠❧② ✇✐t❤♦✉t r❡♣❧❛❝❡♠❡♥t ❢r♦♠ ❛♥ ✉r♥ ❝♦♥t❛✐♥✐♥❣ ✷✵ ❜❛❧❧s ♥✉♠❜❡r❡❞ ✶ t❤r♦✉❣❤ ✷✵✳ ❆ ❜❡t ✐s ♣✉t ❢♦r ❛t ❧❡❛st ♦♥❡ ♦❢ t❤❡ ❜❛❧❧s t❤❛t ❛r❡ ❞r❛✇♥ ❤❛s ❛ ♥✉♠❜❡r ❛s ❧❛r❣❡ ❛s ♦r ❧❛r❣❡r t❤❛♥ ✶✼✱ ✇❤❛t ❡✈❡♥t t❤❛t ✐s ❝♦♥s✐❞❡r❡❞ ❛♥❞ ✐ts ♣r♦❜❛❜✐❧✐t②❄ ❚❤❡s❡ ❡✈❡♥ts ❢r♦♠ t✇♦ ❡①❛♠♣❧❡s ❛❜♦✈❡ ❝❛♥ ❜❡ ❛ss♦❝✐❛t❡❞ t♦ r❛♥❞♦♠ ✈❛r✐❜❧❡✳
❉❡❢✐♥✐t✐♦♥ ✶✳ ❆ r❛♥❞♦♠ ✈❛r✐❛❜❧❡X ✐s ❛ ❢✉♥❝t✐♦♥ t❤❛t ♠❛♣s t❤❡ s❛♠♣❧❡ s♣❛❝❡
t♦ t❤❡ r❡❛❧ ❧✐♥❡❀ t❤❛t ✐s✱ ❢♦r ❡❛❝❤ω∈Ω✱ X(ω)✐s ❛ r❡❛❧ ♥✉♠❜❡r✳
❚❤❡r❡ ❛r❡ t✇♦ t②♣❡s ♦❢ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s ❜❛s❡❞ ✐ts s❛♠♣❧❡ s♣❛❝❡✱ ❞✐s❝r❡t❡ r❛♥✲ ❞♦♠ ✈❛r✐❛❜❧❡ ❛♥❞ ❝♦♥t✐♥✉♦✉s r❛♥❞♦♠ ✈❛r✐❛❜❧❡✳
✶✳✶✳ ❉✐s❝r❡t❡ ❘❛♥❞♦♠ ❱❛r✐❛❜❧❡s
❉❡❢✐♥✐t✐♦♥ ✷✳ ■❢ t❤❡ s❡t ♦❢ ❛❧❧ ♣♦ss✐❜❧❡ ✈❛❧✉❡s ♦❢ ❛ r❛♥❞♦♠ ✈❛r✐❛❜❧❡✱ X✱ ✐s
❛ ❝♦✉♥t❛❜❧❡ s❡t✱ x1, x2, ..., xn ♦r x1, x2, ...✱ t❤❡♥ X ✐s ❝❛❧❧❡❞ ❛ ❞✐s❝r❡t❡ r❛♥❞♦♠ ✈❛r✐❛❜❧❡✳ ❚❤❡ ❢✉♥❝t✐♦♥
f(x) =P(X =x) x=x1, x2, ...
t❤❛t ❛ss✐❣♥s t❤❡ ♣r♦❜❛❜✐❧✐t② t♦ ❡❛❝❤ ♣♦ss✐❜❧❡ ✈❛❧✉❡ x ✇✐❧❧ ❜❡ ❝❛❧❧❡❞ t❤❡ ❞✐s❝r❡t❡
♣r♦❜❛❜✐❧✐t② ❞❡♥s✐t② ❢✉♥❝t✐♦♥ ✭❞✐s❝r❡t❡ ♣❞❢✮✱ ♣r♦❜❛❜✐❧✐t② ♠❛ss ❢✉♥❝t✐♦♥✱ ♣r♦❜❛❜✐❧✐t② ❢✉♥❝t✐♦♥ ♦r ❞❡♥s✐t② ❢✉♥❝t✐♦♥✳
❚❤❡♦r❡♠ ✸✳ ❆ ❢✉♥❝t✐♦♥f(x)✐s ❛ ❞✐s❝r❡t❡ ♣❞❢ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐t s❛t✐s✜❡s ❜♦t❤ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt✐❡s ❢♦r ❛t ♠♦st ❛ ❝♦✉♥t❛❜❧② ✐♥✜♥✐t❡ s❡t ♦❢ r❡❛❧sx1, x2, ...
f(xi)≥0
✶✳✶✳ ❉■❙❈❘❊❚❊ ❘❆◆❉❖▼ ❱❆❘■❆❇▲❊❙ ✷
❢♦r ❛❧❧xi✱ ❛♥❞
X
all xi
f(xi) = 1
Pr♦♦❢✳ ❋r♦♠ ❉❡✜♥✐t✐♦♥ ❄❄✳
❉❡❢✐♥✐t✐♦♥ ✹✳ ❚❤❡ ❝✉♠♠✉❧❛t✐✈❡ ❞✐str✐❜✉t✐♦♥ ❢✉♥❝t✐♦♥ ✭❈❉❋✮ ♦❢ ❛ r❛♥❞♦♠ ✈❛r✐❛❜❧❡X ✐s ❞❡✜♥❡❞ ❢♦r ❛♥② r❡❛❧x❜②
F(x) =P(X ≤x)
❚❤❡ ❣❡♥❡r❛❧ r❡❧❛t✐♦♥s❤✐♣ ❜❡t✇❡❡♥F(x)❛♥❞f(x)❢♦r ❛ ❞✐s❝r❡t❡ ❞✐str✐❜✉t✐♦♥ ✐s ❣✐✈❡♥ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ t❤❡♦r❡♠✳
❚❤❡♦r❡♠ ✺✳ ▲❡t X ❜❡ ❛ ❞✐s❝r❡t❡ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ✇✐t❤ ♣❞❢ f(x) ❛♥❞ ❈❉❋
F(x)✳ ■❢ t❤❡ ♣♦ss✐❜❧❡ ✈❛❧✉❡s ♦❢ X ❛r❡ ✐♥❞❡①❡❞ ✐♥ ✐♥❝r❡❛s✐♥❣ ♦r❞❡r✱x1< x2< x3...✱
t❤❡♥f(x1) =F(x1)❛♥❞ ❢♦r ❛♥②i >1✱
f(xi) =F(xi)−F(xi−1)
❋✉rt❤❡r♠♦r❡✱ ✐❢x < x1 t❤❡♥ F(x) = 0✱ ❛♥❞ ❢♦r ❛♥② ♦t❤❡r r❡❛❧ x F(x) = X
xi≤x
f(xi)
✇❤❡r❡ t❤❡ s✉♠♠❛t✐♦♥ ✐s t❛❦❡♥ ♦✈❡r ❛❧❧ ✐♥❞✐❝❡si s✉❝❤ t❤❛txi≤x
❚❤❡ ❈❉❋ ♦❢ ❛♥② r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ♠✉st s❛t✐s❢② t❤❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ t❤❡♦r❡♠✳
❚❤❡♦r❡♠ ✻✳ ❆ ❢✉♥❝t✐♦♥ F(x) ✐s ❛ ❈❉❋ ❢♦r s♦♠❡ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ X ✐❢ ❛♥❞
♦♥❧② ✐❢ ✐t s❛t✐s✜❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt✐❡s✿
✭✶✳✶✳✶✮ lim
x✙✲∞F(x) = 0
✭✶✳✶✳✷✮ lim
x✙∞F(x) = 1
✭✶✳✶✳✸✮ lim
h→0+F(x+h) =F(x)
✭✶✳✶✳✹✮ a < b implies F(a)≤F(b)
Pr♦♣❡rt② ✭✶✳✶✳✸✮ s❛②s t❤❛t F(x) ✐s ❝♦♥t✐♥✉♦✉s ❢r♦♠ t❤❡ r✐❣❤t✳ ❆♥❞ ♣r♦♣❡rt② ✭✶✳✶✳✹✮ s❛②s t❤❛tF(x)✐s ♥♦♥❞❡❝r❡❛s✐♥❣✳
❙♦♠❡ ✐♠♣♦rt❛♥t ♣r♦♣❡rt✐❡s ♦❢ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥ ✐♥✈♦❧✈❡ ♥✉♠❡r✐❝❛❧ q✉❛♥✲ t✐t✐❡s ❝❛❧❧❡❞ ❡①♣❡❝t❡❞ ✈❛❧✉❡s✳
❉❡❢✐♥✐t✐♦♥ ✼✳ ■❢ X ✐s ❛ ❞✐s❝r❡t❡ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ✇✐t❤ ♣❞❢ f(x)✱ t❤❡♥ t❤❡
❡①♣❡❝t❡❞ ✈❛❧✉❡ ♦❢X ✐s ❞❡✜♥❡❞ ❜②
E(X) =X x
✶✳✷✳ ❈❖◆❚■◆❯❖❯❙ ❘❆◆❉❖▼ ❱❆❘■❆❇▲❊❙ ✸
❚❤❡ ♠❡❛♥ ♦r ❡①♣❡❝t❡❞ ✈❛❧✉❡ ♦❢ ❛ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ✐s ❛ ✏✇❡✐❣❤t ❛✈❡r❛❣❡✱✑ ❛♥❞ ✐t ❝❛♥ ❜❡ ❝♦♥s✐❞❡r❡❞ ❛s ❛ ♠❡❛s✉r❡ ♦❢ t❤❡ ✏❝❡♥t❡r✑ ♦❢ t❤❡ ❛ss♦❝✐❛t❡❞ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥✳
✶✳✷✳ ❈♦♥t✐♥✉♦✉s ❘❛♥❞♦♠ ❱❛r✐❛❜❧❡s
❊❛❝❤ ✇♦r❦ ❞❛② ❛ ♠❛♥ r✐❞❡s ❛ ❜✉s t♦ ❤✐s ♣❧❛❝❡ ♦❢ ❜✉s✐♥❡ss✳ ❍✐s ✇❛✐t✐♥❣ t✐♠❡ ♦♥ ❛♥② ❣✐✈❡♥ ♠♦r♥✐♥❣ t♦ ❜❡ ❛ r❛♥❞♦♠ ✈❛r✐❛❜❧❡X✳ ❙✉♣♣♦s❡ t❤❛t ❤❡ ✐s ✈❡r② ♦❜s❡r✈❛♥t
❛♥❞ ♥♦t✐❝❡❞ ♦✈❡r t❤❡ ②❡❛rs t❤❛t t❤❡ ❢r❡q✉❡♥❝② ♦❢ ❞❛②s ✇❤❡♥ ❤❡ ✇❛✐ts ♥♦ ♠♦r❡ t❤❛♥
x♠✐♥✉t❡s ❢♦r t❤❡ ❜✉s ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦x❢♦r ❛❧❧x✳
P(X ≤x) =...
❆♥❞ ❛ ♥❡✇ ❜✉s ❛rr✐✈❡s ♣r♦♠♣t❧② ❡✈❡r② ✜✈❡ ♠✐♥✉t❡s✳
...≤x≤...
P(...≤X ≤...) =...
❉❡❢✐♥✐t✐♦♥ ✽✳ ❆ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ X ✐s ❝❛❧❧❡❞ ❛ ❝♦♥t✐♥✉♦✉s r❛♥❞♦♠ ✈❛r✐❛❜❧❡
✐❢ t❤❡r❡ ✐s ❛ ❢✉♥❝t✐♦♥f(x)❝❛❧❧❡❞ t❤❡ ♣r♦❜❛❜✐❧✐t② ❞❡♥s✐t② ❢✉♥❝t✐♦♥ ✭♣❞❢✮ ♦❢X✱ s✉❝❤
t❤❛t t❤❡ ❈❉❋ ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❛s
F(x) = x ˆ
−∞
f(t)dt
❆❜♦✈❡ ❞❡✜♥✐t✐♦♥ ♣r♦✈✐❞❡s ❛ ✇❛② t♦ ❞❡r✐✈❡ t❤❡ ❈❉❋ ✇❤❡♥ t❤❡ ♣❞❢ ✐s ❣✐✈❡♥✱ s♣❡❝✐✜❝❛❧❧②✱
❚❤❡ ❢✉♥❝t✐♦♥ F(x)✱ ❛s r❡♣r❡s❡♥t❡❞ ✐♥ ❉❡✜♥✐t✐♦♥ ✽✱ ✐s ✉♥❛✛❡❝t❡❞ r❡❣❡r❞❧❡ss ♦❢
❤♦✇ s✉❝❤ ✈❛❧✉❡s ✐s tr❡❛t❡❞✳ ❚❤✉s✱P(X =c) = 0❛♥❞P(a≤X≤b) =P(a < X ≤b) =
P(a≤X < b) =P(a < X < b)✳
❚❤❡♦r❡♠ ✾✳ ❆ ❢✉♥❝t✐♦♥f(x)✐s ❛ ♣❞❢ ❢♦r s♦♠❡ ❝♦♥t✐♥✉♦✉s r❛♥❞♦♠ ✈❛r✐❛❜❧❡X
✐❢ ❛♥❞ ♦♥❧② ✐❢
❖t❤❡r ♣r♦♣❡rt✐❡s ♦❢ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥ ❝❛♥ ❜❡ ❞❡s❝r✐❜❡❞ ✐♥ t❡r♠s ♦❢ q✉❛♥✲ t✐t✐❡s ❝❛❧❧❡❞ ♣❡r❝❡♥t✐❧❡s✳
❉❡❢✐♥✐t✐♦♥ ✶✵✳ ❯♥❞❡r t❤✐s ❛ss✉♠♣t✐♦♥✱ t❤❡ ✐♥✈❡rs❡ ❢✉♥❝t✐♦♥ F−1 ✐s ✇❡❧❧ ❞❡✲
✜♥❡❞❀x=F−1(y) ✐❢y =F(x)✳ ❚❤❡pt❤ q✉❛♥t✐❧❡ ♦❢ t❤❡ ❞✐str✐❜✉t✐♦♥F ✐s ❞❡✜♥❡❞
t♦ ❜❡ t❤❛t ✈❛❧✉❡xp s✉❝❤ t❤❛tF(xp) =p✱ ♦rP(X < xp) =p✳
❙♣❡❝✐❛❧ ❝❛s❡s ❛r❡p= 12✱ ✇❤✐❝❤ ❝♦rr❡s♣♦♥❞s t♦ t❤❡ ♠❡❞✐❛♥ ♦❢F✱ ❛♥❞p=14 ❛♥❞
p=3
4✱ ✇❤✐❝❤ ❝♦rr❡s♣♦♥❞ t♦ t❤❡ ❧♦✇❡r ❛♥❞ ✉♣♣❡r q✉❛rt✐❧❡s ♦❢F✳
❉❡❢✐♥✐t✐♦♥ ✶✶✳ ■❢X ✐s ❛ ❝♦♥t✐♥✉♦✉s r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ✇✐t❤ ♣❞❢f(x)✱ t❤❡♥ t❤❡
❡①♣❡❝t❡❞ ✈❛❧✉❡s ♦❢X ✐s ❞❡✜♥❡❞ ❜②
E(X) =
∞
ˆ
−∞
xf(x)dx
❙♦♠❡ ♣r♦♣❡rt✐❡s ♦❢ ❡①♣❡❝t❡❞ ✈❛❧✉❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ♦❢X ❛r❡ ✉s❡❢✉❧ t♦ ❝♦♥s✐❞❡r✳
✶✳✸✳ P❘❖❇▲❊▼❙ ✺
✭❛✮ ❋✐♥❞ t❤❡ ♠❡❞✐❛♥ ♦❢X❄
✭❜✮ ❙❦❡t❝❤ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❈❉❋ ❛♥❞ s❤♦✇ t❤❡ ♣♦s✐t✐♦♥ ♦❢ t❤❡ ♠❡❞✐❛♥ ♦♥ t❤❡ ❣r❛♣❤
✭✹✮ ❋✐♥❞ ♠❡❞✐❛♥ ♦❢ X ✇✐t❤ ❈❉❋F(x) = 1−e−(x3) 2
, x >0 ✭✺✮ ❋✐♥❞ ♠♦❞❡ ♦❢ X ✐❢ ♣❞❢ ♦❢X ✐sf(x) = x
10, x= 2,3,5
✭✻✮ ❋✐♥❞ ♠♦❞❡ ♦❢ X∼f(x) =√1 2πe
−1 2(x−2)
2
,−∞< x <∞
✭✼✮ ▲❡t X ❜❡ ❛ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ✇✐t❤ ❞✐s❝r❡t❡ ♣❞❢f(x) = x
8, x= 1,2,5✱ ❛♥❞
③❡r♦ ♦t❤❡r✇✐s❡✳ ❋✐♥❞ ✭❛✮ E(X)
✭❜✮ V ar(X) ✭❝✮ E(2x+ 3)
✭✽✮ ▲❡t X ❜❡ ❛ ❝♦♥t✐♥✉♦✉s r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ✇✐t❤ ♣❞❢f(x) = 3x2,0< x <1✱
❛♥❞ ③❡r♦ ♦t❤❡r✇✐s❡✳ ❋✐♥❞ ✭❛✮ E(X)
✭❜✮ V ar(X) ✭❝✮ E(Xr)
✭❞✮ E 3X−5X2+ 1
✭✾✮ ❙✉♣♣♦s❡ t❤❛tX✐s ❛ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ✇✐t❤ ▼●❋Mx(t) = 18et+14e2t+58e5t ✭❛✮ ❲❤❛t ✐s t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢X
✭❜✮ ❲❤❛t ✐sP[X= 2]
✭✶✵✮ ❆ss✉♠❡ t❤❛tX❜❡ ❛ ❝♦♥t✐♥✉♦✉s r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ✇✐t❤ ❞✐s❝r❡t❡ ♣❞❢f(x) =
e−(x+2),−2< x <∞✱ ❛♥❞ ③❡r♦ ♦t❤❡r✇✐s❡✳ ❋✐♥❞
✭❛✮ ❋✐♥❞ ♠♦♠❡♥t ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ♦❢X
