❈❍❆P❚❊❘ ✶
❙♣❡❝✐❛❧ Pr♦❜❛❜✐❧✐t② ❉✐str✐❜✉t✐♦♥s
❚❤❡ ♣✉r♣♦s❡ ♦❢ t❤✐s ❝❤❛♣t❡r ✐s t♦ ❞❡✈❡❧♦♣ s♦♠❡ s♣❡❝✐❛❧ ❞✐sr✐❜✉t✐♦♥s✳ ❙♣❡❝✐❛❧ ❞✐s❝r❡t❡ ❞✐str✐❜✉t✐♦♥s ✇✐❧❧ ❜❡ ❞❡r✐✈❡❞ ✉s✐♥❣ t❤❡ ❝♦✉♥t✐♥❣ t❡❝❤♥✐q✉❡s✳ ❆♥❞ ❙♣❡❝✐❛❧ ❝♦♥t✐♥✉♦✉s ❞✐str✐❜✉t✐♦♥s ❛❧s♦ ✇✐❧❧ ❜❡ ♣r❡s❡♥t❡❞✱ ❛♥❞ r❡❧❛t✐♦♥s❤✐♣s ❜❡t✇❡❡♥ ✈❛r✐♦✉s s♣❡❝✐❛❧ ❞✐str✐❜✉t✐♦♥s ✇✐❧❧ ❜❡ ❞✐s❝✉ss❡❞✳
✶✳✶✳ ❙♣❡❝✐❛❧ ❉✐s❝r❡t❡ ❉✐str✐❜✉t✐♦♥s
✶✳✶✳✶✳ ❇❡r♥♦✉❧❧✐ ❉✐str✐❜✉t✐♦♥✳ ❙✉♣♣♦s❡ t❤❛t ❛ tr✐❛❧✱ ♦r ❛♥ ❡①♣❡r✐♠❡♥t✱ ✇❤♦s❡ ♦✉t❝♦♠❡ ❝❛♥ ❜❡ ❝❧❛ss✐✜❡❞ ❛s ❡✐t❤❡r ❛ s✉❝❝❡ss ♦r ❛ ❢❛✐❧✉r❡ ✐s ♣❡r❢♦r♠❡❞✳ ▲❡t X = 1
✇❤❡♥ t❤❡ ♦✉t❝♦♠❡ ✐s ❛ s✉❝❝❡ss ❛♥❞X= 0✇❤❡♥ ✐t ✐s ❛ ❢❛✐❧✉r❡✱ t❤❡♥ t❤❡ ♣r♦❜❛❜✐❧✐t②
♠❛ss ❢✉♥❝t✐♦♥ ♦❢X ✐s ❣✐✈❡♥ ❜②
✭✶✳✶✳✶✮ p(1) =P(X = 1) =p
p(0) =P(X= 0) = 1−p
✇❤❡r❡ p✱0 ≤p≤1 ✱ ✐s t❤❡ ♣r♦❜❛❜✐❧✐t② t❤❛t t❤❡ tr✐❛❧ ✐s ❛ s✉❝❝❡ss✳ ❆♥ ❛❧t❡r♥❛t✐✈❡
r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤✐s ❢✉♥❝t✐♦♥ ✐s
f(x) =pxq1−x
, x= 0,1
❛♥❞ ❛ r❛♥❞♦♠ ✈❛r✐❛❜❧❡X ✐s s❛✐❞ t♦ ❜❡ ❛ ❇❡r♥♦✉❧❧✐ r❛♥❞♦♠ ✈❛r✐❛❜❧❡✳
■♥ ❛♣♣❧✐❝❛t✐♦♥s✱ ❇❡r♥♦✉❧❧✐ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s ♦❢t❡♥ ♦❝❝✉r ❛s ✐♥❞✐❝❛t♦rs✳ ■❢ A ✐s ❛♥ ❡✈❡♥t✱ t❤❡♥ t❤❡ ✐♥❞✐❝❛t♦r r❛♥❞♦♠ ✈❛r✐❛❜❧❡✱ IA t❛❦❡s ♦♥ t❤❡ ✈❛❧✉❡ ✶ ✐❢A ♦❝❝✉rs ❛♥❞ t❤❡ ✈❛❧✉❡ ✵ ✐❢A❞♦❡s ♥♦t ♦❝❝✉r
IA(w) = (
1 , if ω∈A
0 , otherwise
❊①❛♠✐♥❡ t❤❡ ♣r♦♣❡rt✐❡s ♦❢ ❛ ❜❡r♥♦✉❧❧✐ r❛♥❞♦♠ ✈❛r✐❛❜❧❡✳
• E(X) =...
• V ar(X) =...
• Mx(t) =...
✶✳✶✳✷✳ ❇✐♥♦♠✐❛❧ ❉✐str✐❜✉t✐♦♥✳ ▲❡t ❛ ♠♦r❡ ❝♦♠♣❧✐❝❛t❡❞ ❡①♣❡r✐♠❡♥t ❛s ❛ s❡✲ q✉❡♥❝❡ ♦❢ n ✐♥❞❡♣❡♥❞❡♥t ❜❡r♥♦✉❧❧✐ tr✐❛❧s✳ ❊❛❝❤ ♦❢ ✇❤✐❝❤ r❡s✉❧ts ✐♥ ❛ s✉❝❝❡ss ✇✐t❤ ♣r♦❜❛❜✐❧✐t②p❛♥❞ ✐♥ ❛ ❢❛✐❧✉r❡ ✇✐t❤ ♣r♦❜❛❜✐❧✐t②1−p✳ ■❢X r❡♣r❡s❡♥ts t❤❡ ♥✉♠❜❡r ♦❢ s✉❝❝❡ss❡s t❤❛t ♦❝❝✉r ✐♥ t❤❡ntr✐❛❧s✱ t❤❡♥X✐s s❛✐❞ t♦ ❜❡ ❛ ❜✐♥♦♠✐❛❧ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ✇✐t❤ ♣❛r❛♠❡t❡rs(n, p)✳ ❚❤✉s✱ ❛ ❇❡r♥♦✉❧❧✐ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ✐s ❥✉st ❛ ❜✐♥♦♠✐❛❧ r❛♥✲
❞♦♠ ✈❛r✐❛❜❧❡ ✇✐t❤ ♣❛r❛♠❡t❡rs(1, p)✳ ❚❤❡ ♣r♦❜❛❜✐❧✐t② ♠❛ss ❢✉♥❝t✐♦♥ ♦❢ ❛ ❜✐♥♦♠✐❛❧
r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ❤❛✈✐♥❣ ♣❛r❛♠❡t❡rs(n, p)✐s ❣✐✈❡♥ ❜②
b(x;n, p) =..., x=...
✶✳✶✳ ❙P❊❈■❆▲ ❉■❙❈❘❊❚❊ ❉■❙❚❘■❇❯❚■❖◆❙ ✷
❊①❛♠♣❧❡ ✶✳ ■t ✐s ❦♥♦✇♥ t❤❛t s❝r❡✇s ♣r♦❞✉❝❡❞ ❜② ❛ ❝❡rt❛✐♥ ❝♦♠♣❛♥② ✇✐❧❧ ❜❡ ❞❡❢❡❝t✐✈❡ ✇✐t❤ ♣r♦❜❛❜✐❧✐t②.01✱ ✐♥❞❡♣❡♥❞❡♥t❧② ♦❢ ❡❛❝❤ ♦t❤❡r✳ ❚❤❡ ❝♦♠♣❛♥② s❡❧❧s t❤❡
s❝r❡✇s ✐♥ ♣❛❝❦❛❣❡s ♦❢10❛♥❞ ♦✛❡rs ❛ ♠♦♥❡②✲❜❛❝❦ ❣✉❛r❛♥t❡❡ t❤❛t ❛t ♠♦st1♦❢ t❤❡10
s❝r❡✇s ✐s ❞❡❢❡❝t✐✈❡✳ ❲❤❛t ♣r♦♣♦rt✐♦♥ ♦❢ ♣❛❝❦❛❣❡s s♦❧❞ ♠✉st t❤❡ ❝♦♠♣❛♥② r❡♣❧❛❝❡❄ ❙♦❧✉t✐♦♥✳
❊①❛♠♣❧❡ ✷✳ ❆ ❝♦♠♠✉♥✐❝❛t✐♦♥ s②st❡♠ ❝♦♥s✐sts ♦❢ ♥ ❝♦♠♣♦♥❡♥ts✱ ❡❛❝❤ ♦❢ ✇❤✐❝❤ ✇✐❧❧✱ ✐♥❞❡♣❡♥❞❡♥t❧②✱ ❢✉♥❝t✐♦♥ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ♣✳ ❚❤❡ t♦t❛❧ s②st❡♠✇✐❧❧ ❜❡ ❛❜❧❡ t♦ ♦♣❡r❛t❡ ❡✛❡❝t✐✈❡❧② ✐❢ ❛t ❧❡❛st ♦♥❡✲❤❛❧❢ ♦❢ ✐ts ❝♦♠♣♦♥❡♥ts ❢✉♥❝t✐♦♥✳ ✭❛✮ ❋♦r ✇❤❛t ✈❛❧✉❡s ♦❢ p ✐s ❛ ✺✲❝♦♠♣♦♥❡♥t s②st❡♠ ♠♦r❡ ❧✐❦❡❧② t♦ ♦♣❡r❛t❡ ❡✛❡❝t✐✈❡❧② t❤❛♥ ❛ ✸✲ ❝♦♠♣♦♥❡♥t s②st❡♠❄
❊①❛♠✐♥❡ t❤❡ ♣r♦♣❡rt✐❡s ♦❢ ❛ ❜✐♥♦♠✐❛❧ r❛♥❞♦♠ ✈❛r✐❛❜❧❡✳
• E(X) =...
• V ar(X) =...
• Mx(t) =...
✶✳✶✳✸✳ ●❡♦♠❡tr✐❝ ❉✐str✐❜✉t✐♦♥✳ ❚❤❡ ❣❡♦♠❡tr✐❝ ❞✐str✐❜✉t✐♦♥ ✐s ❛❧s♦ ❝♦♥str✉❝t❡❞ ❢r♦♠ ✐♥❞❡♣❡♥❞❡♥t ❇❡r♥♦✉❧❧✐ tr✐❛❧s✱ ❜✉t ❢r♦♠ ❛♥ ✐♥✜♥✐t❡ s❡q✉❡♥❝❡✳ ❙✉♣♣♦s❡ t❤❛t ✐♥✲ ❞❡♣❡♥❞❡♥t tr✐❛❧s✱ ❡❛❝❤ ❤❛✈✐♥❣ ❛ ♣r♦❜❛❜✐❧✐t② p✱0 < p < 1✱ ♦❢ ❜❡✐♥❣ ❛ s✉❝❝❡ss✱ ❛r❡
♣❡r❢♦r♠❡❞ ✉♥t✐❧ ❛ s✉❝❝❡ss ♦❝❝✉rs✳ ▲❡tX ❡q✉❛❧ t❤❡ ♥✉♠❜❡r ♦❢ tr✐❛❧s r❡q✉✐r❡❞✱ t❤❡♥ X ✐s s❛✐❞ t♦ ❜❡ ❛ ❣❡♦♠❡tr✐❝ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ✇✐t❤ ♣❛r❛♠❡t❡rs(p)✇✐t❤ t❤❡ ♣r♦❜❛✲
❜✐❧✐t② ♠❛ss ❢✉♥❝t✐♦♥
P(X =x) =..., x=...
❊①❛♠♣❧❡ ✸✳ ❆♥ ✉r♥ ❝♦♥t❛✐♥sN ✇❤✐t❡ ❛♥❞M ❜❧❛❝❦ ❜❛❧❧s✳ ❇❛❧❧s ❛r❡ r❛♥❞♦♠❧② s❡❧❡❝t❡❞✱ ♦♥❡ ❛t ❛ t✐♠❡✱ ✉♥t✐❧ ❛ ❜❧❛❝❦ ♦♥❡ ✐s ♦❜t❛✐♥❡❞✳ ❆ss✉♠❡ t❤❛t ❡❛❝❤ ❜❛❧❧ s❡❧❡❝t❡❞ ✐s r❡♣❧❛❝❡❞ ❜❡❢♦r❡ t❤❡ ♥❡①t ♦♥❡ ✐s ❞r❛✇♥✱ ✇❤❛t ✐s t❤❡ ♣r♦❜❛❜✐❧✐t② t❤❛t ✭❛✮ ❡①❛❝t❧②n ❞r❛✇s ❛r❡ ♥❡❡❞❡❞❄ ✭❜✮ ❛t ❧❡❛stk ❞r❛✇s ❛r❡ ♥❡❡❞❡❞❄
❊①❛♠✐♥❡ t❤❡ ♣r♦♣❡rt✐❡s ♦❢ ❛ ❣❡♦♠❡tr✐❝ r❛♥❞♦♠ ✈❛r✐❛❜❧❡✳
• E(X) =...
• V ar(X) =...
• Mx(t) =...
✶✳✶✳✹✳ ◆❡❣❛t✐✈❡ ❇✐♥♦♠✐❛❧ ❉✐str✐❜✉t✐♦♥✳ ❙✉♣♣♦s❡ t❤❛t ✐♥❞❡♣❡♥❞❡♥t tr✐❛❧s✱ ❡❛❝❤ ❤❛✈✐♥❣ ♣r♦❜❛❜✐❧✐t②p✱0< p <1✱ ♦❢ ❜❡✐♥❣ ❛ s✉❝❝❡ss ❛r❡ ♣❡r❢♦r♠❡❞ ✉♥t✐❧ ❛ t♦t❛❧
♦❢r s✉❝❝❡ss❡s ✐s ❛❝❝✉♠✉❧❛t❡❞✳ ▲❡t X ❡q✉❛❧ t❤❡ ♥✉♠❜❡r ♦❢ tr✐❛❧s r❡q✉✐r❡❞✱ t❤❡♥X ✐s s❛✐❞ t♦ ❜❡ ❛ ♥❡❣❛t✐✈❡ ❜✐♥♦♠✐❛❧ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ✇✐t❤ ♣❛r❛♠❡t❡rs(r, p) ✇✐t❤ t❤❡
♣r♦❜❛❜✐❧✐t② ♠❛ss ❢✉♥❝t✐♦♥
P(X =x) =..., x=...
❊①❛♠♣❧❡ ✹✳ ■❢ ✐♥❞❡♣❡♥❞❡♥t tr✐❛❧s✱ ❡❛❝❤ r❡s✉❧t✐♥❣ ✐♥ ❛ s✉❝❝❡ss ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ♣✱ ❛r❡ ♣❡r❢♦r♠❡❞ ✇❤❛t ✐s t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢rs✉❝❝❡ss❡s ♦❝❝✉rr✐♥❣ ❜❡❢♦r❡m❢❛✐❧✉r❡s❄
❊①❛♠✐♥❡ t❤❡ ♣r♦♣❡rt✐❡s ♦❢ ❛ ♥❡❣❛t✐✈❡ ❜✐♥♦♠✐❛❧ r❛♥❞♦♠ ✈❛r✐❛❜❧❡✳
• E(X) =...
• V ar(X) =...
✶✳✶✳ ❙P❊❈■❆▲ ❉■❙❈❘❊❚❊ ❉■❙❚❘■❇❯❚■❖◆❙ ✸
✶✳✶✳✺✳ ❍②♣❡r❣❡♦♠❡tr✐❝ ❉✐str✐❜✉t✐♦♥✳ ❙✉♣♣♦s❡ t❤❛t ❛ s❛♠♣❧❡ ♦❢ s✐③❡ n ✐s t♦ ❜❡ ❝❤♦s❡♥ r❛♥❞♦♠❧② ✭✇✐t❤♦✉t r❡♣❧❛❝❡♠❡♥t✮ ❢r♦♠ ❛♥ ✉r♥ ❝♦♥t❛✐♥✐♥❣ N ❜❛❧❧s✱ ♦❢ ✇❤✐❝❤m❛r❡ ✇❤✐t❡ ❛♥❞N−m❛r❡ ❜❧❛❝❦✳ ▲❡tX ❞❡♥♦t❡ t❤❡ ♥✉♠❜❡r ♦❢ ✇❤✐t❡ ❜❛❧❧s s❡❧❡❝t❡❞✱ t❤❡♥X ✐s s❛✐❞ t♦ ❜❡ ❛ ❤②♣❡r❣❡♦♠❡tr✐❝ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ✇✐t❤ ♣❛r❛♠❡t❡rs
(n, M, N)✇✐t❤ t❤❡ ♣r♦❜❛❜✐❧✐t② ♠❛ss ❢✉♥❝t✐♦♥
P(X =x) =..., x= 1,2, ...
❊①❛♠♣❧❡ ✺✳ ❆ ♣✉r❝❤❛s❡r ♦❢ ❡❧❡❝tr✐❝❛❧ ❝♦♠♣♦♥❡♥ts ❜✉②s t❤❡♠ ✐♥ ❧♦ts ♦❢ s✐③❡ ✶✵✳ ■t ✐s ❤✐s ♣♦❧✐❝② t♦ ✐♥s♣❡❝t ✸ ❝♦♠♣♦♥❡♥ts r❛♥❞♦♠❧② ❢r♦♠ ❛ ❧♦t ❛♥❞ t♦ ❛❝❝❡♣t t❤❡ ❧♦t ♦♥❧② ✐❢ ❛❧❧ ✸ ❛r❡ ♥♦♥❞❡❢❡❝t✐✈❡✳ ■❢ ✸✵ ♣❡r❝❡♥t ♦❢ t❤❡ ❧♦ts ❤❛✈❡ ✹ ❞❡❢❡❝t✐✈❡ ❝♦♠♣♦♥❡♥ts ❛♥❞ ✼✵ ♣❡r❝❡♥t ❤❛✈❡ ♦♥❧② ✶✱ ✇❤❛t ♣r♦♣♦rt✐♦♥ ♦❢ ❧♦ts ❞♦❡s t❤❡ ♣✉r❝❤❛s❡r r❡❥❡❝t❄
◆♦✇✱❊①❛♠✐♥❡ t❤❡ ♣r♦♣❡rt✐❡s ♦❢ ❛ ❤②♣❡r❣❡♦♠❡tr✐❝ r❛♥❞♦♠ ✈❛r✐❛❜❧❡✳
• E(X) =...
• V ar(X) =...
✶✳✶✳✻✳ P♦✐ss♦♥ ❉✐str✐❜✉t✐♦♥✳ ❆ r❛♥❞♦♠ ✈❛r✐❛❜❧❡X t❤❛t t❛❦❡s ♦♥ ♦♥❡ ♦❢ t❤❡ ✈❛❧✉❡s 0,1,2, ... ✐s s❛✐❞ t♦ ❜❡ ❛ P♦✐ss♦♥ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ✇✐t❤ ♣❛r❛♠❡t❡r λ ✐❢✱ ❢♦r s♦♠❡λ >0✱
P(X =x) =e−λλ
x
x!, x= 0,1,2, ...
❚❤❡ P♦✐ss♦♥ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ❤❛s ❛ tr❡♠❡♥❞♦✉s r❛♥❣❡ ♦❢ ❛♣♣❧✐❝❛t✐♦♥s ✐♥ ❞✐✈❡rs❡ ❛r❡❛s ❜❡❝❛✉s❡ ✐t ♠❛② ❜❡ ✉s❡❞ ❛s ❛♥ ❛♣♣r♦①✐♠❛t✐♦♥ ❢♦r ❛ ❜✐♥♦♠✐❛❧ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ✇✐t❤ ♣❛r❛♠❡t❡rs (n, p)✇❤❡♥n✐s ❧❛r❣❡ ❛♥❞p✐s s♠❛❧❧ ❡♥♦✉❣❤ s♦ t❤❛t np✐s ♦❢ ♠♦❞❡r❛t❡ s✐③❡✳ ❚❤❡ ♥✉♠❜❡r ♦❢ s✉❝❝❡ss❡s ♦❝❝✉rr✐♥❣ ✐s ❛♣♣r♦①✐♠❛t❡❧② ❛ P♦✐ss♦♥ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ✇✐t❤ ♣❛r❛♠❡t❡r λ=np✳ ❊①❛♠✐♥❡ t❤❡ ♣r♦♣❡rt✐❡s ♦❢ ❛ ♣♦✐ss♦♥ r❛♥❞♦♠ ✈❛r✐❛❜❧❡✳
• E(X) =...
• V ar(X) =...
• Mx(t) =...
✶✳✶✳✼✳ ❉✐s❝r❡t❡ ❯♥✐❢♦r♠ ❉✐str✐❜✉t✐♦♥✳ ❆ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ X t❤❛t t❛❦❡s ♦♥ ♦♥❡ ♦❢ t❤❡ ✈❛❧✉❡s 1,2,3, ..., N ✐s s❛✐❞ t♦ ❜❡ ❛ ❞✐s❝r❡t❡ ✉♥✐❢♦r♠ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ✇✐t❤ ♣❛r❛♠❡t❡r N ✐❢✱ ❢♦r s♦♠❡N >0✱
P(X =x) = 1
N, x= 1,2, ..., N ❊①❛♠✐♥❡ t❤❡ ♣r♦♣❡rt✐❡s ♦❢ ❛ ❞✐s❝r❡t❡ ✉♥✐❢♦r♠ r❛♥❞♦♠ ✈❛r✐❛❜❧❡✳
• E(X) =...
• V ar(X) =...
