❈❍❆P❚❊❘ ✶
❚❤❡ ❈❡♥tr❛❧ ▲✐♠✐t ❚❤❡♦r❡♠
❈❡♥tr❛❧ ▲✐♠✐t ❚❤❡♦r❡♠ st❛t❡s t❤❛t t❤❡ s✉♠ ♦❢ ❛ ❧❛r❣❡ ♥✉♠❜❡r ♦❢ ✐♥❞❡♣❡♥❞❡♥t r❛♥❞♦♠ ✈❛r✐❛❜❧❡s ❤❛s ❛ ❞✐str✐❜✉t✐♦♥ t❤❛t ✐s ❛♣♣r♦①✐♠❛t❡❧② ♥♦r♠❛❧✳ ❍❡♥❝❡✱ ✐t ♥♦t ♦♥❧② ♣r♦✈✐❞❡s ❛ s✐♠♣❧❡ ♠❡t❤♦❞ ❢♦r ❝♦♠♣✉t✐♥❣ ❛♣♣r♦①✐♠❛t❡ ♣r♦❜❛❜✐❧✐t✐❡s ❢♦r s✉♠s ♦❢ ✐♥❞❡♣❡♥❞❡♥t r❛♥❞♦♠ ✈❛r✐❛❜❧❡s✱ ❜✉t ❛❧s♦ ❤❡❧♣s ❡①♣❧❛✐♥ t❤❡ r❡♠❛r❦❛❜❧❡ ❢❛❝t t❤❛t t❤❡ ❡♠♣✐r✐❝❛❧ ❢r❡q✉❡♥❝✐❡s ♦❢ s♦ ♠❛♥② ♥❛t✉r❛❧ ♣♦♣✉❧❛t✐♦♥s ❡①❤✐❜✐t ❜❡❧❧✲s❤❛♣❡❞ ✭t❤❛t ✐s✱ ♥♦r♠❛❧✮ ❝✉r✈❡s✳
❚❤❡♦r❡♠ ✶✳ ✭❈❡♥tr❛❧ ▲✐♠✐t ❚❤❡♦r❡♠✮ ▲❡tX1, X2, ..., Xi, ... ❜❡ ❛ s❡q✉❡♥❝❡
♦❢ ✐♥❞❡♣❡♥❞❡♥t r❛♥❞♦♠ ✈❛r✐❛❜❧❡s ✇✐t❤E(Xi) =µ❛♥❞V ar(Xi) =σ2<∞✱ ❛♥❞ t❤❡
❝♦♠♠♦♥ ❞✐str✐❜✉t✐♦♥ ❢✉♥❝t✐♦♥ F ❛♥❞ ♠♦♠❡♥t✲❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ M ❞❡✜♥❡❞ ✐♥ ❛ ♥❡✐❣❤❜♦r❤♦♦❞ ♦❢ ③❡r♦✳ ▲❡t Zn =
n
P
i=1
Xi−nµ
√
nσ ✱ t❤❡♥ t❤❡ ❧✐♠✐t✐♥❣ ❞✐str✐❜✉t✐♦♥ ♦❢ Zn ✐s
t❤❡ st❛♥❞❛r❞ ♥♦r♠❛❧
Pr♦♦❢✳ ❚❤❡ ❧✐♠✐t✐♥❣ ❞✐str✐❜✉t✐♦♥ ♦❢Zn✐s t❤❡ st❛♥❞❛r❞ ♥♦r♠❛❧✱ ✇r✐tt❡♥Zn→d
Z∼N(0,1) ❛sn→ ∞✱ ✐❢ lim
n✙∞MZn(t) =MZ(t) =e
t2
2 ▼♦♠❡♥t✲❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ♦❢Zn ✐s✱
MZn(t) =E e Znt
=E
e
n
P
i=1Xi−
nµ
√
nσ
t
=Ee√tnσ(X−µ)
n
=MX−µ
t
√nσ
n
m(t) =MX−µ
t
√nσ
❤❛s ❛ ❚❛②❧♦r s❡r✐❡s ❡①♣❛♥s✐♦♥ ❛❜♦✉t ③❡r♦✿
m(t) =m(0) +m′(0)t+m′′(0)t
2
2 +εs ✇❤❡r❡ εs
s2 →0 ❛ss→0✳ ❙✐♥❝❡ m(0) = 1✱m′(0) = 0✱ ❛♥❞ m′′(0) =σ
2✳ ❆sn → ∞
❛♥❞ t
√nσ →0 ❛♥❞
MX−µ
t
√
nσ
= 1 +1 2σ
2
t
√
nσ
2
+εn
✶✳ ❚❍❊ ❈❊◆❚❘❆▲ ▲■▼■❚ ❚❍❊❖❘❊▼ ✸
❙♦❧✉t✐♦♥✳ ❚❤❡ ❡①❛❝t ♣r♦❜❛❜✐❧✐t② ✐s
P(Y20≥9) = 1−P(Y20≤8)
= 1− 8 P y=0
20 y
0.5y0.520−y
= 0.7483
❙✐♥❝❡ ❛ ❜✐♥♦♠✐❛❧ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ✐s t❤❡ s✉♠ ♦❢ ✐♥❞❡♣❡♥❞❡♥t ❇❡r♥♦✉❧❧✐ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s✱ ✐ts ❞✐str✐❜✉t✐♦♥ ❝❛♥ ❜❡ ❛♣♣r♦①✐♠❛t❡❞ ❜② ❛ ♥♦r♠❛❧ ❞✐str✐❜✉t✐♦♥ ✭❊①❛♠♣❧❡ ❄❄✮✳ ❚❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ✐s ❜❡st ✇❤❡♥ t❤❡ ❜✐♥♦♠✐❛❧ ❞✐str✐❜✉t✐♦♥ ✐s s②♠♠❡tr✐❝✱ t❤❛t ✐s✱ ✇❤❡♥p= 1
2✳ ❆ ❢r❡q✉❡♥t❧② ✉s❡❞ r✉❧❡ ♦❢ t❤✉♠❜ ✐s t❤❛t t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ✐s r❡❛s♦♥❛❜❧❡
✇❤❡♥ np >5 ❛♥❞ n(1−p) >5✳ ❚❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ✐s ❡s♣❡❝✐❛❧❧② ✉s❡❢✉❧ ❢♦r ❧❛r❣❡ ✈❛❧✉❡s ♦❢n✱ ❢♦r ✇❤✐❝❤ t❛❜❧❡s ❛r❡ ♥♦t r❡❛❞✐❧② ❛✈❛✐❧❛❜❧❡✳ ❆ ♥♦r♠❛❧ ❛♣♣r♦①✐♠❛t✐♦♥ ✐s
P(Y20≥9) = 1−P(Y20≤8)
= 1−Φ
8−20(0.5) √
20(0.5)(0.5)
= 1−Φ8−10
√
5
= 1−Φ (−0.89) = 0.8133
❇❡❝❛✉s❡ t❤❡ ❜✐♥♦♠✐❛❧ ❞✐str✐❜✉t✐♦♥ ✐s ❞✐s❝r❡t❡ ❛♥❞ t❤❡ ♥♦r♠❛❧ ❞✐str✐❜✉t✐♦♥ ✐s ❝♦♥t✐♥✲ ✉♦✉s✱ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ❝❛♥ ❜❡ ✐♠♣r♦✈❡❞ ❜② ♠❛❦✐♥❣ ❛ ❝♦♥t✐♥✉✐t② ❝♦rr❡❝t✐♦♥✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ❡❛❝❤ ❜✐♥♦♠✐❛❧ ♣r♦❜❛❜✐❧✐t②✱ b(y;n, p)❤❛s t❤❡ s❛♠❡ ✈❛❧✉❡ ❛s t❤❡ ❛r❡❛ ♦❢ ❛ r❡❝t❛♥❣❧❡ ♦❢ ❤❡✐❣❤t b(y;n, p) ❛♥❞ ✇✐t❤ t❤❡ ✐♥t❡r✈❛❧[y−0.5, y+ 0.5]❛s ✐ts ❜❛s❡✱ ❜❡❝❛✉s❡ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❜❛s❡ ✐s ♦♥❡ ✉♥✐t✳ ❚❤❡ ❛r❡❛ ♦❢ t❤✐s r❡❝t❛♥❣❧❡ ❝❛♥ ❜❡ ❛♣♣r♦①✐✲ ♠❛t❡❞ ❜② t❤❡ ❛r❡❛ ✉♥❞❡r t❤❡ ♣❞❢ ♦❢Y ∼N(np, npq)✳ ■♥ ❣❡♥❡r❛❧✱ ✐❢Yn ∼BIN(n, p)
❛♥❞a≤b❛r❡ ✐♥t❡❣❡r✱ t❤❡♥P(a≤Yn ≤b) = Φ
b+0.5−np
√npq
−Φa−0.5−np
√npq
✳ ❊①❛♠♣❧❡ ✹✳ ▲❡tX1, X2, ..., Xn ❜❡ ❛ r❛♥❞♦♠ s❛♠♣❧❡ ❢r♦♠ ❛ ✉♥✐❢♦r♠ ❞✐str✐❜✉✲
t✐♦♥✱ Xi ∼ U N IF(0,1) ❛♥❞ ❧❡t Yn = n P i=1
Xi✳ ❋✐♥❞ t❤❡ ❧✐♠✐t✐♥❣ ❞✐str✐❜✉t✐♦♥ ♦❢
Yn✳
❙♦❧✉t✐♦♥✳ ❙✐♥❝❡ E(Xi) = 12 ❛♥❞ V ar(Xi) = 121 ❛♥❞ ❢r♦♠ ❚❤❡♦r❡♠ ✶✱ t❤❡♥
Yn = n P i=1
Xi →d Z ∼ N nµ, nσ2
= N n 2,
n 12
✳ ❆♥❞ ❢♦r ❡①❛♠♣❧❡ n = 12✱ t❤❡♥ Y12−6→d Z∼N(0,1)✳
❊①❛♠♣❧❡ ✺✳ ❙✉♣♣♦s❡ t❤❛t t❤❡ ♥✉♠❜❡r ♦❢ ✐♥s✉r❛♥❝❡ ❝❧❛✐♠s✱N✱ ✜❧❡❞ ✐♥ ❛ ②❡❛r ✐s P♦✐ss♦♥ ❞✐str✐❜✉t❡❞ ✇✐t❤E(N) = 10,000✳ ❯s❡ t❤❡ ♥♦r♠❛❧ ❛♣♣r♦①✐♠❛t✐♦♥ t♦ t❤❡ P♦✐ss♦♥ t♦ ❛♣♣r♦①✐♠❛t❡P(N >10,200)✳
