04 bài giảng số 4 biến ngẫu nhiên, hàm các biến ngẫu nhiên và các định lý giới hạn

✺✷
❈❤➢➡♥❣ ✹✿ ❇✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ❤❛✐ ❝❤✐Ò✉✲❍➭♠ ❝➳❝ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥

✶✳
❚❛ ❦Ý ❤✐Ö✉ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ❤❛✐ ❝❤✐Ò✉ ✭❳✱❨✮✱ tr♦♥❣ ➤ã ❳✱❨ ➤➢î❝ ❣ä✐ ❧➭ ❝➳❝ t❤➭♥❤ ♣❤➬♥
❝ñ❛ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ❤❛✐ ❝❤✐Ò✉✱ ♠ç✐ t❤➭♥❤ ♣❤➬♥ ❧➭ ♠ét ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ♠ét ❝❤✐Ò✉✳ ❈➳❝
❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ❳✱❨ ➤➢î❝ ①Ðt ♠ét ❝➳❝❤ ➤å♥❣ t❤ê✐✳
❚➢➡♥❣ tù✱ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ♥ ❝❤✐Ò✉ ❝ã t❤Ó ①❡♠ ①Ðt ♥❤➢ ❤Ö ♥ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ♠ét ❝❤✐Ò✉✳
❱Ý ❞ô✿ ▼ét ♠➳② s➯♥ ①✉✃t r❛ ♠ét s➯♥ ♣❤➮♠✱ ♥Õ✉ ❦Ý❝❤ ❝ì ❝ñ❛ ♥ã ➤➢î❝ ➤♦ ❜➺♥❣ ❝❤✐Ò✉ ❞➭✐
❳✱ ❝❤✐Ò✉ ré♥❣ ❨✱ t❛ ❝ã ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ❤❛✐ ❝❤✐Ò✉ ✭❳✱❨✮✱ ♥Õ✉ t❤➟♠ ❝➯ ❜Ò ❞➭② ❩ t❛ ❝ã ❜✐Õ♥
♥❣➱✉ ♥❤✐➟♥ ✸ ❝❤✐Ò✉ ✭❳✱❨✱❩✮✳
❇✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ♥❤✐Ò✉ ❝❤✐Ò✉ ❣ä✐ ❧➭ rê✐ r➵❝ ♥Õ✉ ❝➳❝ t❤➭♥❤ ♣❤➬♥ ❝ñ❛ ♥ã ❧➭ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥
rê✐ r➵❝✱ ❣ä✐ ❧➭ ❧✐➟♥ tô❝ ♥Õ✉ ❝➳❝ t❤➭♥❤ ♣❤➬♥ ❝ñ❛ ♥ã ❧✐➟♥ tô❝✳
§

❑❤➳✐ ♥✐Ö♠ ✈Ò ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ❤❛✐ ❝❤✐Ò✉

✷✳
❇➯♥❣ ♣❤➞♥ ♣❤è✐ ①➳❝ s✉✃t ❝ñ❛ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ❤❛✐ ❝❤✐Ò✉ rê✐ r➵❝ ❧✐Öt ❦➟ t✃t ❝➳❝ ❝➳❝ ❣✐➳
trÞ ❝ã t❤Ó ❝ã ❝ñ❛ ♥ã ✈➭ ❝➳❝ ①➳❝ s✉✃t t➢➡♥❣ ø♥❣✳
§

❇➯♥❣ ♣❤➞♥ ♣❤è✐ ①➳❝ s✉✃t ❝ñ❛ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ❤❛✐ ❝❤✐Ò✉

❨ \❳

x1

x2

y1

P (x1 , y1 )

P (x2 , y1 )

y2

P (x1 , y2 )

P (x2 , y2 )

✳✳✳✳

✳✳✳✳

✳✳✳✳

yj

P (x1 , yj )

P (x2 , yj )

✳✳✳✳

✳✳✳✳

✳✳✳✳

ym

P (x1 , ym ) P (x2 , ym )

✳✳✳
✳✳✳
✳✳✳
✳✳✳
✳✳✳
✳✳✳
✳✳✳

xi
P (xi , y1 )
P (xi , y2 )

✳✳✳
P (xi , yj )

✳✳✳
P (xi , ym )

❚r♦♥❣ ➤ã P (x , y ) = P (X = x , Y = y )✱ i = 1, n✱ j = 1, m
❈➳❝ P (x , y ) ♣❤➯✐ t❤á❛ ♠➲♥ ➤✐Ò✉ ❦✐Ö♥
i

i

j

i

j

j



0 ≤ P (xi , yj ) ≤ 1



n
i=1

m
j=1

P (xi , yj ) = 1

✯❇➯♥❣ ♣❤➞♥ ♣❤è✐ ①➳❝ s✉✃t ❜✐➟♥ ❝ñ❛ t❤➭♥❤ ♣❤➬♥ ❳

✳✳✳
✳✳✳
✳✳✳
✳✳✳
✳✳✳
✳✳✳
✳✳✳

xn

P (xn , y1 )
P (xn , y2 )

✳✳
P (xn , yj )

✳✳
P (xn , ym )




P
P (xi ) =

m
j=1

x1

x2

P (x1 ) P (x2 )

xi
P (xi )

xn
P (xn )

P (xi , yj )

ố st ủ t

P
P (xj ) =

n
i=1

Đ



y1

y2

P (y1 )

P (y2 )

yj
P (yj )




ym
P (ym )

P (xi , yj )

ố st ủ ế ề

ị ĩ

ố st ủ ế ề í ệ ị ở
F (x, y) = P (X < x, Y < y)

ề t ì ọ trị ủ ố st t ỗ ể st ể
ế trị t ột ó ó ỉ ì ẽ
í t

í t
0 F (x, y) 1

í t
t từ ố số
ứ sử x < x
ó P (X < x , Y < y) = P (X < x , Y < y) + P (x

1

2

2

1

1

X < x2 , Y < y)

P (X < x2 , Y < y) P (X < x1 , Y < y) = P (x1 X < x2 , Y < y) 0
F (x2 , y) F (x1 , y) 0
F (x2 , y) F (x1 , y)

í t



F (, y) = 0 F (x, +) = 0
F (, +) = 0, F (+, +) = 1


✺✹
✰❚Ý♥❤ ❝❤✃t ✹✳

✲❤➭♠ ♣❤➞♥ ❜è ①➳❝ s✉✃t ❜✐➟♥ ❝ñ❛ r✐➟♥❣ t❤➭♥❤ ♣❤➬♥ ❳✳
F (+∞, y) = F (y)✲❤➭♠ ♣❤➞♥ ❜è ①➳❝ s✉✃t ❜✐➟♥ ❝ñ❛ r✐➟♥❣ t❤➭♥❤ ♣❤➬♥ ❨✳
✰❍Ö q✉➯✿

F (x, +∞) = F1 (x)
2

P (x1 < X < x2 , Y < y) = F (x2 , y) − F (x1 , y)
P (X < x2 , y1 < Y < y2 ) = F (x, y2 ) − F (x, y1 )
P (x1 < X < x2 , y1 < Y < y2 ) = F (x2 , y2 ) − F (x1 , y2 ) − F (x2 , y1 ) + F (x1 , y1 )

✹✳
●✐➯ sö ✭❳✱❨✮ ❧➭ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ❤❛✐ ❝❤✐Ò✉ ❧✐➟♥ tô❝
§

❍➭♠ ♠❐t ➤é ①➳❝ s✉✃t ❝ñ❛ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ❤❛✐ ❝❤✐Ò✉

✶✳➜Þ♥❤ ♥❣❤Ü❛✳

❑Ý ❤✐Ö✉ ❢✭①✱②✮ ❝❤♦ ❜ë✐ ❝➠♥❣ t❤ø❝
f (x, y) =

∂ 2 F (x, y)
∂x∂y

✷✳❚Ý♥❤ ❝❤✃t✳

✰❚Ý♥❤ ❝❤✃t ✶✿ f (x, y) ≥ 0
✰❚Ý♥❤ ❝❤✃t ✷✿ ❳➳❝ s✉✃t ➤Ó ✭❳✱❨✮ ♥❤❐♥ ❣✐➳ trÞ tr♦♥❣ ♠ét ♠✐Ò♥ ❉
P ((x, Y ) ∈ D) =

f (x, y)dxdy
D

✰❚Ý♥❤ ❝❤✃t ✸✿

x

y

−∞

−∞

F (x, y) =

✰❚Ý♥❤ ❝❤✃t ✹✿

+∞

f (x, y)dxdy
+∞

f (x, y)dxdy = 1
−∞

−∞

✰●❤✐ ❝❤ó✿
●ä✐ f (x) ❧➭ ❤➭♠ ♠❐t ➤é ①➳❝ s✉✃t ❜✐➟♥ ❝ñ❛ t❤➭♥❤ ♣❤➬♥ ❳✱ t❛ ❝ã
1

+∞

f1 (x) =

f (x, y)dy
−∞

❚❤❐t ✈❐②✿
dF1 (x)
dF (x, +∞)
d
f1 (x) =
=
=
dx
dx
dx

x

+∞

+∞

f (x, y)dxdy =
−∞

−∞

f (x, y)dy
−∞

✺✺
❚➢➡♥❣ tù✿ f (y) ❧➭ ❤➭♠ ♠❐t ➤é ①➳❝ s✉✃t ❜✐➟♥ ❝ñ❛ t❤➭♥❤ ♣❤➬♥ ❨
2

+∞

f (x, y)dx

f2 (y) =
−∞

§

✺✳

◗✉② ❧✉❐t ♣❤➞♥ ♣❤è✐ ①➳❝ s✉✃t ❝ã ➤✐Ò✉ ❦✐Ö♥

❝ñ❛ ❝➳❝ t❤➭♥❤ ♣❤➬♥ ❝ñ❛ ❤Ö ❤❛✐ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥

❳Ðt ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ rê✐ r➵❝ ✭❳✱❨✮
●ä✐ P (x /y ) ❧➭ ❝➳❝ ①➳❝ s✉✃t ❝ã ➤✐Ò✉ ❦✐Ö♥ ➤Ó t❤➭♥❤ ♣❤➬♥ ❳ ♥❤❐♥ ❣✐➳ trÞ x ✈í✐ ➤✐Ò✉ ❦✐Ö♥
t❤➭♥❤ ♣❤➬♥ ❨ ♥❤❐♥ ❣✐➳ trÞ y ✳
❚❛ ❝ã ❜➯♥❣ ♣❤➞♥ ♣❤è✐ ①➳❝ s✉✃t ❝ã ➤✐Ò✉ ❦✐Ö♥ ❝ñ❛ t❤➭♥❤ ♣❤➬♥ ❳ ✈í✐ ➤✐Ò✉ ❦✐Ö♥ Y = y ❝ã
❞➵♥❣✿
✶✳

i

j

i

j

j

X/yj

P

x1

x2

P (x1 /yj ) P (x2 /yj )

❚r♦♥❣ ➤ã✿
P (xi /yj ) =

❈❤ó ý✿

n

✳✳✳
✳✳✳

xi
P (xi /yj )

✳✳✳
✳✳✳

xn
P (xn /yj )

P (xi , yj )
P (yj )
m

P (xi /yj ) = 1,
i=1

P (yj /xi ) = 1
j=1

❱Ý ❞ô ✶✿
❚õ ❦Õt q✉➯ ♣❤➞♥ tÝ❝❤ sè ❧✐Ö✉ t❤è♥❣ ❦➟ tr♦♥❣ t❤➳♥❣ ✈Ò ❞♦❛♥❤ sè ❜➳♥ ❤➭♥❣ ❳ ✈➭ ❝❤✐ ♣❤Ý
q✉➯♥❣ ❝➳♦ ❨ ✭➤➡♥ ✈Þ tr✐Ö✉ ➤å♥❣✮ ❝ñ❛ ♠ét ❝➠♥❣ t② t❤✉ ➤➢î❝ ❜➯♥❣ ♣❤➞♥ ♣❤è✐ ①➳❝ s✉✃t ➤å♥❣
t❤ê✐✿
Y \X

✶
✶✱✺
✷

✶✵✵
✵✱✶✺
✵✱✵✺
✵✱✵✶

✷✵✵
✵✱✶
✵✱✷
✵✱✵✺

✸✵✵
✵✱✵✹
✵✱✶✺
✵✱✷✺

❛✳ ▲❐♣ ❜➯♥❣ ♣❤➞♥ ♣❤è✐ ①➳❝ s✉✃t ❝ñ❛ ❞♦❛♥❤ sè ❜➳♥ ❤➭♥❣✳
❜✳ ▲❐♣ ❜➯♥❣ ♣❤➞♥ ♣❤è✐ ①➳❝ s✉✃t ❝ñ❛ ❞♦❛♥❤ sè ❜➳♥ ❤➭♥❣ ❦❤✐ ❝❤✐ ♣❤Ý q✉➯♥❣ ❝➳♦ ❧➭ ✶✱✺ tr✐Ö✉✳


✺✻
●✐➯✐✿
❛✳ ❇➯♥❣ ♣❤➞♥ ♣❤è✐ ❝ñ❛ ❞♦❛♥❤ sè ❜➳♥ ❤➭♥❣✿
❳ ✶✵✵ ✷✵✵ ✸✵✵
P ✵✱✷✶ ✵✱✸✺ ✵✱✹✹
❜✳
P (X = 100/Y = 1, 5) =

❚➢➡♥❣ tù t❛ ❝ã ❜➯♥❣ ♣❤➞♥ ♣❤è✐ s❛✉

P (X = 100, Y = 1, 5)
0, 05
=
= 0, 125
P (Y = 1, 5)

0, 4

❳✴❨❂✶✱✺ ✶✵✵ ✷✵✵ ✸✵✵
P ✵✱✶✷✺ ✵✱✺ ✵✱✸✼✺
❳Ðt ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ❧✐➟♥ tô❝ ✭❳✱❨✮
❍➭♠ ♠❐t ➤é ①➳❝ s✉✃t ❝ã ➤✐Ò✉ ❦✐Ö♥ ❝ñ❛ t❤➭♥❤ ♣❤➬♥ ❳ ✈í✐ ❨❂②✱ ❦Ý ❤✐Ö✉ f (x/y)
✷✳

f (x/y) =

f (x, y)
=
f2 (y)

f (x, y)
+∞
−∞

f (x, y)dx

❚➢➡♥❣ tù✱ ❤➭♠ ♠❐t ➤é ❝ã ➤✐Ò✉ ❦✐Ö♥ ❝ñ❛ t❤➭♥❤ ♣❤➬♥ ❨ ✈í✐ ➤✐Ò✉ ❦✐Ö♥ ❳❂①✿
f (y/x) =

f (x, y)
=
f1 (x)

f (x, y)
+∞
−∞

f (x, y)dy

❈❤ó ý✿
◆Õ✉ ❳✱❨ ❧➭ ➤é❝ ❧❐♣ t❤× ♣❤➞♥ ♣❤è✐ ①➳❝ s✉✃t ❝ã ➤✐Ò✉ ❦✐Ö♥ ❜➺♥❣ ♣❤➞♥ ♣❤è✐ ①➳❝ s✉✃t ❦❤➠♥❣
❝ã ➤✐Ò✉ ❦✐Ö♥✱ ❞♦ ➤ã✿
◆Õ✉ ❳✱ ❨ rê✐ r➵❝✿
P (xi , yj ) = P (xi ).P (yj /xi ) = P (yj ).P (xi /yj ) = P (xi ).P (yj ) (1)

◆Õ✉ ❳✱❨ ❧✐➟♥ tô❝✿
f (x, y) = f1 (x).f2 (y) (2)

✭✶✮✱✭✷✮ ❝❤Ý♥❤ ❧➭ ➤✐Ò✉ ❦✐Ö♥ ❝➬♥ ✈➭ ➤ñ ➤Ó ❤❛✐ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ❳✱❨ ➤é❝ ❧❐♣✳
§

✻✳

❈➳❝ t❤❛♠ sè ➤➷❝ tr➢♥❣ ❝ñ❛ ❤Ö ❤❛✐ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥


✺✼
✶✳❑ú ✈ä♥❣✿

◆Õ✉ ❳✱❨ rê✐ r➵❝
n

n

E(X) =

m

xi P (xi ) =
i=1
m

E(Y ) =

xi P (xi , yj )
i=1 j=1
m
n

yj P (yj ) =
j=1

yj P (xi , yj )
j=1 i=1

◆Õ✉ ❳✱❨ ❧✐➟♥ tô❝
+∞

+∞

+∞

−∞
+∞

−∞

+∞

−∞

−∞

xf (x, y)dxdy

xf1 (x)dx =

E(X) =
−∞
+∞

yf (x, y)dxdy

yf2 (y)dy =

E(Y ) =
−∞
✷✳P❤➢➡♥❣ s❛✐✿

◆Õ✉ ❳✱❨ rê✐ r➵❝
n

n

x2i P (xi , yj ) − E 2 (X)

[xi − E(X)] p(xi ) =

V (X) =
i=1

◆Õ✉ ❳✱❨ ❧✐➟♥ tô❝

m

2

i=1 j=1

+∞

+∞

+∞

[x − E(X)]2 f1 (x)dx =

V (X) =
−∞

x2 f (x, y)dxdy − E ( X)
−∞

−∞

❚➢➡♥❣ tù t❛ ❝ã ❝➠♥❣ t❤ø❝ tÝ♥❤ ❱✭❨✮✳
❱Ý ❞ô ✷✿

❚õ ❜➯♥❣ ♣❤➞♥ ♣❤è✐ ①➳❝ s✉✃t ❱❉✶✱ tÝ♥❤ ❣✐➳ trÞ tr✉♥❣ ❜×♥❤ ✈➭ ♣❤➢➡♥❣ s❛✐ ❝ñ❛ ❝❤✐ ♣❤Ý q✉➯♥❣
❝➳♦❄
❚❛ ❝ã ❜➯♥❣ ♣❤➞♥ ♣❤è✐ ①➳❝ s✉✃t ❝ñ❛ ❝❤✐ ♣❤Ý q✉➯♥❣ ❝➳♦
❨ ✶ ✶✱✺ ✷
P ✵✱✷✾ ✵✱✹ ✵✱✸✶
E(Y ) = 1.0, 29 + 1, 5.0, 4 + 2.0, 31 = 1, 51
E(Y 2 ) = 12 .0, 29 + 1, 52 .0, 4 + 22 .0, 31 = 2, 43
V (Y ) = E(Y 2 ) − E 2 (Y ) = 0, 1499
✸✳❍✐Ö♣ ♣❤➢➡♥❣ s❛✐✿


✺✽
❍✐Ö♣ ♣❤➢➡♥❣ s❛✐ ❝ñ❛ ❤❛✐ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ❳✱❨ ❧➭ ❦ú ✈ä♥❣ t♦➳♥ ❝ñ❛ tÝ❝❤ ❝➳❝ s❛✐ ❧Ö❝❤ ❝ñ❛
❝➳❝ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ➤ã ✈í✐ ❦ú ✈ä♥❣ t♦➳♥ ❝ñ❛ ❝❤ó♥❣✱ ❦Ý ❤✐Ö✉ ❈♦✈✭❳✱❨✮
Cov(X, Y ) = E{[X − E(X)][Y − E(Y )]}

❇✐Õ♥ ➤æ✐ t❛ ❝ã ❝➠♥❣ t❤ø❝✿
m

n

xi yj P (xi , yj ) − E(X).E(Y ),

Cov(X, Y ) =

♥Õ✉ ❳✱❨ rê✐ r➵❝

j=1 i=1

♥Õ✉ ❳✱❨ ❧✐➟♥ tô❝

❚õ ➤Þ♥❤ ♥❣❤Ü❛ t❛ t❤✃② ➤➡♥ ✈Þ ❝ñ❛ ❤✐Ö♣ ♣❤➢➡♥❣ s❛✐ ❜➺♥❣ tÝ❝❤ ➤➡♥ ✈Þ ❝ñ❛ ❳ ✈➭ ❨✱ ➤Ó ❦❤➽❝
♣❤ô❝✱ ♥❣➢ê✐ t❛ ➤➢❛ r❛ ❦❤➳✐ ♥✐Ö♠ ❤Ö sè t➢➡♥❣ q✉❛♥
+∞

+∞

x(x, y)dxdy − E(X).E(Y ),

Cov(X, Y ) =

−∞

−∞

✹✳ ❍Ö sè t➢➡♥❣ q✉❛♥✳

ρxy =

Cov(X, Y )
σX σY

❱Ý ❞ô ✸✿
❈❤♦ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ✭❳✱❨✮ tr♦♥❣ ❱❉✶✳ ❚Ý♥❤ ❈♦✈✭❳✱❨✮✱ ρ(X, Y )✳
●✐➯✐✿
❚❛ ❝ã ❊✭❳✮❂✷✷✸✱ ❱✭❳✮❂✺✾✼✶✳
❊✭❨✮❂✶✱✺✶✱ ❱✭❨✮❂✵✱✶✹✾✾
❈♦✈✭❳✱❨✮❂❊✭❳❨✮✲❊✭❳✮❊✭❨✮❂✶✼✱✷✼
ρxy = 0, 577

✯▼ét sè tÝ♥❤ ❝❤✃t✿

✶✳ ρ = ρ
✷✳−1 ≤ ρ ≤ 1
✸✳◆Õ✉ ❳✱❨ ➤é❝ ❧❐♣ t❤× ρ = 0
✹✳ ➜➷❝ ❜✐Öt ♥Õ✉ ρ = ±1 t❤× ❳✱❨ ♣❤ô t❤✉é❝ t✉②Õ♥ tÝ♥❤ ✭❨❂❛❳✰❜✮✳
❍✐Ö♣ ♣❤➢➡♥❣ s❛✐ ✈➭ ❤Ö sè t➢➡♥❣ q✉❛♥ ➤➷❝ tr➢♥❣ ❝❤♦ ♠ø❝ ➤é ❝❤➷t ❝❤Ï ❝ñ❛ ♠è✐ ❧✐➟♥ ❤Ö ♣❤ô
t❤✉é❝ ❣✐÷❛ ❝➳❝ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ❳ ✈➭ ❨✳
◆Õ✉ ρ = 0✱ t❛ ♥ã✐ ❳✱❨ ❦❤➠♥❣ ❝ã ♠è✐ q✉❛♥ ❤Ö t➢➡♥❣ q✉❛♥✳
❈❤ó ý✿
◆Õ✉ ρ = 0 t❤× ❝❤➢❛ ❝❤➽❝ ❳✱❨ ➤é❝ ❧❐♣
xy

yx

xy

xy

xy

xy

xy


✺✾
❍❛✐ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ❣ä✐ ❧➭ t➢➡♥❣ q✉❛♥ ✈í✐ ♥❤❛✉ ♥Õ✉ ❤✐Ö♣ ♣❤➢➡♥❣ s❛✐ ✭❤Ö sè
t➢➡♥❣ q✉❛♥✮ ❦❤➳❝ ❦❤➠♥❣✱ ❣ä✐ ❧➭ ❦❤➠♥❣ t➢➡♥❣ q✉❛♥ ♥Õ✉ ❝❤ó♥❣ ❜➺♥❣ ❦❤➠♥❣✳
◆❤➢ ✈❐② ✷ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ❦❤➠♥❣ t➢➡♥❣ q✉❛♥ t❤× ❝ã t❤Ó ➤é❝ ❧❐♣ ❤♦➷❝ ♣❤ô t❤✉é❝✳
✺✳◆Õ✉ ❳ ✈➭ ❨ ❧➭ ❤❛✐ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ♣❤ô t❤✉é❝ t❤×✿
➜Þ♥❤ ♥❣❤Ü❛✿

V (X + Y ) = V (X) + V (Y ) + 2Cov(X, Y )
V (X − Y ) = V (X) + V (Y ) − 2Cov(X, Y )

❚æ♥❣ q✉➳t✿ V (aX ± bY ) = a V (X) + b V (Y ) ± 2abCov(X, Y )
❱Ý ❞ô ✹✿
❈ã ✷ ❧♦➵✐ ❝æ ♣❤✐Õ✉ ❆✱❇ ➤➢î❝ ❜➳♥ ❝ã ❧➲✐ s✉✃t t➢➡♥❣ ø♥❣ ❧➭ ✷ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ❝ã ❜➯♥❣ ♣❤➞♥
♣❤è✐ ①➳❝ s✉✃t
2

❨❳
✵
✹
✻

2

✲✷
✵
✵✱✵✺
✵✱✶

✵
✵✱✵✺
✵✱✶
✵✱✵✺

✺
✵✱✵✺
✵✱✷✺

✵✱✶

✶✵
✵✱✶
✵✱✶✺
✵

❛✳◆Õ✉ ➤➬✉ t➢ t♦➭♥ ❜é ✈➭♦ ❝æ ♣❤✐Õ✉ ❇ t❤× ❧➲✐ s✉✃t ❦ú ✈ä♥❣ ✈➭ ♠ø❝ ➤é rñ✐ r♦ ❧➭ ❜❛♦ ♥❤✐➟✉✳
❜✳➜➬✉ t➢ t❤❡♦ tû ❧Ö ♥➭♦ t❤× rñ✐ r♦ ✈Ò ❧➲✐ s✉✃t t❤✃♣ ♥❤✃t✳
●✐➯✐✿
❛✳ ❇➯♥❣ ♣❤➞♥ ♣❤è✐ ①➳❝ s✉✃t ❝ñ❛ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ❨
❨ ✵ ✹ ✻
P ✵✱✷ ✵✱✺✺ ✵✱✷✺
❚❛ ❝ã ❊✭❨✮❂✵✳✵✱✷✰✹✳✵✱✺✺✰✻✳✵✱✵✱✷✺❂✸✱✼
E(Y 2 ) = 17, 8
V (Y ) = 4, 11

❜✳ ●ä✐ ♣ ❧➭ tû ❧Ö ➤➬✉ t➢ ✈➭♦ ❆✱ ✶✲♣ ❧➭ tû ❧Ö ➤➬✉ t➢ ✈➭♦ ❇
▲➲✐ s✉✃t t❤✉ ➤➢î❝✿ pX + (1 − p)Y
❉♦ ❳✱❨ ❦❤➠♥❣ ➤é❝ ❧❐♣✱ t❛ ❝ã✿
V (pX + (1 − p)Y ) = p2 V (X) + (1 − p)2 V (Y ) + 2p(1 − p)Cov(X, Y )

✱

E(X) = 4, 2 V (X) = 17, 96


✻✵
Cov(X, Y ) =

xi yj Pij − E(X).E(Y ) = −3, 14

❘ñ✐ r♦ ✈Ò ❧➲✐ s✉✃t t❤✃♣ ♥❤✃t ❦❤✐ ♣❂✵✱✷✺✺✳
➜➬✉ t➢ ✷✺✱✺✪ ✈➭♦ ❝æ ♣❤✐Õ✉ ❆✳
§✼✳

❑ú ✈ä♥❣ t♦➳♥ ❝ã ➤✐Ò✉ ❦✐Ö♥ ✲❍➭♠ ❤å✐ q✉②

✶✳❑ú ✈ä♥❣ t♦➳♥ ❝ã ➤✐Ò✉ ❦✐Ö♥

❑ú ✈ä♥❣ t♦➳♥ ❝ã ➤✐Ò✉ ❦✐Ö♥ ❝ñ❛ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ❨ ✈í✐ ❳❂① ❝❤♦ ❜ë✐ ❝➠♥❣ t❤ø❝
m

E(Y /X = x) =

yj P (yj /x),

♥Õ✉ ✭❳✱❨✮ ❧➭ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ rê✐ r➵❝

j=1

♥Õ✉ ✭❳✱❨✮ ❧➭ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ❧✐➟♥ tô❝
❚➢➡♥❣ tù✱ t❛ ❝ò♥❣ ❝ã ➤Þ♥❤ ♥❣❤Ü❛ ❦ú ✈ä♥❣ t♦➳♥ ❝ã ➤✐Ò✉ ❦✐Ö♥ ❝ñ❛ ❳ ❦❤✐ ❨❂②✿ E(X/Y = y)
+∞

yf (y/x)dy,

E(Y /X = x) =

−∞

✷✳❍➭♠ ❤å✐ q✉②✳

❚❛ ♥❤❐♥ t❤✃② E(Y /X = x) ❧➭ ❤➭♠ ♣❤ô t❤✉é❝ ✈➭♦ ❣✐➳ trÞ ①
◆Õ✉ ➤➷t E(Y /X = x) = f (x)✲➤➢î❝ ❣ä✐ ❧➭ ❤➭♠ ❤å✐ q✉② ❝ñ❛ ❨ ➤è✐ ✈í✐ ❳
E(X/Y = y) = g(y)✲➤➢î❝ ❣ä✐ ❧➭ ❤➭♠ ❤å✐ q✉② ❝ñ❛ ❳ ➤è✐ ✈í✐ ❨
◆❤❐♥ ①Ðt✿❍➭♠ ❤å✐ q✉② ❝❤♦ ❜✐Õt sù ♣❤ô t❤✉é❝ ❝ñ❛ tr✉♥❣ ❜×♥❤ ❝ñ❛ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ♥➭②
✈➭♦ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ❦✐❛✳
❱Ý ❞ô✿ ❚❤è♥❣ ❦➟ ❞➞♥ sè ♠ét ♥➢í❝ ë ➤é t✉æ✐ tr➢ë♥❣ t❤➭♥❤ t❤❡♦ tr×♥❤ ➤é ❤ä❝ ✈✃♥ ❳ ✈➭ ❧ø❛
t✉æ✐ ❨✱ t❤✉ ➤➢î❝ ❜➯♥❣ s❛✉✳
❚×♠ ❤ä❝ ✈✃♥ tr✉♥❣ ❜×♥❤ t❤❡♦ ❧ø❛ t✉æ✐❄
❨ ❳ ❚❤✃t ❤ä❝ ✵ ❚✐Ó✉ ❤ä❝ ✶ ❚r✉♥❣ ❤ä❝ ✷ ➜➵✐ ❤ä❝ ✸
✷✺✲✸✺✭✸✵✮
✵✱✵✶
✵✱✵✸
✵✱✶✽
✵✱✵✼
✸✺✲✺✺✭✹✺✮
✵✱✵✷
✵✱✵✻
✵✱✷✶
✵✱✵✽
✺✺✲✶✵✵✭✼✵✮ ✵✱✵✺
✵✱✶
✵✱✶✺
✵✱✵✹
●✐➯✐
❚×♠ E(X/Y )
❱í✐ ❨❂✸✵
⇒ E(X/Y = 30) = 2, 069

✻✶
❳✴❨❂✸✵
✵
✶
✷
✸
P ✵✱✵✶✴✵✱✷✾ ✵✱✵✸✴✵✱✷✾ ✵✱✶✽✴✵✱✷✾ ✵✱✵✼✴✵✱✷✾
❚➢➡♥❣ tù✿ E(X/Y = 45) = 1, 946✱ E(X/Y = 70) = 1, 529
✭➜å t❤Þ ➤➢ê♥❣ ❤å✐ q✉②✮
§✽✳

◗✉② ❧✉❐t ♣❤➞♥ ♣❤è✐ ①➳❝ s✉✃t ❝ñ❛ ❤➭♠ ❝➳❝ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥

✶✳◗✉② ❧✉❐t ♣❤➞♥ ♣❤è✐ ①➳❝ s✉✃t ❝ñ❛ ❤➭♠ ♠ét ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥✳

●✐➯ sö ❨ ❧➭ ❤➭♠ ❝ñ❛ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ❳✿ Y = ϕ(X)
◆Õ✉ ❳ ❧➭ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ rê✐ r➵❝ t❤× ø♥❣ ✈í✐ ❝➳❝ ❣✐➳ trÞ ❦❤➳❝ ♥❤❛✉ ❝ñ❛ ❳✱ t❛ ❝ã ❝➳❝ ❣✐➳
trÞ ❦❤➳❝ ♥❤❛✉ ❝ñ❛ ❨ ✈➭ ❝➳❝ ①➳❝ s✉✃t t➢➡♥❣ ø♥❣ ✈í✐ ❝➳❝ ❣✐➳ trÞ ➤ã ❜➺♥❣ ♥❤❛✉✳
❱Ý ❞ô✶✿
●✐➯ sö ❳ ❝ã ❜➯♥❣ ♣❤➞♥ ♣❤è✐ ①➳❝ s✉✃t
❳ ✷ ✸
P ✵✱✻ ✵✱✹
Y = X2

❨ ♥❤❐♥ ❝➳❝ ❣✐➳ trÞ ✿ y = 2 = 4✱ y = 3
✈➭ t❛ ❝ã ❜➯♥❣ ♣❤➞♥ ♣❤è✐ ①➳❝ s✉✃t ❝ñ❛ ❨✿
1

2

2

2

=9

❨ ✹ ✾
P ✵✱✻ ✵✱✹
❱Ý ❞ô ✷✿
❈ò♥❣ ❧➭ ❤➭♠ Y = X ♥❤➢♥❣ ♥Õ✉ ❳ ❝ã ❜➯♥❣ ♣❤➞♥ ♣❤è✐ ①➳❝ s✉✃t✿
2

❳ ✲✷ ✷ ✸
P ✵✱✹ ✵✱✺ ✵✱✶
❑❤✐ ➤ã✿ P (y ) = P (Y = 4) = P (X = 2) + P (X = −2) = 0, 9
1

P (y2 ) = P (Y = 9) = P (X = 3) = 0, 1

❉♦ ➤ã ❜➯♥❣ ♣❤➞♥ ♣❤è✐ ①➳❝ s✉✃t ❝ñ❛ ❨✿


✻✷
❨ ✹ ✾
P ✵✱✾ ✵✱✶
✯ ◆Õ✉ ❳ ❧➭ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ❧✐➟♥ tô❝ ✈í✐ ❤➭♠ ♠❐t ➤é ①➳❝ s✉✃t ❢✭①✮✱ ♥❣➢ê✐ t❛ ❝❤ø♥❣ ♠✐♥❤
➤➢î❝ r➺♥❣ ♥Õ✉ Y = ϕ(X) ❧➭ ❦❤➯ ✈✐✱ ➤➡♥ ➤✐Ö✉ t❤× ❤➭♠ ♠❐t ➤é ❣✭②✮ ❝ñ❛ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥
❨ ❝❤♦ ❜ë✐

g(y) = f (ψ(y))|ψ (y)|

✭tr♦♥❣ ➤ã X = ψ(Y ) ❧➭ ❤➭♠ ♥❣➢î❝ ❝ñ❛ Y = ϕ(X)✮
❱Ý ❞ô ✸✿
❈❤♦ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ X ∼ N (µ, σ )✱ Y = aX + b✱ ❚×♠ ♣❤➞♥ ♣❤è✐ ①➳❝ s✉✃t ❝ñ❛ ❨✳
●✐➯✐✿
❚❛ ❝ã✿ ψ(y) = x = ⇒ |ψ (y)| = | |
2

y−b
x

1
a

y−b
(y−(aµ+b)2
( a −µ)2
1
1
1
−
√ e 2(aσ)2
g(y) = √ e− 2σ2 .
=
|a|
σ 2π
σ|a| 2π

❑▲✿ ◆❤➢ ✈❐② ♥Õ✉ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ❳ ♣❤➞♥ ♣❤è✐ ❝❤✉➮♥ t❤× ♠ét ❤➭♠ t✉②Õ♥ tÝ♥❤ ❜✃t ❦× ❝ñ❛

♥ã ❝ò♥❣ ♣❤➞♥ ♣❤è✐ t❤❡♦ q✉② ❧✉❐t ❝❤✉➮♥✳
✷✳◗✉② ❧✉❐t ♣❤➞♥ ♣❤è✐ ①➳❝ s✉✃t ❝ñ❛ ❤➭♠ ❤❛✐ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥✳

❈❤♦ ❤➭♠ ❤❛✐ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ Z = ϕ(X, Y )
❚❛ ①Ðt tr➢ê♥❣ ❤î♣ ➤➷❝ ❜✐Öt✿ Z = X + Y
❱Ý ❞ô✿
❈❤♦ ❜➯♥❣ ♣❤➞♥ ♣❤è✐ ①➳❝ s✉✃t ❝ñ❛ ❝➳❝ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ❳✱❨✿
❳ ✶ ✷
P ✵✱✹ ✵✱✻
❨ ✸ ✹
P ✵✱✷ ✵✱✽
▲❐♣ ❜➯♥❣ ♣❤➞♥ ♣❤è✐ ①➳❝ s✉✃t ❝ñ❛ Z = X + Y


✻✸
●✐➯✐✿
❚❛ ♣❤➯✐ t×♠ t✃t ❝➯ ❝➳❝ ❣✐➳ trÞ ❝ã t❤Ó ❝ã ❝ñ❛ Z ✈➭ ❝➳❝ ①➳❝ s✉✃t t➢➡♥❣ ø♥❣
z = 1 + 3 = 4✱z = 1 + 4 = 5✱z = 2 + 3 + 5✱z = 2 + 4 = 6
1

2

3

4

P (Z = 4) = P (X = 1).P (Y = 3) = 0, 4.0, 2 = 0, 08
P (Z = 5) = P (X = 1).P (Y = 4) + P (X = 2).P (Y = 3) = 0, 44
P (Z = 6) = P (X = 2).P (Y = 4) = 0, 48

❩ ✹ ✺ ✻
P ✵✱✵✽ ✵✱✹✹ ✵✱✹✽
✸✳❈➳❝ t❤❛♠ sè ➤➷❝ tr➢♥❣ ❝ñ❛ ❤➭♠ ❝➳❝ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥✳

✯●✐➯ sö ❳ ❧➭ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ rê✐ r➵❝✱ P (X = x ) = p ✱ i = 1, n
❑ú ✈ä♥❣ ✈➭ ♣❤➢➡♥❣ s❛✐ ❝ñ❛ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ Y = ϕ(X) ➤➢î❝ ①➳❝ ➤Þ♥❤ ❜ë✐ ❝➠♥❣ t❤ø❝✿
i

i

n

E(Y ) = E(ϕ(X)) =

ϕ(xi )pi
i=1
n

{ϕ(xi ) − E[ϕ(X)]}2 pi

V (Y ) = V (ϕ(X)) =
i=1
n

ϕ2 (xi )pi − {E[ϕ(X)]}2

=
i=1

◆Õ✉ ❳ ❧➭ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ❧✐➟♥ tô❝ ✈í✐ ❤➭♠ ♠❐t ➤é ①➳❝ s✉✃t f (x)

+∞

ϕ(x)f (x)dx

E(Y ) = E(ϕ(X)) =
−∞
+∞

{ϕ(x) − E[ϕ(X)]}2 f (x)dx

V (Y ) = V (ϕ(X)) =
−∞
+∞

ϕ2 (x)f (x)dx − {E[ϕ(X)]}2

=
−∞

❱Ý ❞ô✿
❈❤♦ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ❳ ❝ã ❜➯♥❣ ♣❤➞♥ ♣❤è✐ ①➳❝ s✉✃t
❚×♠ ❦ú ✈ä♥❣ ✈➭ ♣❤➢➡♥❣ s❛✐ ❝ñ❛ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ Y = ϕ(X) = X
●✐➯✐✿
❚❛ ❝ã ϕ(1) = 2, ϕ(3) = 10, ϕ(5) = 26

2

+1

✻✹
❳ ✶ ✸ ✺
P ✵✱✷ ✵✱✺ ✵✱✸
❉♦ ➤ã✿
E(Y ) = 2.0, 2 + 10.0, 5 + 26.0, 3 = 13, 2
V (Y ) = 22 .0, 2 + 102 .0, 5 + 262 .0, 3 − (13, 2)2 = 79, 36


✻✺
❈❤➢➡♥❣ ✺✿ ❈➳❝ ➤Þ♥❤ ❧ý ❣✐í✐ ❤➵♥

✶✳
◆Õ✉ ❳ ❧➭ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ❝ã E(X) < +∞, V (X) < +∞ t❤× ✈í✐ ♠ä✐
§

❝ã✿

❇✃t ➤➻♥❣ t❤ø❝ ❚r➟❜➢s❡♣

P (|X − E(X)| < ) ≥ 1 −

>0

tï② ý✱ t❛

V (X)
2

❈❤ø♥❣ ♠✐♥❤✿
❚❛ sÏ ❝❤ø♥❣ ♠✐♥❤ ❝❤♦ tr➢ê♥❣ ❤î♣ ❳ ❧➭ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ rê✐ r➵❝✱ P (X = x ) = p

●✐➯ sö ❝ã ❦ ❣✐➳ trÞ x , x , ...x t❤á❛ ♠➲♥ |x − E(X)| < ❀ ♥✲❦ ❣✐➳ trÞ x ❝ß♥ ❧➵✐ t❤á❛ ♠➲♥✿
i

1

2

k

i

i

i

|xi − E(X)| ≥

❚❛ ❝ã✿

P (|X − E(X)| < ) = 1 − P (|X − E(X)| ≥ )
n

(xi − E(X)) pi ≥

V (X) =

❍➡♥ ♥÷❛✿ p

n
2

i=1
k+1

n
2

(xi − E(X)) pi ≥
i=k+1

+ pk+2 + ... + pn

n
2

k+1

pi =

2

pi
i=k+1

❧➭ ①➳❝ s✉✃t ➤Ó ❳ ♥❤❐♥ ♠ét tr♦♥❣ ❝➳❝ ❣✐➳ trÞ x

k+1 , ...xn

⇒ pk+1 + pk+2 + ... + pn = P (|X − E(X)| ≥ )
⇒ V (X) ≥

2

P (|X − E(X)| ≥ )

⇒ P (|X − E(X)| ≥ ) ≤

V (X)
2

❤❛② P (|X − E(X)| < ) ≥ 1 −
✯◆❤❐♥ ①Ðt✿
✰ ❇✃t ➤➻♥❣ t❤ø❝ ❝❤♦ ♣❤Ð♣ ➤➳♥❤ ❣✐➳ ❝❐♥ tr➟♥ ❤♦➷❝ ❝❐♥ ❞➢í✐ ①➳❝ s✉✃t ➤Ó ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥
❳ ♥❤❐♥ ❣✐➳ trÞ s❛✐ ❧Ö❝❤ s♦ ✈í✐ ❦ú ✈ä♥❣ t♦➳♥ ❝ñ❛ ♥ã ❧í♥ ❤➡♥ ❤♦➷❝ ♥❤á ❤➡♥ ✶ sè
✰❇✃t ➤➻♥❣ t❤ø❝ ➤➢î❝ ➳♣ ❞ô♥❣ ♠➭ ❦❤➠♥❣ ❝➬♥ ❜✐Õt q✉② ❧✉❐t ♣❤➞♥ ♣❤è✐ ①➳❝ s✉✃t ❝ñ❛ ❳✳
❱Ý ❞ô✿
❚❤✉ ♥❤❐♣ tr✉♥❣ ❜×♥❤ ❤➭♥❣ ♥➝♠ ❝ñ❛ ❞➞♥ ❝➢ ♠ét ✈ï♥❣ ❧➭ ✼✵✵ ❯❙❉✱ ➤é ❧Ö❝❤ ❝❤✉➮♥ ✶✷✵❯❙❉✳
❳➳❝ ➤Þ♥❤ ❦❤♦➯♥❣ t❤✉ ♥❤❐♣ ❤➭♥❣ ♥➝♠ ①✉♥❣ q✉❛♥❤ ❣✐➳ trÞ tr✉♥❣ ❜×♥❤ ❝ñ❛ Ýt ♥❤✃t ✾✺✪ ❞➞♥
❝➢ ✈ï♥❣ ➤ã✳
●✐➯✐✿
●ä✐ ❳ ❧➭ t❤✉ ♥❤❐♣ ❤➭♥❣ ♥➝♠ ❝ñ❛ ❞➞♥ ❝➢ ✈ï♥❣ ➤ã
V (X)
2

P (|X − E(X)| < ) ≥ 0, 95


✻✻
⇒
P (|X − 700| < ) ≥ 1 −

1202
2

= 0, 95

⇒ = 536, 656 ⇒ X ∈ (700 − 536, 656; 700 + 536, 656) = (163, 344; 1236, 656)

❱❐② Ýt ♥❤✃t ✾✺✪ ❞➞♥ ❝➢ ✈ï♥❣ ➤ã ❝ã t❤✉ ♥❤❐♣ tr♦♥❣ ❦❤♦➯♥❣ (163, 344; 1236, 656)✳
§

✷✳

➜Þ♥❤ ❧ý ❚r➟❜➢s❡♣

◆Õ✉ ❝➳❝ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ X , X , ..X ➤é❝ ❧❐♣ tõ♥❣ ➤➠✐✱ ❝➳❝ ❦ú ✈ä♥❣ ❤÷✉ ❤➵♥✱ ❝➳❝
♣❤➢➡♥❣ s❛✐ V (X ) ≤ C, ∀ > 0 ❜Ð tï② ý✱ t❛ ❝ã✿
1

2

n

i

lim P (|

n→∞

X1 + X2 + ... + Xn E(X1 ) + E(X2 ) + ... + E(Xn )

−
|< )=1
n
n

❈❤ø♥❣ ♠✐♥❤✿
❳Ðt X =
❚❛ ❝ã✿ E(X) =
E(X) =

X1 +X2 +...+Xn
n
n
1
i=1 E(Xi )
n
n
1
i=1 V (Xi )
n2

➳♣ ❞ô♥❣ ❜✃t ➤➻♥❣ t❤ø❝ ❚r➟❜➢s❡♣ ➤è✐ ✈í✐ X
P (|X − E(X)| < ) ≥ 1 −

V (X)
2

=1−

V (Xi )

n2 2

⇒
P (|X − E(X)| < ) ≥ 1 −

nC
C
=1− 2
2
2
n
n

⇒
lim P (|X − E(X)| < ) ≥ 1

n→∞

❤❛②

lim P (|X − E(X)| < ) = 1

n→∞

✯➜➷❝ ❜✐Öt✱ ❦❤✐ ❝➳❝ X ❝ï♥❣ ❦ú ✈ä♥❣ E(X ) = m✱ t❛ ❝ã✿
i

i

lim P (|

n→∞

X1 + X2 + ... + Xn
− m)| < ) = 1
n

❑Õt ❧✉❐♥ tr➟♥ ➤➢î❝ ❣ä✐ ❧➭ ➜Þ♥❤ ❧ý ▲✉❐t sè ❧í♥ ❝ñ❛ ❚r➟❜➢s❡♣✳
❇➯♥ ❝❤✃t ❝ñ❛ ➤Þ♥❤ ❧ý ❚r➟❜➢s❡♣ ❧➭ ❝❤ø♥❣ ♠✐♥❤ sù ❤é✐ tô t❤❡♦ ①➳❝ s✉✃t ❝ñ❛ tr✉♥❣ ❜×♥❤ sè
❤ä❝ ❝ñ❛ ♠ét sè ❧í♥ ❝➳❝ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ✈Ò tr✉♥❣ ❜×♥❤ sè ❤ä❝ ❝ñ❛ ❝➳❝ ❦ú ✈ä♥❣ t♦➳♥✳



Đ



ị ý r

ọ t st t ệ ế ố tr
é tử ộ st t ệ ế ố ó tr ỗ é tử ớ > 0 ó

ị ýị ý t số ớ ủ r

lim P (|f p| < ) = 1

n

ứ ụ t tứ rs ế f =
X

n

E(f ) = n1 E(X), V (f ) =

1
V
n2

(X)

số t ệ ế ố tr é tử X B(n, p) E(X) = np
V (X) = np(1 p) t t tứ rs t ó
ét ị ý r ứ sự ộ tụ t st ủ t st t ệ
ế ố tr é tử ộ ề st t ệ ế ố ó tr ỗ é tử
số é tử t ứ tỏ sự ổ ị ủ t st q trị st ủ
ế ố ó
Đ



trị ý ớ tr t

tr
ị ĩ

eitX

ý ệ

tr ủ ế ỳ ọ ủ ế

X (t)

X (t) = E(eitX ) = E(cos(tX)) + iE(sin(tX))

ế ế rờ r P (X = x ) = p
ế ế tụ t ộ
i

i

X (t)

=

X (t)

=

n
itxi
pi
i=1 e
+ itx
e f (x)dx


tí t ủ tr

| (t)| 1

ế Y = aX + b tì (t) = e (at)
ế X , X , ...X ế ộ tì
X

ibt

Y

1

2

X

n

n

X1 +X2 +...+Xn (t) =

Xk (t)
k=1

ế tồ t E|X| tì (t) ũ tồ t ế t ọ t
F (x) ị t tr (t)
k

X

X

✻✽
❢✳ ❈❤♦ {F (x)} ❧➭ ❞➲② ❤➭♠ ♣❤➞♥ ♣❤è✐ ①➳❝ s✉✃t t➢➡♥❣ ø♥❣ ✈í✐ ❞➲② ❤➭♠ ➤➷❝ tr➢♥❣ ϕ (t)
❑❤✐ ➤ã {F (x)} ❤é✐ tô tí✐ F (x) ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ ϕ (t) ❤é✐ tô tí✐ ❤➭♠ ➤➷❝ tr➢♥❣ ϕ(t) t➢➡♥❣
ø♥❣ ✈í✐ F (x)✳
❱Ý ❞ô ✶✿ ❈❤♦ X ∼ A(p)✳ ❚×♠ ❤➭♠ ➤➷❝ tr➢♥❣ ϕ (t)
❚❛ ❝ã ❜➯♥❣ ♣❤➞♥ ♣❤è✐ ①➳❝ s✉✃t ❝ñ❛ ❳✿
n

n

n

n

X

❳ ✵ ✶
P ✶✲♣ ♣
ϕX (t) = eit0 (1 − p) + eit1 p = peit + 1 − p

❱Ý ❞ô ✷✿ ❈❤♦ X ∼ B(n, p)✳ ❚×♠ ❤➭♠ ➤➷❝ tr➢♥❣✳
❚❛ ❝ã✿
n

n
itx

e

ϕX (t) =

Cnx px (1

n−x

− p)

Cnx (peit )x (1 − p)n−x = (peit + 1 − p)n

=
x=0

x=0

◆❤❐♥ ①Ðt✿ ❚õ ❤❛✐ ✈Ý ❞ô tr➟♥ t❛ ❝ò♥❣ ❝ã t❤Ó s✉② r❛ ➤➢î❝ ♠è✐ q✉❛♥ ❤Ö ❣✐÷❛ q✉② ❧✉❐t A(p)
✈➭ B(n, p)✳
❱Ý ❞ô ✸✿ ❈❤♦ X ∼ N (0, 1)✳ ❚×♠ ❤➭♠ ➤➷❝ tr➢♥❣✳
●✐➯✐✿
1
ϕX (t) = √
2π
t2
1
= √ e− 2
2π
t2
1
= √ e− 2

2π

+∞

eitx e
−∞
+∞

−x2
2

1

2

e− 2 (x−it) dx
−∞
+∞

+∞

1
dx = √
2π

1

e− 2 (x

2 −2itx−t2 )− t2

2

dx

−∞

✭➤➷tu = x − it)

u2

e− 2 du
−∞

2

=e

− t2

✷✳➜Þ♥❤ ❧ý ▲✐♥❞❡r❜❡r❣✲▲❡✇✐✿

✲◆Õ✉ X , X , ...X , .... ❧➭ ♠ét ❞➲② ❝➳❝ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥ ➤é❝ ❧❐♣ ❝ï♥❣ t✉➞♥ t❤❡♦ ✶ q✉② ❧✉❐t
♣❤➞♥ ♣❤è✐ ①➳❝ s✉✃t ♥➭♦ ➤ã ✈í✐ ❦ú ✈ä♥❣ t♦➳♥ ✈➭ ♣❤➢➡♥❣ s❛✐ ❤÷✉ ❤➵♥ E(X ) = a, V (X ) =
1

2

n

i

σ 2 , ∀i

t❤× q✉② ❧✉❐t ♣❤➞♥ ♣❤è✐ ①➳❝ s✉✃t ❝ñ❛ ❜✐Õ♥ ♥❣➱✉ ♥❤✐➟♥
Un − E(Un )
U˜n =
,
V (Un )

✈í✐U

n
n

=

Xi
i=1

i


✻✾
❤é✐ tô tí✐ q✉② ❧✉❐t ❝❤✉➮♥ ❤ã❛✱ ❦❤✐ n −→ ∞
❚ø❝ ❧➭✿

1
P (U˜n < x) = √
2π

x

t2

e− 2 dt
−∞

❚❛ t❤õ❛ ♥❤❐♥ ➤Þ♥❤ ❧ý tr➟♥✳
❱Ò ♠➷t t❤ù❝ ❤➭♥❤✱ ❝ã t❤Ó sö ❞ô♥❣ ➤Þ♥❤ ❧ý ✈í✐ ♥ ➤ñ ❧í♥
1
P (U˜n < x) ≈ √
2π
P (a < Un < b) ≈ φ0 (

+∞

t2

e− 2 dt = φ(x)
−∞

b − E(Un )
V (Un )

) − φ0 (

a − E(Un )
V (Un )

)

❱Ý ❞ô ✹✿ ❈❤ä♥ ♥❣➱✉ ♥❤✐➟♥ ✶✾✷ sè tr➟♥ ➤♦➵♥ ❬✵✱✶❪✳ ❚×♠ ①➳❝ s✉✃t ➤Ó tæ♥❣ sè ➤✐Ó♠ t❤✉ ➤➢î❝
❳ ♥➺♠ tr♦♥❣ ❦❤♦➯♥❣ (88, 104)✳
X ✱ X ❧➭ sè ❝❤✃♠ t❤✉ ➤➢î❝ ë ❧➬♥ ❝❤ä♥ t❤ø ✐✱ X ∼ U (0, 1)✳
●✐➯✐✿ ❚❛ ❝ã X =
i

192
i=1

E(Xi ) = 21 , V (Xi ) =

i

i

i

1
12

E(X) = nE(Xi ) = 96; V (X) = nV (Xi ) = 16

❚❛ ❝ã

P (88 < X < 104) ≈ φ0 (

104 − 96
88 − 96
√

) − φ0 ( √
) = 2φ0 (2) = 0, 9544
16
16