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Bài giảng Chương 3. Vectơ ngẫu nhiên môn Xác suất thống kê | Đại học Duy Tân

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Bài giảng Chương 3. Vectơ ngẫu nhiên môn Xác suất thống kê | Đại học Duy Tân. Tài liệu được sưu tầm giúp bạn tham khảo, ôn tập và đạt kết quả cao. Mời bạn đọc đón xem.

Môn: Xác xuất thống kê (STA 151) 144 tài liệu

Trường: Đại học Duy Tân 1.9 K tài liệu

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X
1
, ..., X
n
n
n (X
1
, ..., X
n
) n
n = 2
(X, Y ) F (x, y)
F
X,Y
(x, y) = P (X < x, Y < y)
(X, Y )
0 ≤ F
X,Y
(x, y) ≤ 1, ∀x, y ∈ R F
X,Y
(x, y)
F
X,Y
(x, y)
lim
x→+∞
y→+∞
F
X,Y
(x, y) = 1, lim
x→−∞
F
X,Y
(x, y) = 0, lim
y→−∞
F
X,Y
(x, y) = 0
X {x
i
, i ∈ I} Y
{y
j
, j ∈ J}
p
ij
= P (X = x
i
, Y = y
j
), i ∈ I, j ∈ J
❱➼ ❞# ✶
▼!" ❤!♣ ❝& ✷ ❜✐ "*➢♥❣ ✈➔ ✸ ❜✐ ✤❡♥✳ ▲➜② ♥❣➝✉ ♥❤✐➯♥ ✶ ✈✐➯♥ ❦❤<♥❣ ❤♦➔♥ ❧↕✐✱ *A✐ "✐➳♣ "C❝
❧➜② ♥❣➝✉ ♥❤✐➯♥ ✷ ✈✐➯♥ ❦❤<♥❣ ❤♦➔♥ ❧↕✐✳ ●E✐
X
✈➔
Y
❧➔ FG ❜✐ ✤❡♥ ❧➜② ✤HI❝ J ❧➛♥ ✶ ✈➔ ❧➛♥ ✷✳
❛✳ ▲➟♣ ❜↔♥❣ ♣❤➙♥ ♣❤G✐ ✤A♥❣ "❤P✐ ❝Q❛
X
✈➔
Y
✳
❜✳ ❚➼♥❤ ①→❝ F✉➜"
P (X + Y < 2)
✳
●✐↔✐✳
❛✳ ❚❛ ❝$
X
♥❤➟♥ ❝→❝ ❣✐→ +,-✿ ✵✱ ✶ ✈➔
Y
♥❤➟♥ ❝→❝ ❣✐→ +,-✿ ✵✱ ✶✱ ✷✳
P (X = 0, Y = 0) = P (X = 0)P (Y = 0|X = 0) = 2/5 ∗ 0 = 0
P (X = 0, Y = 1) = P (X = 0)P (Y = 1|X = 0) = 2/5 ∗
C
1
3
C
1
1
C
2
4
= 1/5
P (X = 0, Y = 2) = P (X = 0)P (Y = 2|X = 0) = 2/5 ∗
C
2
3
C
2
4
= 1/5
P (X = 1, Y = 0) = P (X = 1)P (Y = 0|X = 1) = 3/5 ∗
C
2
2
C
2
4
= 1/10
P (X = 1, Y = 1) = P (X = 1)P (Y = 1|X = 1) = 3/5 ∗
C
1
2
C
1
2
C
2
4
= 2/5
❚!♥ ❚❤➜% ❚& ✹✴✷✸
P (X = 1, Y = 2) = P (X = 1)P (Y = 2|X = 1) = 3/5 ∗
C
2
2
C
2
4
= 1/10
❇↔♥❣ ♣❤➙♥ ♣❤9✐ ✤;♥❣ +❤<✐✿
X
Y
✵ ✶ ✷
✵ ✵ ✶✴✺ ✶✴✺
✶ ✶✴✶✵ ✷✴✺ ✶✴✶✵
❜✳ ❳→❝ A✉➜+✿
P (X + Y < 2) = P(X = Y = 0) + P (X = 0, Y = 1) + P (X = 1, Y = 0)
= 0 + 1/5 + 1/10 = 3/10
❚!♥ ❚❤➜% ❚& ✺✴✷✸
❱➼ ❞# ✷
❈❤♦ ❤❛✐ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥
X , Y
✤!❝ ❧➟♣ ❝& ❜↔♥❣ ♣❤➙♥ ♣❤G✐✿
X
✶ ✷
P
✵✱✹ ✵✱✻
Y
✶ ✷ ✸
P
✵✱✸ ✵✱✸ ✵✱✹
▲➟♣ ❜↔♥❣ ♣❤➙♥ ♣❤G✐ ✤A♥❣ "❤P✐ ❝Q❛
(X, Y )
✳
●✐↔✐✳
❱➻
X , Y
✤F❝ ❧➟♣ ♥➯♥✿
P (X = 1, Y = 1) = 0, 4 ∗ 0, 3 = 0, 12; P (X = 1, Y = 2) = 0, 4 ∗ 0, 3 = 0, 12
P (X = 1, Y = 3) = 0, 4 ∗ 0, 4 = 0, 16; P (X = 2, Y = 1) = 0, 6 ∗ 0, 3 = 0, 18
P (X = 2, Y = 2) = 0, 6 ∗ 0, 3 = 0, 18; P (X = 2, Y = 3) = 0, 6 ∗ 0, 4 = 0, 24
❇↔♥❣ ♣❤➙♥ ♣❤9✐ ✤;♥❣ +❤<✐✿
X
Y
✶ ✷ ✸
✶ ✵✱✶✷ ✵✱✶✷ ✵✱✶✻
✷ ✵✱✶✽ ✵✱✶✽ ✵✱✷✹
❚!♥ ❚❤➜% ❚& ✻✴✷✸
p
i.
= P (X = x
i
) =
X
j
P (X = x
i
, Y = y
j
) =
X
j
p
ij
X X
X x
1
x
2
x
i
P p
1.
p
2.
p
i.
Y
P (x
i
|y
j
) = P (X = x
i
|Y = y
j
) =
P (X = x
i
, Y = y
j
)
P (Y = y
j
)
=
p
ij
p
.j
X x
i
Y = y
j
y
j
X
Y = y
j
X|Y = y
j
x
1
x
2
P P (x
1
|y
j
) P(x
2
|y
j
)
x
i
Y
X = x
i
❱➼ ❞# ✸
❱❡❝#♦ ♥❣➝✉ ♥❤✐➯♥
(X, Y )
❝, ♣❤➙♥ ♣❤/✐ ✤12❝ ❝❤♦ 3 ❜↔♥❣ 6❛✉✿
❨
❳
✶ ✷ ✸
✺ ✵✳✶ ✵✳✷ ✵✳✸
✻ ✵✳✵✽ ✵✳✶✻ ✵✳✶✻
❚➻♠ ♣❤➙♥ ♣❤/✐ ❜✐➯♥ ❝F❛
X , Y
✈➔ #➼♥❤
P (Y = 6|X = 1)
✳
●✐↔✐✳
❚❛ ❝#✿
❨
❳
✶ ✷ ✸
P (Y = y
j
)
✺ ✵✳✶ ✵✳✷ ✵✳✸ ✵✱✻
✻ ✵✳✵✽ ✵✳✶✻ ✵✳✶✻ ✵✱✹
P (X = x
i
)
✵✳✶✽ ✵✳✸✻ ✵✳✹✻
❚!♥ ❚❤➜% ❚& ✶✵✴✷✸
1❤➙♥ ♣❤6✐ ❜✐➯♥ ❝:❛
X
✿
X
✶ ✷ ✸
P
✵✱✶✽ ✵✱✸✻ ✵✱✹✻
1❤➙♥ ♣❤6✐ ❜✐➯♥ ❝:❛
Y
✿
Y
✺ ✻
P
✵✱✻ ✵✱✹
❳→❝ <✉➜?✿
P (Y = 6|X = 1) =
P (X = 1, Y = 6)
P (X = 1)
=
0, 08
0, 18
=
4
9
❚!♥ ❚❤➜% ❚& ✶✶✴✷✸
✷✳✷ *❤➙♥ ♣❤/✐ ✤1♥❣ 3❤4✐ ❧✐➯♥ 3#❝
❛✳ ✣:♥❤ ♥❣❤➽❛✿
❱❡❝?B ♥❣➝✉ ♥❤✐➯♥
(X, Y )
✤FG❝ ❣H✐ ❧➔ ❝# ♣❤➙♥ ♣❤6✐ ✤K♥❣ ?❤L✐ ❧✐➯♥ ?M❝
♥➳✉ ❤➔♠ ♣❤➙♥ ♣❤6✐ ✤K♥❣ ?❤L✐ ❝:❛
(X, Y )
❝# ?❤➸ ❜✐➸✉ ❞✐➵♥ S ❞↕♥❣
F
X,Y
(x, y) =
x
Z
−∞
y
Z
−∞
f
X,Y
(u, v)dudv
❍➔♠
f
X,Y
(u, v)
✤FG❝ ❣H✐ ❧➔ ❤➔♠ ♠➟? ✤W ✤K♥❣ ?❤L✐ ❝:❛
X , Y
✳
❚➼♥❤ ❝❤➜3✿
i)f
X,Y
(u, v) ≥ 0, ii)f
X,Y
(u, v) =
∂
2
F
X,Y
(u, v)
∂u∂v
, iii)
+∞
Z
−∞
+∞
Z
−∞
f
X,Y
(u, v)dudv = 1
❚!♥ ❚❤➜% ❚& ✶✷✴✷✸
❜✳ "❤➙♥ ♣❤'✐ ❜✐➯♥✿
f
X
(x) =
+∞
Z
−∞
f
X,Y
(x, y)dy
f
Y
(y) =
+∞
Z
−∞
f
X,Y
(x, y)dx
❝✳ ▼➟. ✤0 ❝1 ✤✐➲✉ ❦✐➺♥✿
❍➔♠
f
X|Y
(x|y) =
f
X,Y
(x, y)
f
Y
(y)
❧➔ ♠➟% ✤' ✤✐➲✉ ❦✐➺♥ ❝/❛
X
✈2✐ ✤✐➲✉ ❦✐➺♥
Y = y
✳
❚!♥ ❚❤➜% ❚& ✶✸✴✷✸
❱➼ ❞9 ✹
❱❡❝#$ ♥❣➝✉ ♥❤✐➯♥
(X, Y )
❝, ❤➔♠ ♠➟# ✤1
f
X,Y
(x, y) =
(
C(x + xy), (x, y) ∈ [0, 1] × [0, 1]
0, (x, y) /∈ [0, 1] × [0, 1]
❚➻♠
C, f
X
(x)
✈➔
f
Y |X
(y|x)
✳
●✐↔✐✳
❚❤❡♦ %➼♥❤ ❝❤➜% ❝/❛ ❤➔♠ ♠➟% ✤'✿
Z
+∞
−∞
Z
+∞
−∞
f
X,Y
(x, y)dxdy =
Z
1
0
Z
1
0
C(x + xy)dxdy =
C
2
Z
1
0
(y + 1)dy =
3C
4
= 1
❙✉② =❛✿
C = 4/3
✳ ▼➟% ✤' ❜✐➯♥ ❝/❛
X
✿
f
X
(x) =
Z
+∞
−∞
f
X,Y
(x, y)dy =
Z
1
0
4
3
(x + xy)dy = 2x, x ∈ [0, 1].
❚!♥ ❚❤➜% ❚& ✶✹✴✷✸
❉♦ ✤B✿
f
X
(x) =
(
2x, x ∈ [0, 1]
0, x /∈ [0, 1]
▼➟% ✤' ❝B ✤✐➲✉ ❦✐➺♥✿
f
Y |X
(y|x) =
f
X,Y (x,y)
f
X
(x)
=
4/3(x + xy)
2x
=
2
3
(y + 1), (x, y) ∈ [0, 1] × [0, 1]
❚!♥ ❚❤➜% ❚& ✶✺✴✷✸
✷✳✸ ❙$ ✤&❝ ❧➟♣ ❝+❛ ❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥
❍❛✐ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥
X
✈➔
Y
✤.❝ ❧➟♣ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐✿
F
X,Y
(x, y) = F
X
(x).F
Y
(y)
✲ ◆➳✉
X
✈➔
Y
❝8 ♣❤➙♥ ♣❤:✐ ✤;♥❣ <❤=✐ >=✐ >↕❝ <❤➻ ✤✐➲✉ ❦✐➺♥ <>➯♥ <>C <❤➔♥❤✿
P (X = x
i
, Y = y
j
) = P (X = x
i
).P (Y = y
j
), ∀i, j
✲ ◆➳✉
X
✈➔
Y
❝8 ♣❤➙♥ ♣❤:✐ ✤;♥❣ <❤=✐ ❧✐➯♥ <D❝ <❤➻ ✤✐➲✉ ❦✐➺♥ <>➯♥ <>C <❤➔♥❤✿
f
X,Y
(x, y) = f
X
(x).f
Y
(y)
❚!♥ ❚❤➜% ❚& ✶✻✴✷✸
✸✳ ❑8 ✈:♥❣ ❝; ✤✐➲✉ ❦✐➺♥
❑F ✈G♥❣ ❝8 ✤✐➲✉ ❦✐➺♥ ❝H❛ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥
Y
✈I✐ ✤✐➲✉ ❦✐➺♥
X = x
✱ ❦➼ ❤✐➺✉
E(Y |X = x)
✤LM❝ ①→❝ ✤P♥❤✿
✲ ◆➳✉
X
✈➔
Y
❝8 ♣❤➙♥ ♣❤:✐ ✤;♥❣ <❤=✐ >=✐ >↕❝ <❤➻✿
E(Y |X = x) =
X
j
y
j
P (Y = y
j
|X = x)
✲ ◆➳✉
X
✈➔
Y
❝8 ♣❤➙♥ ♣❤:✐ ✤;♥❣ <❤=✐ ❧✐➯♥ <D❝ <❤➻✿
E(Y |X = x) =
+∞
Z
−∞
yf
Y |X
(y|x)dy
❚!♥ ❚❤➜% ❚& ✶✼✴✷✸
✹✳ ❍✐➺♣ ♣❤AB♥❣ C❛✐ ✈➔ ❤➺ CE FAB♥❣ G✉❛♥
❛✳ ❍✐➺♣ ♣❤AB♥❣ C❛✐✿
✲ ❍✐➺♣ ♣❤LQ♥❣ R❛✐ ❝H❛ ❤❛✐ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥
X
✈➔
Y
✱ ❦➼ ❤✐➺✉
Cov(X, Y )
✤LM❝ ①→❝ ✤P♥❤
<❤❡♦ ❜✐➸✉ <❤V❝✿
Cov(X, Y ) = E
(X −EX)(Y − EY )
✲ ❍❛✐ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥
X
✈➔
Y
✤LM❝ ❣G✐ ❧➔ <LQ♥❣ W✉❛♥ ✈I✐ ♥❤❛✉ ♥➳✉
Cov(X, Y ) 6= 0
✈➔
❦❤X♥❣ <LQ♥❣ W✉❛♥ ♥➳✉
Cov(X, Y ) = 0
✳
❚➼♥❤ ❝❤➜F✿
✐✮
Cov(X, Y ) = Cov(Y, X)
✐✐✮
Cov(aX + b, cY + d) = acCov(X, Y ), a, b, c, d = const
✐✐✐✮
Cov(X, Y ) = E(XY ) − E(X).E(Y )
✐✈✮
D(X ±Y ) = D(X) + D(Y ) ± 2Cov(X, Y )
◆❤➟♥ ①➨F✿
✐✮ ◆➳✉
X , Y
✤.❝ ❧➟♣ <❤➻
Cov(X, Y ) = 0
✳ ❉♦ ✤8✱ <\ ✧<➼♥❤ ❝❤➜< ✤.❝ ❧➟♣✧ R✉② >❛ ✧<➼♥❤
❝❤➜< ❦❤X♥❣ <LQ♥❣ W✉❛♥✧✱ <✉② ♥❤✐➯♥ ✤✐➲✉ ♥❣LM❝ ❧↕✐ ♥8✐ ❝❤✉♥❣ ❦❤X♥❣ ✤`♥❣✳
❚!♥ ❚❤➜% ❚& ✶✽✴✷✸
✐✐✮ ❚#♦♥❣ '❤)❝ ❤➔♥❤✱ '❛ '➼♥❤
Cov(X, Y ) = E(XY ) − E(X).E(Y )
✱ '#♦♥❣ ✤0✿
E(XY ) =
P
x
i
y
j
p
ij
, X , Y
❝0 ♣❤➙♥ ♣❤4✐ ✤5♥❣ '❤6✐ #6✐ #↕❝
+∞
R
−∞
+∞
R
−∞
xyf
XY
(x, y)dxdy, X, Y
❝0 ♣❤➙♥ ♣❤4✐ ✤5♥❣ '❤6✐ ❧✐➯♥ ':❝
❜✳ ❍➺ $% &'(♥❣ +✉❛♥
❍➺ =4 '>?♥❣ @✉❛♥ ❝B❛ ❤❛✐ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥
X
✈➔
Y
✱ ❦➼ ❤✐➺✉
ρ(X, Y )
✤>H❝ ①→❝ ✤K♥❤ ❜L✐
❜✐➸✉ '❤N❝✿
ρ(X, Y ) =
Cov(X, Y )
p
D(X)D(Y )
❚➼♥❤ ❝❤➜&✿
✐✮
ρ(X, Y ) = ρ(Y, X)
✐✐✮
|ρ(X, Y )| ≤ 1
4 ♥❣❤➽❛ ❤➺ $% &'(♥❣ +✉❛♥✿
❍➺ =4 '>?♥❣ @✉❛♥ ✤♦ ♠N❝ ✤P ♣❤: '❤✉P❝ '✉②➳♥ '➼♥❤ ❣✐R❛
X
✈➔
Y
✳ ❑❤✐
|ρ(X, Y )|
❝➔♥❣ ❣➛♥ ✶ '❤➻ @✉❛♥ ❤➺ '✉②➳♥ '➼♥❤ ❣✐R❛
X
✈➔
Y
❝➔♥❣ ♠↕♥❤✳
❚!♥ ❚❤➜% ❚& ✶✾✴✷✸
❱➼ ❞8 ✺
❱❡❝#♦ ♥❣➝✉ ♥❤✐➯♥
(X, Y )
❝, ♣❤➙♥ ♣❤/✐ ✤12❝ ❝❤♦ 3 ❜↔♥❣ 6❛✉✿
❨
❳
✶ ✷ ✸
✶ ✵✳✶ ✵✳✶ ✵✳✷
✷ ✵✳✷ ✵✳✶ ✵✳✸
❚➼♥❤
E(Y |X = 1), Cov(X, Y ), ρ(X, Y )
✳
●✐↔✐✳
X❤➙♥ ♣❤4✐ ❜✐➯♥✿
X
✶ ✷ ✸
P
✵✱✸ ✵✱✷ ✵✱✺
Y
✶ ✷
P
✵✱✹ ✵✱✻
❳→❝ =✉➜' ❝0 ✤✐➲✉ ❦✐➺♥✿
P (Y = 1|X = 1) =
P (X = 1, Y = 1)
P (X = 1)
=
0, 1
0, 3
=
1
3
❚!♥ ❚❤➜% ❚& ✷✵✴✷✸
P (Y = 2|X = 1) =
P (X = 1, Y = 2)
P (X = 1)
=
0, 2
0, 3
=
2
3
X❤➙♥ ♣❤4✐ ❝0 ✤✐➲✉ ❦✐➺♥✿
Y |X = 1
✶ ✷
P
✶✴✸ ✷✴✸
❉♦ ✤0✿
E(Y |X = 1) =
P
y
i
P (Y = y
i
|X = 1) = 5/3
✳
❍✐➺♣ ♣❤>?♥❣ =❛✐✿
Cov(X, Y ) = E(XY ) − E(X)E(Y )
✱ '#♦♥❣ ✤0✿
E(X) =
P
p
i
x
i
= 2, 2; E(Y ) =
P
p
i
y
i
= 1, 6; E(XY ) =
P
p
ij
x
i
y
i
= 3, 5
⇒ Cov(X, Y ) = 3, 5 − 2, 2 ∗ 1, 6 = −0, 02
X❤>?♥❣ =❛✐✿
D(X) = E(X
2
) − (EX)
2
=
P
p
i
x
2
i
− 2, 2
2
= 0, 76
D(Y ) = E(Y
2
) − (EY )
2
=
P
p
i
y
2
i
− 1, 6
2
= 0, 24
⇒ ρ(X , Y ) =
Cov(X, Y )
p
D(X)D(Y )
=
−0, 02
√
0, 76 ∗ 0, 24
= −0, 0468
❚!♥ ❚❤➜% ❚& ✷✶✴✷✸
❱➼ ❞# ✻
❈❤♦ ✈❡❝&' ♥❣➝✉ ♥❤✐➯♥
(X, Y )
❝. ❤➔♠ ♠➟& ✤3 ✤4♥❣ &❤5✐✿
f(x, y) =
(
x + y, (x, y) ∈ [0, 1]
2
0, (x, y) /∈ [0, 1]
2
❚➼♥❤
Cov(X, Y ), D(X + 2Y )
✳
●✐↔✐✳
❤➙♥ ♣❤%✐ ❜✐➯♥✿
f
X
(x) =
Z
+∞
−∞
f(x, y)dy =
Z
1
0
(x + y)dy = x + 1/2, x ∈ [0, 1]
f
Y
(y) =
Z
+∞
−∞
f(x, y)dx =
Z
1
0
(x + y)dx = y + 1/2, y ∈ [0, 1]
E(X) =
Z
+∞
−∞
xf
X
(x)dx =
Z
1
0
x(x + 1/2)dx = 7/12; E(Y ) = 7/12
❚!♥ ❚❤➜% ❚& ✷✷✴✷✸
E(XY ) =
Z
+∞
−∞
Z
+∞
−∞
xyf(x, y)dxdy =
Z
1
0
Z
1
0
xy(x + y)dxdy = 1/3
❙✉② -❛✿
Cov(X, Y ) = E(XY ) − E(X)E(Y ) = 1/3 − (7/12)
2
= −1/144
✳
▼➦2 ❦❤→❝✱
D(X + 2Y ) = D(X) + D(2Y ) + 2Cov(X, 2Y ) = D(X) + 4D(Y ) + 4Cov(X , Y )
E(X
2
) =
Z
+∞
−∞
x
2
f
X
(x)dx =
Z
1
0
x
2
(x + 1/2)dx = 5/12; E(Y
2
) = 5/12
D(X) = E(X
2
) − (EX)
2
= 5/12 − (7/12)
2
= 11/144; D(Y ) = 11/144
❱➟②✱
D(X + 2Y ) = 11/144 + 4 ∗11/144 −4 ∗1/144 = 17/48
❚!♥ ❚❤➜% ❚& ✷✸✴✷✸

Tài liệu khác của Đại học Duy Tân

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X1, ..., Xn n n (X1, ..., Xn) n n = 2 (X, Y ) F (x, y)
FX,Y (x, y) = P (X < x, Y < y) (X, Y )
0 ≤ FX,Y (x, y) ≤ 1, ∀x, y ∈ R FX,Y (x, y) FX,Y (x, y) lim FX,Y (x, y) = 1, lim FX,Y (x, y) = 0, lim FX,Y (x, y) = 0 x→+∞ x→−∞ y→−∞ y→+∞ X {xi, i ∈ I} Y {yj, j ∈ J}
pij = P (X = xi, Y = yj), i ∈ I, j ∈ J ❱➼ ❞# ✶
▼!" ❤!♣ ❝& ✷ ❜✐ "*➢♥❣ ✈➔ ✸ ❜✐ ✤❡♥✳ ▲➜② ♥❣➝✉ ♥❤✐➯♥ ✶ ✈✐➯♥ ❦❤<♥❣ ❤♦➔♥ ❧↕✐✱ *A✐ "✐➳♣ "C❝
❧➜② ♥❣➝✉ ♥❤✐➯♥ ✷ ✈✐➯♥ ❦❤<♥❣ ❤♦➔♥ ❧↕✐✳ ●E✐ X ✈➔ Y ❧➔ FG ❜✐ ✤❡♥ ❧➜② ✤HI❝ J ❧➛♥ ✶ ✈➔ ❧➛♥ ✷✳
❛✳ ▲➟♣ ❜↔♥❣ ♣❤➙♥ ♣❤G✐ ✤A♥❣ "❤P✐ ❝Q❛ X ✈➔ Y ✳
❜✳ ❚➼♥❤ ①→❝ F✉➜" P (X + Y < 2)✳
●✐↔✐✳ ❛✳ ❚❛ ❝$ X ♥❤➟♥ ❝→❝ ❣✐→ +,-✿ ✵✱ ✶ ✈➔ Y ♥❤➟♥ ❝→❝ ❣✐→ +,-✿ ✵✱ ✶✱ ✷✳
P (X = 0, Y = 0) = P (X = 0)P (Y = 0|X = 0) = 2/5 ∗ 0 = 0 C1
P (X = 0, Y = 1) = P (X = 0)P (Y = 1|X = 0) = 2/5 ∗ 3 C11 = 1/5 C24 C2
P (X = 0, Y = 2) = P (X = 0)P (Y = 2|X = 0) = 2/5 ∗ 3 = 1/5 C24 C2
P (X = 1, Y = 0) = P (X = 1)P (Y = 0|X = 1) = 3/5 ∗ 2 = 1/10 C24 C1
P (X = 1, Y = 1) = P (X = 1)P (Y = 1|X = 1) = 3/5 ∗ 2 C12 = 2/5 C24 ❚!♥ ❚❤➜% ❚& ✹✴✷✸ C2
P (X = 1, Y = 2) = P (X = 1)P (Y = 2|X = 1) = 3/5 ∗ 2 = 1/10 C24
❇↔♥❣ ♣❤➙♥ ♣❤9✐ ✤;♥❣ +❤<✐✿ Y ✵ ✶ ✷ X ✵ ✵ ✶✴✺ ✶✴✺ ✶
✶✴✶✵ ✷✴✺ ✶✴✶✵ ❜✳ ❳→❝ A✉➜+✿
P (X + Y < 2) = P (X = Y = 0) + P (X = 0, Y = 1) + P (X = 1, Y = 0) = 0 + 1/5 + 1/10 = 3/10 ❚!♥ ❚❤➜% ❚& ✺✴✷✸ ❱➼ ❞# ✷
❈❤♦ ❤❛✐ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ X, Y ✤!❝ ❧➟♣ ❝& ❜↔♥❣ ♣❤➙♥ ♣❤G✐✿ X ✶ ✷ Y ✶ ✷ ✸ P ✵✱✹ ✵✱✻ P ✵✱✸ ✵✱✸ ✵✱✹
▲➟♣ ❜↔♥❣ ♣❤➙♥ ♣❤G✐ ✤A♥❣ "❤P✐ ❝Q❛ (X, Y )✳
●✐↔✐✳ ❱➻ X, Y ✤F❝ ❧➟♣ ♥➯♥✿
P (X = 1, Y = 1) = 0, 4 ∗ 0, 3 = 0, 12;
P (X = 1, Y = 2) = 0, 4 ∗ 0, 3 = 0, 12
P (X = 1, Y = 3) = 0, 4 ∗ 0, 4 = 0, 16;
P (X = 2, Y = 1) = 0, 6 ∗ 0, 3 = 0, 18
P (X = 2, Y = 2) = 0, 6 ∗ 0, 3 = 0, 18;
P (X = 2, Y = 3) = 0, 6 ∗ 0, 4 = 0, 24
❇↔♥❣ ♣❤➙♥ ♣❤9✐ ✤;♥❣ +❤<✐✿ Y ✶ ✷ ✸ X ✶
✵✱✶✷ ✵✱✶✷ ✵✱✶✻ ✷
✵✱✶✽ ✵✱✶✽ ✵✱✷✹ ❚!♥ ❚❤➜% ❚& ✻✴✷✸ X X pi. = P (X = xi) = P (X = xi, Y = yj) = pij j j X X X x1 x2 xi P p1. p2. pi. Y P (X = x p P (x i, Y = yj ) ij i|yj ) = P (X = xi|Y = yj ) = = P (Y = yj) p.j X xi Y = yj yj X Y = yj X|Y = yj x1 x2 P P (x1|yj) P (x2|yj) xi Y X = xi ❱➼ ❞# ✸
❱❡❝#♦ ♥❣➝✉ ♥❤✐➯♥ (X, Y ) ❝, ♣❤➙♥ ♣❤/✐ ✤12❝ ❝❤♦ 3 ❜↔♥❣ 6❛✉✿ ❳ ✶ ✷ ✸ ❨ ✺ ✵✳✶ ✵✳✷ ✵✳✸ ✻
✵✳✵✽ ✵✳✶✻ ✵✳✶✻
❚➻♠ ♣❤➙♥ ♣❤/✐ ❜✐➯♥ ❝F❛ X, Y ✈➔ #➼♥❤ P (Y = 6|X = 1)✳ ●✐↔✐✳ ❚❛ ❝#✿ ❳ ✶ ✷ ✸ P (Y = y ❨ j ) ✺ ✵✳✶ ✵✳✷ ✵✳✸ ✵✱✻ ✻
✵✳✵✽ ✵✳✶✻ ✵✳✶✻ ✵✱✹ P (X = xi)
✵✳✶✽ ✵✳✸✻ ✵✳✹✻ ❚!♥ ❚❤➜% ❚& ✶✵✴✷✸
1❤➙♥ ♣❤6✐ ❜✐➯♥ ❝:❛ X✿ X ✶ ✷ ✸ P
✵✱✶✽ ✵✱✸✻ ✵✱✹✻
1❤➙♥ ♣❤6✐ ❜✐➯♥ ❝:❛ Y ✿ Y ✺ ✻ P ✵✱✻ ✵✱✹ ❳→❝ <✉➜?✿ P (X = 1, Y = 6) 0, 08 4 P (Y = 6|X = 1) = = = P (X = 1) 0, 18 9 ❚!♥ ❚❤➜% ❚& ✶✶✴✷✸
✷✳✷ *❤➙♥ ♣❤/✐ ✤1♥❣ 3❤4✐ ❧✐➯♥ 3#❝
❛✳ ✣:♥❤ ♥❣❤➽❛✿ ❱❡❝?B ♥❣➝✉ ♥❤✐➯♥ (X, Y ) ✤FG❝ ❣H✐ ❧➔ ❝# ♣❤➙♥ ♣❤6✐ ✤K♥❣ ?❤L✐ ❧✐➯♥ ?M❝
♥➳✉ ❤➔♠ ♣❤➙♥ ♣❤6✐ ✤K♥❣ ?❤L✐ ❝:❛ (X, Y ) ❝# ?❤➸ ❜✐➸✉ ❞✐➵♥ S ❞↕♥❣ x y Z Z FX,Y (x, y) = fX,Y (u, v)dudv −∞ −∞
❍➔♠ fX,Y (u, v) ✤FG❝ ❣H✐ ❧➔ ❤➔♠ ♠➟? ✤W ✤K♥❣ ?❤L✐ ❝:❛ X, Y ✳ ❚➼♥❤ ❝❤➜3✿ +∞ +∞ ∂2F Z Z i)f X,Y (u, v) X,Y (u, v) ≥ 0, ii)fX,Y (u, v) = , iii) f ∂u∂v X,Y (u, v)dudv = 1 −∞ −∞ ❚!♥ ❚❤➜% ❚& ✶✷✴✷✸
❜✳ "❤➙♥ ♣❤'✐ ❜✐➯♥✿ +∞ Z fX(x) = fX,Y (x, y)dy −∞ +∞ Z fY (y) = fX,Y (x, y)dx −∞
❝✳ ▼➟. ✤0 ❝1 ✤✐➲✉ ❦✐➺♥✿ ❍➔♠ f f X,Y (x, y) X|Y (x|y) = fY (y)
❧➔ ♠➟% ✤' ✤✐➲✉ ❦✐➺♥ ❝/❛ X ✈2✐ ✤✐➲✉ ❦✐➺♥ Y = y✳ ❚!♥ ❚❤➜% ❚& ✶✸✴✷✸ ❱➼ ❞9 ✹
❱❡❝#$ ♥❣➝✉ ♥❤✐➯♥ (X, Y ) ❝, ❤➔♠ ♠➟# ✤1
(C(x + xy), (x, y) ∈ [0, 1] × [0, 1] fX,Y (x, y) = 0, (x, y) / ∈ [0, 1] × [0, 1]
❚➻♠ C, fX(x) ✈➔ fY |X(y|x)✳
●✐↔✐✳ ❚❤❡♦ %➼♥❤ ❝❤➜% ❝/❛ ❤➔♠ ♠➟% ✤'✿ Z +∞ Z +∞ Z 1 Z 1 C Z 1 3C fX,Y (x, y)dxdy = C(x + xy)dxdy = (y + 1)dy = = 1 −∞ −∞ 0 0 2 0 4
❙✉② =❛✿ C = 4/3✳ ▼➟% ✤' ❜✐➯♥ ❝/❛ X✿ Z +∞ Z 1 4 fX(x) = fX,Y (x, y)dy = (x + xy)dy = 2x, x ∈ [0, 1]. −∞ 0 3 ❚!♥ ❚❤➜% ❚& ✶✹✴✷✸ ❉♦ ✤B✿ (2x, x ∈ [0, 1] fX(x) = 0, x / ∈ [0, 1]
▼➟% ✤' ❝B ✤✐➲✉ ❦✐➺♥✿ f 4/3(x + xy) 2 f X,Y (x,y) Y |X (y|x) = =
= (y + 1), (x, y) ∈ [0, 1] × [0, 1] fX(x) 2x 3 ❚!♥ ❚❤➜% ❚& ✶✺✴✷✸
✷✳✸ ❙$ ✤&❝ ❧➟♣ ❝+❛ ❝→❝ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥
❍❛✐ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ X ✈➔ Y ✤.❝ ❧➟♣ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐✿ FX,Y (x, y) = FX(x).FY (y)
✲ ◆➳✉ X ✈➔ Y ❝8 ♣❤➙♥ ♣❤:✐ ✤;♥❣ <❤=✐ >=✐ >↕❝ <❤➻ ✤✐➲✉ ❦✐➺♥ <>➯♥ <>C <❤➔♥❤✿
P (X = xi, Y = yj) = P (X = xi).P (Y = yj), ∀i, j
✲ ◆➳✉ X ✈➔ Y ❝8 ♣❤➙♥ ♣❤:✐ ✤;♥❣ <❤=✐ ❧✐➯♥ ➯♥ <>C <❤➔♥❤✿ fX,Y (x, y) = fX(x).fY (y) ❚!♥ ❚❤➜% ❚& ✶✻✴✷✸
✸✳ ❑8 ✈:♥❣ ❝; ✤✐➲✉ ❦✐➺♥
❑F ✈G♥❣ ❝8 ✤✐➲✉ ❦✐➺♥ ❝H❛ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ Y ✈I✐ ✤✐➲✉ ❦✐➺♥ X = x✱ ❦➼ ❤✐➺✉ E(Y |X = x)
✤LM❝ ①→❝ ✤P♥❤✿
✲ ◆➳✉ X ✈➔ Y ❝8 ♣❤➙♥ ♣❤:✐ ✤;♥❣ <❤=✐ >=✐ >↕❝ <❤➻✿ X E(Y |X = x) = yjP (Y = yj|X = x) j
✲ ◆➳✉ X ✈➔ Y ❝8 ♣❤➙♥ ♣❤:✐ ✤;♥❣ <❤=✐ ❧✐➯♥ +∞ Z E(Y |X = x) = yfY |X(y|x)dy −∞ ❚!♥ ❚❤➜% ❚& ✶✼✴✷✸
✹✳ ❍✐➺♣ ♣❤AB♥❣ C❛✐ ✈➔ ❤➺ CE FAB♥❣ G✉❛♥
❛✳ ❍✐➺♣ ♣❤AB♥❣ C❛✐✿
✲ ❍✐➺♣ ♣❤LQ♥❣ R❛✐ ❝H❛ ❤❛✐ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ X ✈➔ Y ✱ ❦➼ ❤✐➺✉ Cov(X, Y ) ✤LM❝ ①→❝ ✤P♥❤
<❤❡♦ ❜✐➸✉ <❤V❝✿
Cov(X, Y ) = E (X − EX)(Y − EY )
✲ ❍❛✐ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ X ✈➔ Y ✤LM❝ ❣G✐ ❧➔ ❦❤X♥❣ ❚➼♥❤ ❝❤➜F✿ ✐✮ Cov(X, Y ) = Cov(Y, X)
✐✐✮ Cov(aX + b, cY + d) = acCov(X, Y ), a, b, c, d = const
✐✐✐✮ Cov(X, Y ) = E(XY ) − E(X).E(Y )
✐✈✮ D(X ± Y ) = D(X) + D(Y ) ± 2Cov(X, Y ) ◆❤➟♥ ①➨F✿
✐✮ ◆➳✉ X, Y ✤.❝ ❧➟♣ <❤➻ Cov(X, Y ) = 0✳ ❉♦ ✤8✱ <\ ✧<➼♥❤ ❝❤➜< ✤.❝ ❧➟♣✧ R✉② >❛ ✧<➼♥❤
❝❤➜< ❦❤X♥❣ ❚!♥ ❚❤➜% ❚& ✶✽✴✷✸
✐✐✮ ❚#♦♥❣ '❤)❝ ❤➔♥❤✱ '❛ '➼♥❤ Cov(X, Y ) = E(XY ) − E(X).E(Y )✱ '#♦♥❣ ✤0✿ P x  iyj pij ,
X, Y ❝0 ♣❤➙♥ ♣❤4✐ ✤5♥❣ '❤6✐ #6✐ #↕❝  E(XY ) = +∞ +∞ R R xyf  XY (x, y)dxdy,
X, Y ❝0 ♣❤➙♥ ♣❤4✐ ✤5♥❣ '❤6✐ ❧✐➯♥ ':❝ −∞ −∞
❜✳ ❍➺ $% &'(♥❣ +✉❛♥
❍➺ =4 '>?♥❣ @✉❛♥ ❝B❛ ❤❛✐ ❜✐➳♥ ♥❣➝✉ ♥❤✐➯♥ X ✈➔ Y ✱ ❦➼ ❤✐➺✉ ρ(X, Y ) ✤>H❝ ①→❝ ✤K♥❤ ❜L✐ ❜✐➸✉ '❤N❝✿ Cov(X, Y ) ρ(X, Y ) = pD(X)D(Y ) ❚➼♥❤ ❝❤➜&✿ ✐✮ ρ(X, Y ) = ρ(Y, X) ✐✐✮ |ρ(X, Y )| ≤ 1
4 ♥❣❤➽❛ ❤➺ $% &'(♥❣ +✉❛♥✿ ❍➺ =4 '>?♥❣ @✉❛♥ ✤♦ ♠N❝ ✤P ♣❤: '❤✉P❝ '✉②➳♥ '➼♥❤ ❣✐R❛
X ✈➔ Y ✳ ❑❤✐ |ρ(X, Y )| ❝➔♥❣ ❣➛♥ ✶ '❤➻ @✉❛♥ ❤➺ '✉②➳♥ '➼♥❤ ❣✐R❛ X ✈➔ Y ❝➔♥❣ ♠↕♥❤✳ ❚!♥ ❚❤➜% ❚& ✶✾✴✷✸ ❱➼ ❞8 ✺
❱❡❝#♦ ♥❣➝✉ ♥❤✐➯♥ (X, Y ) ❝, ♣❤➙♥ ♣❤/✐ ✤12❝ ❝❤♦ 3 ❜↔♥❣ 6❛✉✿ ❳ ✶ ✷ ✸ ❨ ✶ ✵✳✶ ✵✳✶ ✵✳✷ ✷ ✵✳✷ ✵✳✶ ✵✳✸
❚➼♥❤ E(Y |X = 1), Cov(X, Y ), ρ(X, Y )✳
●✐↔✐✳ X❤➙♥ ♣❤4✐ ❜✐➯♥✿ X ✶ ✷ ✸ Y ✶ ✷ P ✵✱✸ ✵✱✷ ✵✱✺ P ✵✱✹ ✵✱✻
❳→❝ =✉➜' ❝0 ✤✐➲✉ ❦✐➺♥✿ P (X = 1, Y = 1) 0, 1 1 P (Y = 1|X = 1) = = = P (X = 1) 0, 3 3 ❚!♥ ❚❤➜% ❚& ✷✵✴✷✸ P (X = 1, Y = 2) 0, 2 2 P (Y = 2|X = 1) = = = P (X = 1) 0, 3 3
X❤➙♥ ♣❤4✐ ❝0 ✤✐➲✉ ❦✐➺♥✿ Y |X = 1 ✶ ✷ P ✶✴✸ ✷✴✸
❉♦ ✤0✿ E(Y |X = 1) = P yiP (Y = yi|X = 1) = 5/3✳
❍✐➺♣ ♣❤>?♥❣ =❛✐✿ Cov(X, Y ) = E(XY ) − E(X)E(Y )✱ '#♦♥❣ ✤0✿ E(X) = P pixi = 2, 2; E(Y ) = P piyi = 1, 6; E(XY ) = P pijxiyi = 3, 5
⇒ Cov(X, Y ) = 3, 5 − 2, 2 ∗ 1, 6 = −0, 02 X❤>?♥❣ =❛✐✿
D(X) = E(X2) − (EX)2 = P pix2i − 2, 22 = 0, 76
D(Y ) = E(Y 2) − (EY )2 = P piy2i − 1, 62 = 0, 24 Cov(X, Y ) −0, 02 ⇒ ρ(X, Y ) = = √ = −0, 0468 pD(X)D(Y ) 0, 76 ∗ 0, 24 ❚!♥ ❚❤➜% ❚& ✷✶✴✷✸ ❱➼ ❞# ✻
❈❤♦ ✈❡❝&' ♥❣➝✉ ♥❤✐➯♥ (X, Y ) ❝. ❤➔♠ ♠➟& ✤3 ✤4♥❣ &❤5✐✿ (x + y, (x, y) ∈ [0, 1]2 f (x, y) = 0, (x, y) / ∈ [0, 1]2
❚➼♥❤ Cov(X, Y ), D(X + 2Y )✳
●✐↔✐✳ ❤➙♥ ♣❤%✐ ❜✐➯♥✿ Z +∞ Z 1 fX(x) = f (x, y)dy =
(x + y)dy = x + 1/2, x ∈ [0, 1] −∞ 0 Z +∞ Z 1 fY (y) = f (x, y)dx =
(x + y)dx = y + 1/2, y ∈ [0, 1] −∞ 0 Z +∞ Z 1 E(X) = xfX(x)dx =
x(x + 1/2)dx = 7/12; E(Y ) = 7/12 −∞ 0 ❚!♥ ❚❤➜% ❚& ✷✷✴✷✸ Z +∞ Z +∞ Z 1 Z 1 E(XY ) = xyf (x, y)dxdy = xy(x + y)dxdy = 1/3 −∞ −∞ 0 0
❙✉② -❛✿ Cov(X, Y ) = E(XY ) − E(X)E(Y ) = 1/3 − (7/12)2 = −1/144✳ ▼➦2 ❦❤→❝✱
D(X + 2Y ) = D(X) + D(2Y ) + 2Cov(X, 2Y ) = D(X) + 4D(Y ) + 4Cov(X, Y ) Z +∞ Z 1 E(X2) = x2fX(x)dx =
x2(x + 1/2)dx = 5/12; E(Y 2) = 5/12 −∞ 0
D(X) = E(X2) − (EX)2 = 5/12 − (7/12)2 = 11/144; D(Y ) = 11/144
❱➟②✱ D(X + 2Y ) = 11/144 + 4 ∗ 11/144 − 4 ∗ 1/144 = 17/48 ❚!♥ ❚❤➜% ❚& ✷✸✴✷✸

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