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y
m
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y
1
X Y
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1
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x
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k
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=
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m
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i
, 6)
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f(u, v) = 6(1/12) = 1/2 , P(X ≤ 2, Y ≤ 6) = 1
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4
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4
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? &
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83.2
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P(X’= 0, Z = 2) = 1/2
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2
, p)
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Y
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λ
1
λ
2
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1
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2
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Y
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X ~ N(µ²
1
, σ²
1
)
Y ~ N(µ²
2
, σ²
2
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:
X + Y ~ B(µ²
1
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2
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1
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2
)
X – Y ~ B(µ²
1
– µ²
2
, σ²
1
+σ²
2
)
5
-
2
-
2
X
Y
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5
f(x, y)
6 7 3 7- 78
.
µ
x
= E(X) = ∑
x
∑
y
x f(x, y) ,
µ
y
= E(Y) = ∑
x
∑
y
y f(x, y)
σ²
x
= E[(x – µ
x
)²] = ∑
x
∑
y
(x – µ
x
)² f(x, y) , σ²
y
= E[(y – µ
y
)²] = ∑
x
∑
y
(y – µ
y
)² f(x, y)
9
71
X
Y
!
! *
:
µ
x
= E(X) = ∫
-∞
+∞
∫
-∞
+∞
x f(x, y) dx dy , µ
y
= ∫
-∞
+∞
∫
-∞
+∞
y f(x, y) dx dy .
σ²
x
= E[(x – µ
x
)²] = ∫
-∞
+∞
∫
-∞
+∞
(x – µ
x
)² f(x, y) dx dy ,
:;<
V
.
72= 72"3 >?
10
σ²
y
= E[(y – µ
y
)²] = ∫
-∞
+∞
∫
-∞
+∞
(y – µ
y
)² f(x, y) dx dy
.
@ 3? %&'" =&A
B& C ?
:
*1 "4?
:
E(y), E(x)
5
σ²
x
, σ²
y
E(x) = ∑
x
∑
y
x f(x, y) = 1(1/8 + ¼ + 1/8) – 5(1/4 + 1/8 + 1/8) = 1/2 – 5/2 = –4/2 = –2
E(Y) = ∑
x
∑
y
y f(x, y) = –4 (1/8 + ¼) – 2 (1/4 + 1/8) + 7 (1/8 + 1/8) = –1/2
σ²
x
= E[(x – µ
x
)²] = ∑
x
∑
y
(x – µ
x
)² f(x, y)
= (1 + 2)² (1/8 + ¼ + 1/8) + (–5 + 2)² (1/4 + 1/8 + 1/8) = 9 (1/2) + 9 (1/2) = 9
σ²
y
= E[(y – µ
y
)²] = ∑
x
∑
y
(y – µ
y
)² f (x, y)
= (–4 + 1/2)² (1/8 + 1/4) + (–2 + 1/2)² (1/4 + 1/8) + (7 + 1/2)² (1/8 + 1/8)
= 49/4 (3/8) + 9/2 (3/8) + (15/2)² (2/8) = 651 / 32 = 20,34
7D6 EA FAG#$ 7)$* ) : *1
f
1
(x)
f
2
(y)
.
E(x) = ∑
x
x f
1
(x) = 1(4/8) – 5(4/8) = –2
V(x) = E(x²) – E²(x)
= [1²(4/8) + (– 5)²(4/8)] – (–2)² = 9
5
-
2
-
3
5
-
2
-
3
-
1
@ 3? &( C &
(Covariance)
72=
(X, Y)
B&
:
Cov (X, Y) = σ
xy
= E[(X – µ
x
)(Y – µ
y
)]
71 9
X
Y
!4) !
:
σ
xy
= ∑
x
∑
y
(x – µ
x
)(y – µ
y
)f(x, y)
71 9
X
Y
!
! *
:
σ
xy
= ∫
-∞
+∞
∫
-∞
+∞
(x – µ
x
) (y – µ
y
) f(x, y) dx dy
5
-
2
-
3
-
2
1
.
&( H&
!
I=*
:
Cov(X, Y) = E[(X – µ
x
)(Y – µ
y
)]
= E[XY – Xµ
y
– µ
x
Y + µ
x
µ
y
]
= E(XY) – E(X)E(Y) – E(X)E(Y) + µ
x
µ
y
7
-2
-4
y
X
1/8
1/4
1/8
1
1/8
1/8
1/4
-5
f
1
(x)
7
-2
-4
Y
X
4/8
1/8
1/4
1/8
1
4/8
1/8
1/8
1/4
-5
1
2/8
3/8
3/8
f
2
(y)
JK1K L :MA
BN& O;1P
.
Q
1
.
11
=> Cov(X, Y) = E(XY) – E(X) E(Y)
2
.
71 9
X
Y
!)* F
1
! BN& %S" T2;M
E(XY) = E(X) E(Y)
U=
:
Cov(X, Y) = E(XY) – E(X) E(Y) = E(XY) – E(XY) =>
Cov (X, Y) = 0
3
.
71 9
X
Y
)* V !)* !
:
2 Cov (X, Y)
±
Var (X ± Y) = V(X) + V(Y)
4
.
&W? - XK OAY Z !" K @ 3? &( 7)4? 7)
:
|σ
xy
| ≤ σ
x
σ
y
5
.
71 9
X
Y
[ \ !4( !
Y = X
!,-
:
Cov(X , Y) = σ
xy
= σ
x
σ
y
6
.
Cov(X + a, Y) = Cov(X , Y)
7
.
Cov (X, X) = V(X)
8
.
Cov(aX + bY, Z) = aCov(X , Z) + bCov(Y , Z)
.
X
Y
!)* + F
/1
E(X) = 100, V(X) = 100, E(Y) = 100, V(Y) = 100
.
Z
/1 + F
Z = 3X – 10
•
: ]*1
:
E(Z), V(Z), E(Z²), V(Z + Y), Cov ( Z + Y, X)
•
]*1
V(Z + X), V(Z) + V(X)
I=#
Cov (Z, X)
.
E(Z) = 3E(X) -10 = 290
V(Z) = 3²V(X) = 900
E(Z²) = V(Z) + E(Z)² = 900+290² = 85000
V(Z + Y) = V(3X – 10 + Y) = 3²V(X) + V(Y) = 900 + 100 = 1000 ( X et Y ind.)
(propriété 8)
Cov (Z + Y, X) = Cov(Z, X) + Cov(Y, X)
(propriété 2 et 6)
= Cov(3X – 10, X) + 0
= 3Cov(X, X) (propriété 8)
= 3V(X) = 3(100) = 300 (pro. 7)
5
-
2
-
4
^_
7
)
2
(
* ! I=*
y
x
xy
σ
σ
σ
b*&
0
9
71
X
Y
!)*
7^_
)
5
(
U I=*
71 9
X
Y
\ !4( !
b*& * !,-
1
.
1
!"& A)- 5cd^ 8 >W e$ f f
[- !)* !? !"& ! V <; &* @ 3? &(
.
g D f U= g D : \ 7))h 9 Bi 78?
-
.
? $ 58"Y" ! 5j(WK HN 7KA @ 3? &( 7A 7k E# :$)?$
.
j 3
9 F'[
W"l? "i E[)#K
E[)#K H& 9 )$#
.
:;<
V
.
72= 72"3 >?
12
7^_ m M 7nY
4
! I=*
7(*=
S o
n
$
)
p
1
(
)
1
(
:
1
1
≤
≤
−
y
x
xy
σ
σ
σ
.
l0i :0Y 0
7(*= :*
:
y
x
xy
r
σ
σ
σ
=
K : * 5? $ j(WK q)
j(W
.
71 9
r
FA
! E")
)* r Fst ! V 5
!
.
.
u$* E ? 9 W"l? @ 3? %&'" j(WK @ 3? &( AY
.
Cov(X, Y) = E(XY) – E(X) E(Y)
E(XY) = 1(–4)(1/8) + (1)(–2)(2/8) + (1)(7)(1/8) + (–5)( –4)(2/8) + (–5)( –2)(1/8) +
(–5)(7)(1/8) = 1.75
E(X) = 1(4/8) + (–5)(4/8) = –2, E(Y) = –4(3/8) – 2(3/8) + 7(2/8) = –1/2.
Cov(X, Y) = 1.75 – (–2)( –1/2) = 0.74
V(X) = E(X²) – E²(X) = 1(4/8) + (–5)²(4/8) – (–2)² = 9 => σ
x
= 3,
V(Y) = E(Y²) – E²(Y) = 20.34 => σ
y
= 4.5.
05
.
0
)
5
.
4
(
3
75
.
0
=
=
=
y
x
xy
r
σ
σ
σ
.
:?
r
v";_$ B4_ j(WK f)&
.
5
-
2
-
5
6 7 3? %&'" 78 7$ !)* r ! E")
)
7 3? 7- 78
(
8 OAY :D 9
3i
)
7- 3i 8
:(
f(x, y) = f
1
(x) f
2
(y)
! b
:
P(X = x, Y = y) = P(X = x) P(Y = y)
@ 3? &( !"& 7h wli 9 5j(WK : E[M $ j(WK f) m M 7nY
Cov(X, Y)
= E(XY) – E(X) E(Y)
!,0- E[)0#K 71 9 BN& %S" T2;_ ( Ux A
E(XY)
=
E(X)
E(Y)
cd^ >W e$ f f
.
j(WK : FAG*&
y
x
xy
r
σ
σ
σ
=
7&8Az L n(= !"& ! ]{ B2;1P 5? $ j(WK D|
D|? li
.