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8/18/2019 Var Infinie s
1/5
X
X
X
(An) (Ω, T , P ) ∀n ∈ N An ⊂ An+1 ∀n ∈ N An+1 ⊂ An
(An) P
+∞n=0
An
= limn→+∞
P (An)
(An) P
+∞n=0
An
= lim
n→+∞P (An)
100 000
50
100 000
An
n (An)
n + 1
n
p =
1
501 000
P (An) = (1 − p)n
P
+∞n=0
An
= lim
n→+∞P (An) = 0 p > 0 1 − p < 1
1
1
A
Ω
A
A
8/18/2019 Var Infinie s
2/5
X
Ω
X : Ω → Z
X
6
X
X (Ω) = {2;3; . . .}
X = i
X i
X
Y
Ω
X + Y
XY
max(X, Y )
min(X, Y )
X
Ω
g : Z → Z
g(X ) : ω → g(X (ω))
g(X ))
X
X
P (X = k)
k
X (Ω)
P (X = k) =
5
6
k−1×
1
6
X = k
k
1
3
k−1
(k − 1)
2
3
k−2×
1
3
P (X = k) = (k − 1)1
32
2
3k−2
k∈X (Ω)
P (X = k) = 1
+∞k=1
5
6
k−1×
1
6 =
1
6
+∞k=0
5
6
k=
1
6 ×
1
1 − 56= 1
+∞k=2
(k −
1)
1
3
223
k−2=
1
9
+∞
k=1k
2
3
k−1=
1
9 ×
1
1 − 232
= 1
8/18/2019 Var Infinie s
3/5
X
F X : R → R F X (x) = P (X x)
F X
X
kP (X = k) X E (X ) =k∈X (Ω)
kP (X = k)
6
E (X ) =
+∞k=2
k(k − 1)
1
3
223
k−2=
1
9 ×
21 − 23
3 = 19 × 54 = 6
X Y
Ω
X + Y
E (X + Y ) = E (X ) + E (Y )
X
E (X )
0
X Y
X Y
E (X ) E (Y )
X
g : R → R
g(X )
E (g(X )) =
k∈X (Ω)
g(k)P (X = k)
X
r
X
r X r
mr(X ) = E (X
r)
X
m
(X − m)2
V (X ) = E ((X − m)2)
X
X
X 2
X
p ∈ [0; 1]
X (Ω) = N
∀k ∈ N∗
P (X = k) = p(1 − p)k−1
X ∼ G ( p)
n
X ∼ G ( p)
X
E (X ) =
1
p V (X ) =
1 − p
p2
8/18/2019 Var Infinie s
4/5
E (X ) =
+∞k=1
pk(1 − p)k−1 = p × 1
(1 − (1 − p))2 =
p
p2 =
1
p
E (X )2 = 1
p2
X (X − 1)
E (X (X − 1)) =
+∞k=1
pk(k − 1)(1 − p)k−1 = p(1 − p) 2
(1 − (1 − p))3 =
2(1 − p)
p2
V (X ) = E (X (X − 1)) + E (X ) − E (X )2 = 2(1 − p)
p2 +
1
p −
1
p2 =
2(1 − p) + p − 1
p2 =
1 − p
p2
X ∼ G ( p)
∀(n, m) ∈ N∗2
P X>n(X >
n + m) = P (X > m)
P (X > m) =k>m
P (X = k) =+∞
k=m+1
pq k−1 = p+∞ j=0
q j+m =
pq m+∞ j=0
q j = p q m
1 − q = q m P (X > n) = q n P (X > n + m) = q n+m
P X>n(X > n + m) = P (X > n + m)
P (X > n) = q m
X
λ ∈ R+
X (Ω) = N ∀k ∈ N P (X = k) = e−λ
λk
k!
X ∼ P (λ)
1 P (X = k) 0 k
+∞
X ∼ P (λ)
X
E (X ) = V (X ) = λ
E (X ) =
+∞k=0
ke−λλk
k! = λe−λ
+∞k=1
λk−1
(k − 1)! = λe−λeλ = λ
E (X (X −1)) =
+∞k=0
k(k − 1)e−λ
λk
k! = λ2
+∞k=2
λk−2
(k − 2)! = λ2 V (X ) = E (X (X − 1)) + E (X ) − E (X )2 =
λ2 + λ − λ2 = λ
n
p
λ ∈ R
+ (X
n)
X
n ∼ B n, λ
n
∀k ∈ N limn→+∞
P (X n = k) = e−λλ
k
k!
8/18/2019 Var Infinie s
5/5
k ∈
N X n ∼ B
n,
λ
n
n k
P (X n =
k) =
n
k
λ
n
k 1 −
λ
n
n−k
λk
nk
n
k
=
n!
k!(n − k)! =
n(n − 1)(n − 2) . . . (n − k + 1)
k!
n
k
n
nk
nk
n
k
∼
nk
k!
1 −
λ
n
n−k= e(n−k) ln(1−
λ
n)
lim
n→+∞
λ
n = 0
ln
1 −
λ
n
∼ −
λ
n
(n − k) ln
1 −
λ
n
∼
n − k
n (−λ) →
n→+∞−λ
limn→+∞
1 − λ
n
= e−λ
P (X n = k) ∼ n
k
k! × λk
nk × e−λ ∼ e−λλk
k!