A Lea To Ire 2
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- Words: 6,176
- Pages: 32
. 2
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X E(X)
E(X) :=
x pX (x)dx E(X) :=
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x pX (x) X
pX X E=R
X≥0 xp(x)
X
• X : Ω→E ⊂ R
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b E(X) =
k≥1
kp(1 − p)k−1 = 1/p E(T ) =
Z
Z
[a, b] X •
T •
X
P •E(1A ) = 0 × P(1A = 0) + 1P(1A = 1) = P(A) E
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E(f (X)) =
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f (x)pX (x)dx X
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f (x)PX (x) x
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E(f (X)) =
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f (x)PX (dx) f (X(ω))P(dω) =
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E(Xi ) = 0P(Xi = 0) + 1P(Xi = 1) = p.
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E(Xi ) = np =
k=0 k=0
E(Sn ) =
n X
. kCnk pk (1 − p)n−k kPSn (k) =
n X n X
S ∼ B(n, p), • S n = X1 + . . . + X n
Xi : Ω→{0, 1} i •
Ω = {0, 1}n . n
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p σ(X) := Var(X)
Var(X) := E (X − E(X))
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Var(X) = E(X 2 ) − E(X)2 = E(X) − E(X)2 = p(1 − p).
P(X = 1) = p = 1 − P(X = 0).
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E(X 2 ) =
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σ 2 . Y ∼ N (µ, σ)
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µ
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Z = (X − E(X))2
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1 Var(X) 2 a ∀a > 0, P (|X − E(X)| ≥ a) ≤
X X
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Xi , = • P Xi ,
X, = • PX ,
X = (X1 , . . . , Xn ) : Ω 7→ E1 × . . . × En •
E i = R ni Xi : Ω 7→ Ei , i = 1, . . . , n
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PX (x1 , x2 , . . . , xn )
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Z Z PX1 (A) = [ A
pX (x1 , . . . , xn )dx2 . . . dxn ]dx1 E2 ×...×En
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pX1 (x1 ) =
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pX (x1 , . . . , xn )dx2 . . . dxn E2 ×...×En
PX1 (x1 ) =
X
PXi (A) = PX (E1 × . . . × Ei−1 × A × Ei+1 × . . . × En ).
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1 p(X,Y ) (x, y) = 1{x2 +y2 ≤1} π
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(R, Θ) X = Rcos(Θ), Y = R sin(Θ)
√ − 1−x2
p(X,Y ) (x, y)dy =
X
√ 1−x2
1[−1,1] (x) 2p dy = 1 − x2 1[−1,1] (x) π π pX (x) =
(X, Y ) =
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Y = ϕ(X)?
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pY (y) = pX (ϕ−1 (y))|Jac ϕ−1 (y)|1O0 (y)
∂ϕi )i,j ∂xj
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( Jac ϕ =
Y = ϕ(X)
ϕ−1 ϕ ϕ ϕ : O 7→ O 0
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PX X •
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pY (y) = pX (ϕ−1 )(y)|Jac ϕ−1 (y)|1O0 (y)
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1 Jac ϕ(ϕ−1 (y)) .
&
ϕ ◦ ϕ−1 = Id ⇒ Jac ϕ−1 (y) =
h(y)pX (ϕ−1 (y))|Jac ϕ−1 (y)|dy =
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h(ϕ(x))pX (x)dx E[h(Y )] =
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1 π 1{|u|≤1}
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pX,Y (x, y) = q(x2 + y 2 )
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(X, Y )
ϕ : R2 \ {x ≥ 0, y = 0} 7→]0, +∞[×]0, 2π[, (x, y) → (r, θ) cos(θ) −r sin(θ) −1 , Jac(ϕ−1 )(r, θ) = r. Dϕ (r, θ) = sin(θ) r cos(θ) M
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(R, Θ) : pR,Θ (r, θ) = rq(r 2 )1{r>0} × 1θ∈]0,2π[
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R : pR (r) = 2πrq(r 2 )1{r>0}
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= E(XY ) − E(X)E(Y ) .
cov(X, Y ) = E[(X −E(X))(Y −E(Y ))]
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1≤i<j≤n
i=1
i=1
Xi ) = Var(
• Var(X + Y ) = Var(X) + Var(Y ) + 2cov(X, Y )
cov(Xi , Xj ) Var(Xi ) + 2
X n X n X
Y X
E(Y 2 ) < ∞ E(X 2 )
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p(1 − p) Sn Var( ) = n n
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i=1
i 6= j 0 cov(Xi , Xj ) = E(Xi Xj ) − p2 = p − p2
Var(Xi ) = np(1 − p) Var(Sn ) =
Xi ∈ {0, 1} =
n X
Xi . i=1
Sn =
Pn
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Sn − p| ≥ a) = 0. limn→∞ P(| n
1 Sn − p| ≥ a) ≤ p(1 − p) P(| n na2
⇒
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P(X ∈ I, Y ∈ J) = P(X ∈ I)P(Y ∈ J). =
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PX,Y (I × J) = PX (I)PY (J) <
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PX,Y (x, y) = PX (x)PY (y) < A 6/ 3 76 0 ,+
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pX,Y (x, y) = pX (x)pY (y).
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P(Xi = ωi ) P(X1 = ω1 , . . . , Xn = ωn ) =
Y
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(1 − p) Ω = {0, 1} , P(ω) = p
X1 , . . . , X n
ωi P n− ωi P n
Xi (ω) = ωi
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1 1]0,2π] . pR (r) = 2πrq(r )1{r>0} , pΘ (θ) = 2π
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pR,Θ (r, θ) = rq(r 2 )1{r>0} × 1θ∈]0,2π[
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1 pX,Y (x, y) = 1{x2 +y2 ≤1} π
2p pX (x) = pY (x) = 1 − x2 1[−1,1] (x) π R, Θ M
Y {x2 + y 2 ≤ 1} X M = (X, Y )
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E(f (X)g(Y )) =
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f (x)g(y)PX,Y (dxdy)
PX,Y (A × B) = PX (A)PY (B) Z Z Z Z f (x)g(y)PX (dx)PY (dy) f (x)g(y)PX,Y (dxdy) =
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=
Z
f (x)PX (dx)
Z
g(y)PY (dy).
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cov(X, Y ) = 0
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Xn ∼ m
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n
X1 +X2 + . . . Xn n X n :=
•
m := E(X1 )
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=
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Var(Xn ) σ2 P [|X n − m| > ε] ≤ = 2 −→ 0 2 nε
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X 1 1 Var(X n ) = 2 Var( Xi ) = σ 2 . n n
σ 2 = Var(X1 ) < ∞
∀ > 0, limn→∞ P |Xn − m| > ε = 0. E(X12 ) < ∞
m = E(X1 ) (X n )
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A : X n →P(A).
Xn = 1{ωn ∈A} E(Xn ) = P(A)
ω1 , ω 2 , . . . •
A ⊂ Ω,
ω∈Ω •
m = E(X1 ) (X n )
P limn→+∞ Xn = m = 1
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E(Xi4 ) < ∞
E(Xi ) = 0
E[(Xi1 . . . Xi4 ]
i1 ,...,i4
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−4
X
E(Xi4 )
+
i
E
E(Xi Xj3 )
+
i6=j
=n "
X
X
4 Xn
n
#
−4
X
E(Xi2 Xj2 )
+
i6=j
X
E(Xi2 )
+
i
<∞
⇒
X i6=j
X n
4 Xn
X
i1 6=i2 6=i3 6=i4
E[(Xi1 . . . Xi4 ]
E(Xi2 Xj2 ) = O(1/n2 )
< ∞ p.s.
⇒
X n → 0 p.s.
X
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