Feuille d`exercices # 1 : Conditionnement

Z X (Ω,F,P)
Z σ(X)−h:R→R
Z=h(X)
(Ω,F,P)G F Y, Z
E(Y2),E(Z2)<+∞E(Y Z|G)2≤E(Y2|G)E(Z2|G)
E[(Y+θZ)2|G]≥0θ∈R
(Ω,F,P)G F Y, Z
EYE(Z|G)=EZE(Y|G)
(Ω,F,P)G F Y
{E(Y|G)>0} G−
{Y > 0}
(Ω,F,P)G F Y
Var(Y|G) := E(Y−E(Y|G))2|GY
GVar(Y) = E(Var(Y|G)) + Var(E(Y|G))
(Ω,F,P)X, Y
E(Y|X) = XE(X|Y) = Y X =Y
x∈R E(X−Y)1{Y≤x<X}=E(Y−X)1{x<X,x<Y }≥0
p, q > 11 /p+1 /q= 1
x, y ≥0xy ≤xp /p+yq/qY∈LpZ∈LqG F
G:= {E(|Y|p|G)>0} ∩ {E(|Z|q|G)>0}G
Y Z
E(|Y|p|G)1
/pE(|Z|q|G)1 /q≤|Y|p
pE(|Y|p|G)+|Z|q qE(|Z|q|G).
GY:= {E(|Y|p|G)=0}GYY
E(|Y Z||G)≤E(|Y|p|G)1
/pE(|Z|q|G)1 /q
X1, ..., Xn
Sn:= X1+... +XnE(X1|Sn) = E(Xi|Sn)i∈ {1, ..., n}
E(X1|Sn)

(Ω,F,P) = (]0,1[,B(]0,1[), λ)λ
X X(ω) = cos(πω)G
A⊆]0,1[ A Ac
G G ]0,1[
E[X|G] = 0
X1, . . . , XnXjλj
X1+X2X1+. . . +Xn
P(Xj=`|X1+. . . +Xn=k)
(Ω,F,P) = (]0,1[,B(]0,1[), λ)λ
X=1[0,3
/4[Y=1[0,1 /2[Z=1[1 /4,3 /4[
Y Z E(Y|Z) = E(Y)E(Y|X)E(Y|X, Z)
E(Y|X, Z)6=E(Y|X)Y Z
(X, Y ) (Ω,F,P)
E(Y|X) = cX
c=E(XY ) /E(X2)
(X1, . . . , Xd) (a1, . . . , ad) (b1, . . . , bd)∈Rd
Y=a1X1+. . . +adXdZ=b1X1+. . . +bdXdE(Z|Y)
(Xn)n≥0m=E[X1]N
(Xn)n≥0G(1 /2)SN:= PN
i=1 Xi
E[SN|N]E[SN]
X1, . . . , Xn
λ > 0T:= X1+. . .+XnE(h(X1)|T)h
n= 2
n≥1p1, p2, p3p1+p2+p3= 1
pij =(n!pi
1pj
2pn−i−j 3
i!j!(n−i−j)! , i +j≤n
0, .
(X, Y )P(X=i, Y =j) = pij
X Y Y X
E(XY )

X1X2
(n1, p) (n2, p)X1X1+X2=n
E(X1|X1+X2)
X1, . . . , Xp
λ1, . . . , λp(X1, . . . , Xp)X1+X2+. . . +Xp=n
ε(Ω,F,P)
±11 /2G F εG
E(ε|G)=0
Y fY(y) := y−21[1,∞[(y)X=Y
E(Y)=+∞E(Y|X)<∞
(X, Y )pij := [i(i+
1)]−1i=j= 1,2, . . .
Y X
(X, Y )
f(x, y) = x e−x(y+1)1{x,y≥0}
f(x, y) = 4x(y−x) exp(−(x+y))1{0≤x≤y}
(X1, . . . , Xn)n−
f(x) (X(1), . . . , X(n))X(1) ≤. . . ≤X(n)
X(1) X(n)=xnE[X(1)|X(n)]
Xi[0,1]
X, Y Y
X Y =y
y
(X, Y )X Y X
E(X−Y)2= 1 Y
X
X1[0,1] X1=x1X2
[x1, x1+ 1] X2=x2X3
[x2, x2+ 1] X4, . . . , Xn
n≥4E(Xn)

[X, Y ]0m= [1,−1]0
Γ = 1 1
1 4.
[X, Y ]0X Y X +Y
E[X|Y]
(U, V )E[U|U+V]
X= (X1, X2, X3)0
Γ := 
2 2 −2
251
−2 1 5 
.
X
E[X1|X2]E[X1|X2, X3]
Γ := 
4 1 2
1 9 −3
2−3 4 
.