Probabilité conditionnelle, indépendance d`événements Variables

(Ω,P) (H1, . . . , Hn) Ω n
i= 1, . . . , n A
P(Hi|A) = P(A|Hi)P(Hi)
Pn
j=1 P(A|Hj)P(Hj).
A B C
A B C
A B C
(Ω,P)A1, . . . , An
Ac
1, A2, . . . , An
B1∈ {A1, Ac
1}, . . . , Bn∈ {An, Ac
n}
B1, . . . , Bn
CNn≥2
Ω = Cn=C × · · · × C PΩi= 1, . . . , n
Xi: Ω → C
ω= (c1, . . . , cn)7→ Xi(ω) = ci.
X1, . . . , XnC
(Ω,P)X Y Ω
R
f g R R f(X)g(Y)
X Y XY E[XY ] = E[X]E[Y]
X2Y2Var(X+Y) = Var(X) + Var(Y)
n X1, . . . , Xn

(Ω,P)A⊂Ω1A: Ω → {0,1}
A
ω∈Ω1A(ω) = 1ω∈A
0ω /∈A.
A B 1Ac1A∩B1A1B
AP(A) = E[1A]
A1, . . . , An
1A1. . . , 1An
A1, . . . , An
P(A1∪ · · · ∪ An) = X
S⊂{1,...,n}
(−1)1+Card(S)P \
i∈S
Ai!
=
n
X
k=1
(−1)k+1 X
1≤i1<···<ik≤n
P(Ai1∩ · · · ∩ Aik).
XN
E[X] =
∞
X
n=1
P(X≥n).
p
1−p
p
n
n p
(Ω,P)X, Y
ΩN
X+Y X Y
s∈]−1,1[ GX+Y(s) = GX(s)GY(s)
n X1, . . . , Xn
n p
(n, p) (m, p)
λ µ
{2, . . . , 12}
X1X2{1, . . . , 6}
GX1GX2X1+X2{2, . . . , 12}
GXi(s) = sϕi(s)ϕi