(un exemple simple : lancer de des).

Lp

i= 1,2Xii
Xi
X1X1+X2
{1,···, n}λ > 0
(Xi)i≥1m σ2
¯
Xn=1
n
n
X
i=1
Xi
S2
n=1
n−1
n
X
i=1
(Xi−¯ Xn)2
X, Y, Z
[0,1]

X=x∈[0,1] Y fY|X=x
fY|X=x(y) = (y−x)e−(y−x)1y>x,
X=x∈[0,1] Y=y > x Z fZ|X=x,Y =y
fZ|X=x,Y =y(z) = (y−x)e−z(y−x)1z>0.
(X, Y, Z)
Z
(X, Y )Z=z
U=Y−X V =Z(Y−X) (X, U, V )X, U
V
1n p p ≤n Ap
p
App n
p1,···, pkn Ap1,···, Apk
φ
φ(n)n n
φ(n) = nY
pdiviseur premier de n
(1 −1
p).
µRF:R→[0,1]
F F −1u∈]0,1[
F−1(u) = inf{x∈Rtel que F(x)≥u}.
F u ∈]0,1[
Du={x∈R|F(x)≥u}
F−1(u)
Du= [F−1(u),+∞[.
x∈Ru∈]0,1[ u≤F(x)F−1(u)≤x
U[0,1] F−1(U)µ
{1,···, n}

U, V [0,1]
X=√−2 log Ucos(2πV )Y=√−2 log Usin(2πV )
(Ui)i≥0[0,1] λ > 0
N= min{n≥0|U0.···.Un≤exp(−λ)}
N
n≥0Vn=−ln(Un)n≥1Sn=Pn−1
k=0 Vn
P(N= 0) P(N= 1)
n≥1{N=n}={Sn< λ et Sn+1 ≥λ}
Sn
fn(x) = xn−1
(n−1)!e−x1x≥0.
(Ω,F,P)X∈L1(Ω,F,P)G F
E(X|G)
E(E(X|G)) = E(X)
XGE(X|G) = X
E(a1X1+a2X2|G) = a1E(X1|G) + a2E(X2|G)
X≥0E(X|G)≥0
σ(X)GE(X|G) = E(X)
X
(Ω,F,P)G F
(X|G) = E[(X−E[X|G])2|G].
(X) = E[ (X|G)] + (E[X|G]).
L2

X Y (Ω,F,P)
Y X xi, i ≥1
Y X =xE(Y|X)
(Xi)i≥1µ σ2
NN∗Xim s2
SN=
N
X
i=1
Xi.
SNN=n Sn
E[SN|N]E[SN]
(SN)
X m σ2(Xm
r)m,r∈N
X
ZnZ0= 1
Zn+1 =X(n+1)
1+. . . +X(n+1)
Zn.
E(Zn+1)E(Zn)E(Zn)
n
Znn
Sn
n S0= 1
Sn+1 Sn=k
E(Sn+1|Sn)E(Sn)
Sn
Wn=Sn(Sn+1)
(n+2)(n+3) E(Wn+1|Wn) = Wn
TN
m∈NT T −m T ≥m