TD 4

◦4
n1
n X 6Y
(X, Y )
n
U6V
(U, V )
p, q ∈[0; 1] n∈N\{0}X
n p Y k ∈ {0, . . . , n}
Y X =k k q
Y
p
q
20
X Y
p∈[0; 1]
U= min(X, Y )U
V=X−Y U V
X Y
λ µ
(X, Y )
T=X+Y
X X +Y=n n ∈N
(X, Y )A={(i, j)∈N2,1≤i<j},
P{(X, Y )=(i, j)}=apj∀(i, j)∈A,
a0<p<1
A X a
a

E(Y)E(XY ) (Y) = 2 (X)
X Y −X X Y −X
n∈NXn0n X0= 0
λ > 0 (m, n)∈N20≤m≤n
Xn−Xmλ(n−m)
XmXn−Xm
(m, n)∈N20≤m≤n
(Xm, Xn)
(Xm, Xn)
n6= 0 k∈NXmXn=k
N k
1 0 k N
A1A2A3
A1A2
A3
A3
i∈ {1,2,3}Xi
AiX1X2X3
(Ω,P)
n≥1P(Xi=n) = pqn−10<p<1q= 1 −p
n≥0P(X1−X2=n)
T=|X1−X2|
{X3> T }
j∈ {1,2}Tj
j T1=X1X1> X2T1=X1+X3X1≤X2
P(X1≤X2)
n≥1P(T1−X1=n)
E(T1) = E(X1)(1 + P(X1≤X2)) E(T1)
(X, Y )
∀(i, j)∈N, j ≤i, P((X, Y )=(i, j)) = (1 −a)(b−a)aibj−i−1,
a b 0<a<1a<b
G(X,Y )(X, Y )
(X, Y ) (s, t)∈]−1; 1[2,
G(X,Y )(s, t) = EsXtY.
GXGYX Y
X Y
X Y