TD9. Variables aléatoires

e
(R2,BR2)C= [−1,1]×[−1,1] X(x, y) =
x S(x, y) = x+y
a p =aλ|C λR2
(R2,BR2)X S RBR
pX( ] − ∞, t[ ) = p(X < t)t∈R
FpXX X
S X
FpSRS pS
λRR
X S C C0=
[0,2] ×[−1,1]
X
XNp
E(X) =
+∞
X
k=1
p(X≥k), EX(X−1)= 2
+∞
X
k=2
+∞
X
l=k
p(X≥l).
XN
n, a, b n ≤a+b
p(X=k) = a
kb n−k
a+b
nmax{0, n −b} ≤ k≤min{a, n}, p(X=k) = 0

a+b
nn a +b
min{a,n}
X
k=max{0,n−b}a
kb n−k=a+b n.
RNdpX
E(X) =
min{a,n}
X
k=max{1,n−b}
ka
kb n−k
a+b
nk0=
k−1n0=n−1a0=a−1b0=b E(X) = an
a+b
min{a,n}
X
k=max{0,n−b}
k(k−1)a
kb n−k
a+b
n=a(a−1)n(n−1)
(a+b)(a+b−1) ;
m2(X) = an(an−n+b)
(a+b)(a+b−1)
V ar(X) = abn(a+b−n)
(a+b)2(a+b−1) .
N=a+b p =a
a+bq=b a+b
n k
lim
Np(X=k) = n
kpkqn−k.
p(X=k)N p q n k
N→+∞N!
(N−r)!
4% Pk
k k = 0,1,2,3
N(0,1)
XN(−1,2) P(X < 1,91) P(X > 0,82) P(0,82 <
X <1,91) P(−0,82< X <1,91) P(X2<1,44) P(log |X|<0)
a P (X >a) = 0,400
Y Z E(Y)
V ar(Y) = 4 P(Y >2) = 0,4V ar(Z)E(Z) = −2,5P(Z <0) = 0,9
X
P(X≥x)>0x≥0
≥b
a b
∀a, b ≥0, p(X≥a+b|X≥a) = p(X≥b) ;

σ > 0
X f(x) = 1
σ−x/σ x≥0f(x) = 0
F X F (0) = 0
p(X≤a)a≥0p(X≥a)
G= 1 −F
G(a+b) = G(a)G(b)a, b ≥0
σ > 0G(1) = −1/σ
G(q) = −q/σ q∈Q+
U, V W = max{U, V }
t∈R
At={(x, y)∈R2|x > t , y < t}, Bt={(x, y)∈R2|x < t , y > t}.
t∈R
p(U<t)p(V <t)p(W<t)
p(U<t)p(V <t)p(W <t)p(U,V )
AtBtp(U,V )
∆ = {(x, x)∈R2, x ∈R}
AtBtt∈R
t
p(U,V )∆
U V
x≤y Qx,y =] − ∞, x[×]− ∞, y[ ∆
p(U,V )Qx,y F(x)F(y) = F(x)2F
U V
U V W