Feuille de TD No3 : Variables aléatoires à densité

o3
F x ∈R
F(x) =
∞
X
i=1
1
2i1[1
i,+∞[(x).
R
A= [1,+∞[
B= [ 1
10 ,+∞[
C={0}
D= [0,1
2[
E=] − ∞,0[
G=]0,+∞[
X F
X Y =F(x)Y
[0,1]
X=F−1(Y)X
Y
X FX
a, b a 6= 0 Y:= aX +b FYFX
X Y
X3,log X, eX, X2−4X+ 2.
F
X[1,5]
X
5−X
12
X λ x, y > 0
P(X > x +y|X > x) = P(X > y).

r > 0Y=rX
Un{1,··· , n}Un
n
[0,1]
x∈RFn(x) = P(Un
n≤x)Un n
x∈Rlimn→+∞Fn(x) = F(x)F(x)
[0,1]
X λ
P(X > x +y|X > x) = P(X > y).
Xnpnpn→0
npn→λ x ∈RGn(x) = P(Xn/n ≤x)
Xn/n
x∈Rlimn→+∞Gn(x) = G(x)G(x)
λ pn=λ/n
pn→0an→0pn
an→λ
anXn
λ
C f
f(x) = C
√x(1−x),0<x<1.
f(x) = Cexp(−x−exp(−x)), x ∈R,
f(x) = C1
1+x2