TD1 - Jean-Romain HEU

M
F¯ BM
F BM ¯ F¯ BM F ¯ BM ¯ F BM F B ¯ M F ¯ B¯ M¯ F¯ B¯ M¯ F B ¯ M
99%
0,2%
A:444411 B:552222 C:663300
A B A C
B C

3×3
p
T T
λ > 0
R+t7→ λe−λt
∀b≥a≥0,P([a, b]) = Zb
a
λe−λtdt.
t≥0Att
t≥0P(At)
t2≥t1≥0P(At2|At1)
[0,1] [0,1]
t7→ 1
x y a, b, c d
0≤a < b ≤1 0 ≤c<d≤1
P(a<x<b c<y<d)
(x, y)
[0,1] ×[0,1]

a, b > 0
P(x∈[a, b]y∈[a, b]),P(x<y),P(x2< y),
P(x+y < a),P(y
x< a),P(x2+y2≤1).
X f [0,1]
f(x) = 5(1 −x)4.
f
X
f(x) = (0x < 1
2
x3x≥1
f
a > 1P(X > a)
mP(X < m) = P(X > m)
b > 1P(X > a +b|X > a)
a
αP(X≥α) = 1
4α
R+∞
αxfX(x)dx
R+∞
αxfX(x)dx
R+∞
1xfX(x)dx

X1
X2
a∈ {0,...,36}X= 35
a X =−1
X
fX
X
E[X]
Y
Y=0X < 0
1X > 0
Y
X|Y= 1

f1(x) = 6x(1 −x)x∈[0,1]
f2(x) = 3x−4x∈[1,+∞[
f3(x) = 1
π
1
1+x2x∈R
n X
a b
X1
X2a>b 0a=b−1a < b
X3b a < 1
2a
x λ X x
x > 1 0
XE(X−a)2
a=E(X)
a
n∈N∗n
n X1, . . . , Xn
Yn=1
nPn
i=1 Xim σ2
Xi
Yn
n+∞(Yn)
X Y G(p)G(q)U=
min(X, Y )
U
U
p q
λ µ