Variables aléatoires à densité

U([a, b])
E(λ)
N(µ, σ2)
(Ω,T, p)
1 6
0 1
0
1
2ln 2 '0,69
1
2
1
2α∈[0,1]
α α
X
α∈[0,1] p(X≤α) = α
X F
F(α) = p(X≤α)
X
F

F(2,3) = p(X≤2,3) 2
6=1 3
1 2
F(α) = p(X≤α) = α
α∈[0,1]
f
fR
f(x)≥0x∈R
Z+∞
−∞
f(t)dt = 1
f F F
f

X f F F R
x0f F F 0(x0) = f(x0)
f(x) = cos x x ∈0,π
2
0x /∈0,π
2
•fR∗f(x)≥0x∈RZ+∞
−∞
f(t)dt =Zπ
2
0
cos tdt = 1 f
•F(x) = Zx
−∞
f(t)dt x ∈R
F(x) = 
0x≤0
sin(x) 0 < x ≤π
2
1x≥π
2
X F F
R R X
RF0F0
X
X f Y =aX +b a 6= 0

∀x∈RG(x) = p(Y≤x)
=pX≤x−b
aa > 0
pX≥x−b
aa < 0
=Fx−b
aa > 0
1−Fx−b
aa < 0
GRFC1R
Y g
g(x) = 1
afx−b aa > 0
−1
afx−b aa < 0.
g(x) = 1
|a|fx−b a
X f X2
g(x) = 0x≤0
1
2√x[f(√x) + f(−√x)] x > 0.
X f Y =eX
F
lim
x→−∞ F(x) = 0 lim x→+∞F(x)=1 F
F
]− ∞, x]
X f
Z+∞
−∞
tf(t)dt E(X) = Z+∞
−∞
tf(t)dt
Z+∞
−∞
t2f(t)dt E(X2) = Z+∞
−∞
t2f(t)dt
E(X)Z+∞
−∞
(t−E(X))2f(t)dt V (X) = Z+∞
−∞
(t−E(X))2f(t)dt
E(X)V(X)
X E(X)V(X)
E(X2)

V(X) = Z+∞
−∞
(t−E(X))2f(t)dt
=Z+∞
−∞
t2f(t)dt −2E(X)Z+∞
−∞
tf(t)dt
| {z }
E(X)
+E(X)2Z+∞
−∞
f(t)dt
=Z+∞
−∞
t2f(t)dt −E(X)2.
V(X)E(X2)V(X) =
E(X2)−E(X)2
f:x7−→ 1
π(1+x2)x∈R
fR
fRf(x)≥0x∈RZ+∞
−∞
dt
π(1 + t2)=1 π[x]+∞
−∞ =
1f
ϕ(X) = ZX
0
tf(t)dt =1
2πln(1 + t2)X
0=1
2πln(1 + X2)7−→
X→+∞+∞Z+∞
0
tf(t)dt
E(X)
f(x) = (2
x3x≥1
0x < 1
f
X Y
f(x, y) = x+y(x, y)∈[0,1]2
f1(x)f2(y)f(x, y)6=f1(x)×f2(y)
X
Z(Ω,T, p)
Z
∀a > 0p(Z≥a)≤E(Z)
a.
Ω
1