Correction TD no 3.

X F
x1, . . . , xnf X f(x) = F0(x)x∈R\{x1, . . . , xn}f(xi)
x f F x F 0(x) = f(x)
f1X f2x∈R\ {x1, . . . , xn}f1(x) = f2(x)
f2X
FYY FXX t ∈R
FY(t) = P(Y≤t) = P(−4X+ 3 ≤t) = PX≥3−t
4= 1 −FX3−t 4
f0 4 FXR0
4x6∈ {0,4}F0
X(x) = f(x)fYY
t6∈ {−13,3}fY(t) = F0
Y(t) = 1
4f03−t 4.
fY(t) = 1
16 −13 <t<3
0
Y
fY
fY(t) = 1
16 −13 ≤t≤3
0
fY(t) = 1
16 −13 ≤t < 3
0
fY(t) = 1
16 −13 < t ≤3
0
FZZ FXX Z
FZ(t)=0 t < 0t≥0
FZ(t) = P(Z≤t)
=P(X2≤t)
=P−√t≤X≤√t
=FX(√t)−FX(−√t)
=FX√t+ 0
t6= 0 t6= 16 fZZ fZ(t) = F0
Z(t)
t > 0t6= 16
fZ(t) = 1
2√tF0
X√t=1
2√tf√t.
fZ(t) = 1
8√t.0<t<16
0

x∈R
FX(x) = Zx
−∞
f(t)dt
x≤0FX(x)=0 x > 0
FX(x) = Zx
0
t
α2exp −t2 2α2dt.
t=u√2α
FX(x) = Zx
√2α
0
2uexp −u2du.
=h−e−u2i
x
√2α
0
= 1 −ex2
2α2.
p∈ {0,1,2,...,100}xp∈RFX(xp) = p/100
p6= 0
xp=r−2α2ln 1−p
100,
p= 0 xp≤0
Z∞
0
e−x2dx=√π
2.
X t 7→ t2/α2e−t2/(2α2)[0,+∞[
E(X) = Z∞
0
t2
α2exp −t2 2α2dt
= 2√2αZ∞
0
x2e−x2dx
x=t/(√2α)
E(X) = −√2αZ∞
0
x−2xe−x2dx
E(X) = √2αZ∞
0
e−x2dx=αpπ/2.
E(X2)
var(X) = E(X2)−(E(X))2.

fRRf(x)dx= 1 f
c≥0
ZR
f(x)dx=Z∞
1
cx−α−1=c−1
αx−α∞
1
=c
α.
c=α
F X x ≤1F(x)=0 x > 1
F(x) = Zx
1
αt−α−1dt= 1 −x−α.
FZZ Z > t X > t Y > t
FZ(t) = P(Z≤t)
= 1 −P(Z > t)
= 1 −P(X > t, Y > t)
= 1 −P(X > t)P(Y > t).
X Y X Y
P(X > t) = P(Y > t)
FZ(t)=1−P(X > t)2
1−(1 −F(t))2.
FZ(t) = 0t < 1
1−t−2αt≥1
fZt6= 1 fZ(t) = F0
Z(t)fZ(1)
fZ(t) = 0t < 1
2αt−2α−1t≥1
x7→ x−α[1,+∞[α > 1E(X)
E(X) = Z∞
1
αx−α=α
α−1.
x7→ x−α+1 α−1>1E(X2)
E(X2) = Z∞
1
αx−α+2 =α
α−2.
varX=E(X2)−(E(X))2=α
α−2−α α−12
.
E(X)=4/3α/(α−1) = 4/3α= 4
X λ > 0
f(x) = λe−λx x≥0f(x)=0 x < 0
XP(X≥0) = 1
T T (Ω) = {1,2,3, . . .}=N?T
P(T=k)k∈T(Ω) k∈N?
P(T=k) = P([X] + 1 = k)
=P([X] = k−1)
=P(k−1≤X < k)
=Zk
k−1
λe−λxdx
=e−λ(k−1) −e−λk
=qk−1p.

Z Z = min(X, Y )FZF
λ t ∈R
FZ(t) = P(min(X, Y )≤t)
= 1 −P(min(X, Y )> t)
= 1 −P(X > t, Y > t)
= 1 −P(X > t)P(Y > t)
= 1 −(1 −F(t))2
X Y X Y
λ F (t)=1−e−λt
t≥0F(t)=0 Z FZ
FZ(t) = 0t < 0
1−e−2λt t≥0
fZZ
fZ(t) = 0t < 0
2λe−2λt t≥0
Z0= max(X, Y )FZ0
Z0t∈R
FZ0(t) = P(Z0≤t) = P(X≤t, Y ≤t) = F(t)2.
FZ0(t) = 0t < 0
1−e−λt2t≥0
fZ0Z0
fZ0(t) = 0t < 0
λe−λt 1−e−λtt≥0
FYY t ∈R
FY(t) = P(Y≤t) = P(aX +b≤t).
a > 0
FY(t) = PX≤t−b
a=Ft−b a.
a < 0
FY(t) = PX≥t−b
a= 1 −PX < t−b a= 1 −PX≤t−b a= 1 −Ft−b a
X
f F a < 0a > 0
fYY
fY(t) = F0
Y(t) = 1
|a|ft−b a
FZZ t ∈RFZ(t) = P(Z≤t) = P(|X| ≤ t)
t < 0FZ(t) = 0 |X|P(|X| ≥ 0) = 1 = 1 −FZ(0) FZ(0) = 0
FZFZ(t)=0 t < 0t≥0
FZ(t) = P(−t≤X≤t) = F(t)−F(−t).
fZZ
fZ(t) = 0t < 0
f(t) + f(−t)t≥0

FTT t ∈R
FT(t) = P(T≤t) = P(|X| ≤ et) = F(et)−F(−et).
fTR
fT(t) = etf(et) + etf(−et),∀t∈R
F
RF(R) =]0,1[ FUU
t∈R
FU(t) = P(F(X)≤t).
t≤0FU(t)=0 t≥1FU(t)=1 t∈]0,1[
FU(t) = PX≤F−1(t).
F F −1
FU(t) = F(F−1(t)) = t.
FU
FU(t) = 
0t≤0
t0<t<1
1t≥1
[0,1] U[0,1]
x∈R[x]x
V V (Ω) = ZV
V(Ω) = Zk∈Z
P(V=k) = P(k≤X < k + 1) = F(k+ 1) −F(k).
t∈R
FV(t) = X
k∈Z,k≤t
P(V=k)
=
[t]
X
k=−∞
P(V=k)
=
[t]
X
k=−∞
F(k+ 1) −F(k)
=F(1 + [t]) .
XN(m, σ2)a, b aX +b
N(am +b, a2σ2)FN(0,1)
t∈RF(t)=1−F(−t)YN(0,1)
F(t) = P(Y≤t) = P(−Y≥ −t)=1−P(−Y < −t)Y∼ N(0,1) −Y∼ N(0,1)
a=−1b= 0) F(t) = 1 −F(−t)
X∼ N(1,25) Y= (X−1)/5N(0,1)
a≥1
P(1 < X < a) = P(0 < Y < (a−1)/5) = F((a−1)/5) −F(0) = F((a−1)/5) −1/2
a F ((a−1)/5) = 1.45 F(x)≤1x