Variables aléatoires : loi et espérance (suite).

X λ ∈R
eλX E[eλX ]
X[a, b]
X θ > 0
XN(0,1)
λ∈ReλX X
[a, b]
E[eλX ] = Zb
a
eλxdx =λ−1[eλx]b
a=λ−1[eλb −eλa].
E[eλX ] = Z+∞
0
e(λ−θ)xdx =+∞λ≥θ,
1
θ−λλ < θ.
y=x−λ/2
1
√2πZ+∞
−∞
eλx−x2/2dx =1
√2πZ+∞
−∞
eλy2/2−λ2/4dy =e−λ2/4.
XNX2
X X
m≥1X
NXkk1mE[Xm+1] = +∞
|x| ≤ 1 + x2x
E[|X|]≤1 + E[X2]
X
X

s > 1, ζ(s) = Pn≥1
1
nsX:
(Ω,F,P)→N∗n≥1
P(X=n) = 1
ζ(m+ 2)
1
nm+2 .
E[Xm] = 1
ζ(m+ 2) X
n≥1
1
n2=ζ(2) ζ(m+ 2) <+∞,
X m
E[Xm+1] = 1
ζ(m+ 2) X
n≥1
1
n= +∞,
X m + 1
f:R→Rf(t) = 1
π
1
1 + t2
fR
X f
X
X
Y= arctan(X)
t= arctan xarctan0=1
1+arctan2
Z+∞
−∞
f(x)dx =1
πZπ/2
−π/2
dθ = 1.
f
x
1+x2∼x X
P(X≤a) = Za
−∞
f(x)dx =1
πZarctan a
−π/2
dθ = arctan a+π/2.
b∈[−π/2, π/2]
P(Y≤b) = P(X≤tan b) = Ztan b
−∞
f(x)dx =1
πZb
−π/2
dθ =b+π/2.
Y[−π/2, π/2]

XN(0,1)
n∈NXnE[Xn]
n≥0E[Xn]n
{1, . . . , n}
f(x) = 1
√2πe−x2
2n≥1x7→ |x|ne−x2 20
x+∞ −∞ n≥1x7→ |x|n+2e−x2
2
0|x|ne−x2
2=O(1
x2) +∞ −∞ R+∞
−∞ |x|ne−x2
2dx
n≥0
mn=1
√2πZ+∞
−∞
xne−x2
2dx.
n mnmn= 0
y=−x
mn=−mn
n= 0 m0m0= 1
n≥2n= 2p
R > 0
1
√2πZ+R
−R
x2pe−x2
2dx =1
√2πZ+R
−R
x2p−1
|{z}
u
xe−x2
2
| {z }
v0
dx
=1
√2πh−x2p−1e−x2
2iR
−R+ (2p−1) 1
√2πZ+R
−R
x2p−2e−x2
2dx.
R+∞m2p= (2p−1)m2p−2
m2p= (2p−1)(2p−3) . . . 3.1 (2p)!! (2p)!
2pp!
∀n≥1, mn=0n
(2p)!! = (2p)!
2pp!n= 2p.
n n −1
n−2
n= 2
n n
n
θ > 0X
θ n ≥1Xn
E[Xn]θ= 1

n= 0 E[X0] = 1 n≥1
E[Xn] = θZ+∞
0
xne−θxdx =θxne−θx
−θ+∞
0
+θn Z+∞
0
xn−1e−θxdx =n
θE[Xn−1].
E[Xn] = n!θ−n
θ= 1 E[Xn] = n!n
λ > 0X
λ k ≥1X(X−1) . . . (X−k+ 1)
E[Xm]m∈ {1,2,3,4}
mE[Xm]m
λ= 1 m≥1
Yk:= X(X−1) . . . (X−k+ 1)
Yk= 0 X= 0, . . . , k −1
E[Yk] = e−λX
i≥k
i(i−1) . . . (i−k+ 1)λi
i!=e−λλkX
i≥k
λi−k
(i−k)! =λk<+∞.
YkXk
X
X=Y1, X2=Y2+Y1, X3=Y3+ 3X2−2X=Y3+ 3Y2+Y1,
X4=Y4+ 6X3−11X2+ 6X=Y4+ 6Y3+ 7X2−6X=Y4+ 6Y3+ 7Y2+Y.
E[Yk] = λk
X
E[X] = λ, E[X2] = λ2+λ, E[X3] = λ3+ 3λ2+λ, E[X4] = λ4+ 6λ3+ 7λ2+λ.
λ= 1
E[X]=1,E[X2] = 3,E[X3] = 5,E[X4] = 15.
E[Xm]
m λk
k m = 4 k= 4
k= 3

k= 2
k= 1
X
E[X]>0P(X > 0) >0
X: (Ω,F,P)→R+
n≥1An∈FAn={X≥1
n}
(An)n≥1Sn≥1An={X > 0}
P(X > 0) P(X≥1
n)n
P(X > 0) >0n≥1P(X≥1
n)>0
E[X]≥EhX{X≥1
n}i≥1
nEh{X≥1
n}i≥1
nPX≥1 n>0.
X
λ, µ > 0 Ω = N2F=P(N2)
(Ω,F)P
∀(n, m)∈N2,P({(n, m)}) = e−(λ+µ)λn
n!
µm
m!.
(Ω,F,P)X(n, m) = n Y (n, m) = m
P(Ω) = 1
X Y
X+Y
m n
µ
X
m≥1
P({n, m}) = e−λλn
n!X
m≥1
e−µµm
m!=e−λλn n!,
X
n≥1X m≥1
P({n, m}) = X
n≥1
e−λλn
n!= 1.