resumé des lois de probabilités

E(X) var (X)E(zX)
P(X= 0) = q, P(X= 1) = p
q= 1 −p
p pq pz +q
B(n, p)P(X=k) = Ck
npkqn−k
q= 1 −p, k = 0,1, . . . , n
np npq (pz +q)n
P(λ)P(X=k) = e−λλk
k!
k= 0,1, . . .
λ λ eλ(z−1)
G(p)P(X=k) = pqk−1
q= 1 −p, k = 1,2, . . .
1
p
q
p2
pz
1−qz
H(N, n, p)
P(X=k) = Ck
Np Cn−k
Nq
Cn
N
q= 1 −p
max(0, n −Nq)6k6min(Np, n)
np npq N−n
N−1
Cn
Nq
Cn
N
F(−n, −Np;Nq −n+ 1; z)
P(X=k) = Cr−1
k+r−1prqk
q= 1 −p, k = 0,1, . . .
rq
p
rq
p2p 1−qz r
P(X=k) = Cr−1
k−1prqk−r
q= 1 −p, k =r, r + 1, . . .
r
p
rq
p2pz 1−qz r
F(a, b;c;z) =
+∞
n=0
a(a+ 1) . . . (a+n−1) b(b+ 1) . . . (b+n−1)
c(c+ 1) ...(c+n−1)
zn
n!
•n p B(n, p)
• B(m, p)B(n, p)B(m+n, p)
• P(λ)P(µ)P(λ+µ)
•(r, p) (s, p)
(r+s, p)
•rG(p) (r, p)

E(X) var (X)E(eitX )
U(a, b)1
b−a1[a,b](x)a+b 2
(b−a)2
12
eibt −eiat
i(b−a)t
E(λ)λe−λx1R+(x)1
λ
1
λ2
λ
λ−it
N(m, σ2)1
√2π σ exp −(x−m)2
2σ2m σ2eimt−1
2σ2t2
W(λ, a)λaxa−1e−λxa1]0,+∞[(x)λ−1
aΓ1
a+ 1λ−2
a[Γ 2
a+ 1
−Γ1
a+ 12]
C(a, b)a
π(a2+ (x−b)2)eibt−a|t|
Γ(a, λ)λa
Γ(a)xa−1e−λx1]0,+∞[(x)a
λ
a
λ2λ λ−it a
B(a, b)1
B(a, b)xa−1(1 −x)b−11]0,1[(x)a
a+b
ab
(a+b)2(a+b+ 1) M(a, a +b;it)
χ2(n)1
2n
2Γ(n
2)xn
2−1e−x 21]0,+∞[(x)n2n(1 −2it)−n/2
T(n)Γ(n+1
2)
√πn Γ(n
2)1 + x2
n−n+1
2
0n > 1n
n−2n > 22 Γ(n
2)|t|√n
2
n
2
Kn
2(|t|√n)
F(m, n)mm
2nn 2
B(m
2,n 2)
xm
2−1
(mx +n)m+n
2
1]0,+∞[(x)
n
n−2
n > 2
2n2(m+n−2)
m(n−4)(n−2)2
n > 4
Mm
2;−n 2;−n mit
Γ(a) = +∞
0
xa−1e−xdx
B(a, b) = 1
0
xa−1(1 −x)b−1dx
M(a;b;z) =
+∞
n=0
a(a+ 1) . . . (a+n−1)
b(b+ 1) . . . (b+n−1)
zn
n!
Kν(z) = π
2
I−ν(z)−Iν(z)
sin πν Iν(z) = z 2ν+∞
n=0
1
n! Γ(n+ν+ 1) z2 4n
•nE(λ) Γ(n, λ)
•Γ(a, λ) Γ(b, λ) Γ(a+b, λ)
•X Y Γ(a, λ) Γ(b, λ)X
X+YB(a, b)
• N(m1, σ2
1)N(m2, σ2 2)
N(m1+m2, σ2
1+σ2 2)
• N(0,1) C(1,0) = T(1)
•nN(0,1) χ2(n) = Γ(n
2,1 2)
•X Y N(0,1) χ2(n)X
√Y/n
T(n)
•X Y χ2(m)χ2(n)mX
nY F(m, n)