Lois de Probabilités Discrètes Usuelles : Résumé Structuré

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B(p)
B(n, p)
H(N, a, n)
P(λ)
G(p)

B(p)
S p ∈]0,1[
E q = 1 −p
X: Ω → {0,1}X(S) = 1 X(E) = 0
X∼ B(p)
xi
piq= 1 −p p
E(X) = pV(X) = pq =p(1 −p)σ(X) = √pq
▷
p= 0,5
p= 0 p= 1
X= 1
X∼ B1
10 =⇒E(X)=0,1 ; V(X) = 0,09
B(n, p)
n
p X =X1+X2+··· +Xn
X∼ B(n, p)
P(X=k) = n
kpk(1 −p)n−k, k ∈ {0,1, . . . , n}

▷
k n
n
kpkqn−k
E(X) = np V(X) = npq σ(X) = √npq
P(X=k+ 1) P(X=k)
P(X=k+ 1) = n−k
k+ 1 ·p q·P(X=k)
X1∼ B(n1, p)X2∼ B(n2, p)
X1+X2∼ B(n1+n2, p)
p
Fn=X/n
E(Fn) = pV(Fn) = pq
n
n→+∞p
⋆
n
p
p= 0,1
n= 1000
X∼ B(1000,0,1) =⇒E(X) = 100,V(X) = 90, σ(X)≈9,49

r≥2n
r p1, p2, . . . , prPpi= 1
miiPmi=n
P=n!
m1!m2!··· mr!pm1
1pm2 2··· pmr
r
▷
n!
m1!m2!···mr!
n r m1, . . . , mr
r= 2 p1=p
p2=q
p= 0,3p= 0,4p= 0,3
P=36!
15! 18! 3! (0,3)15(0,4)18(0,3)3
H(N, a, n)
N a b =N−a
n n ≤N X
H(N, a, n)
P(X=k) = a
k b n−k
N
n,max(0, n −b)≤k≤min(n, a)
p=a/N q = 1 −p
E(X) = np V(X) = npq ·N−n
N−1
| {z }
ρ: