"de x" k

XP=C(λ)λk
k!λ > 0
X GX(z) := Pk>0zkP(X=k)
GX(z)
GX(z)=Pk>0zkP(X=k)
=C(λ)Pk>0
(λz)k
k!
=C(λ)eλz
G0
X(1) + G00 X(1) −G0 X(1)2
G0
X(z) = λGX(z)G00 X(z) = λ2GX(z)G0 X(1) + G00 X(1) −G0 X(1)2=GX(1)(λ+λ2−λ2GX(1))
C(λ) = e−λ
XPk>0P(X=k)=1 GX(1) = Pk>0P(X=k) = C(λ)eλ
C(λ) = e−λ
XE(X) Var (X)
X GX(z) = eλ(z−1)
G0
X(z) = X
k>1
kzk−1P(X=k)G0
X(1) = E(X)
G00
X(z) = X
k>2
k(k−1)zk−2P(X=k) = G00
X(1) = EX2−E(X)
•E(X) = G0
X(1) = λ
•Var (X) = EX2−E(X)2=G0
X(1) + G00 X(1) −G0 X(1)2=λ
XB(n, p)
X(n, p)
GX(z) = Pk>0zkP(X=k)
=Pk>0zkn
kpk(1 −p)n−k
= (pz + (1 −p))n
G0
X(z) = np(zp + 1 −p)n−1G00 X(z) = n(n−1)p2(zp + 1 −p)n−2
E(X) = np Var (X) = np(1 −p)
z7→ Ee−zX 
X7→ EezX 
z7→ EzX=Pk>0zkP(X=k)

N λ
p X
(X, N )
P(X=x N =n) = P(N=n)P(X=x|N=n)
=λn
n!e−λn xpx(1 −p)n−x
X
P(X=x) = Pn>0P(N=n)P(X=x|N=n)
=Pn>x(λn
n!e−λ)(n xpx(1 −p)n−x)
= e−λPn>xn
x(λp)x(λ−λp)n−x
=(λp)x
x!e−λPn>x
(λ−λp)n−x
(n−x)!
=(λp)x
x!e−λPk>0
(λ−λp)k
(k)!
=(λp)x
x!e−λp
X λp
λ=m
k m −16k6m
P(X=k) = mk
k!e−m
R=P(X=k+ 1)
P(X=k)=m k+ 1
R > 1⇐⇒ k < m −1k=bnc
∃k Rk= 1 m∈N