sviluppare i due logaritmi in serie di McLaurin:
ln
(
1
+
x
)
=
x
−
x
2
2
+
x
3
3
−
x
4
4
+
⋯
{\displaystyle \ln(1+x)\;=\;x-{\frac {x^{2}}{2}}+{\frac {x^{3}}{3}}-{\frac {x^{4}}{4}}+\cdots }
Il primo termine di
ln
κ
{\displaystyle \ln \kappa }
−
(
N
p
+
λ
+
1
2
)
(
λ
N
p
−
λ
2
2
N
2
p
2
+
λ
3
3
N
3
p
3
−
⋯
)
=
{\displaystyle -{\biggl (}Np+\lambda +{\frac {1}{2}}{\biggr )}\left({\frac {\lambda }{Np}}-{\frac {\lambda ^{2}}{2\,N^{2}p^{2}}}+{\frac {\lambda ^{3}}{3\,N^{3}p^{3}}}-\cdots \right)=}
=
−
λ
+
(
λ
2
2
N
p
−
λ
2
N
p
)
−
(
λ
3
3
N
2
p
2
−
λ
3
2
N
2
p
2
+
λ
2
N
p
)
+
⋯
{\displaystyle =-\lambda +\left({\frac {\lambda ^{2}}{2\,Np}}-{\frac {\lambda ^{2}}{Np}}\right)-\left({\frac {\lambda ^{3}}{3\,N^{2}p^{2}}}-{\frac {\lambda ^{3}}{2\,N^{2}p^{2}}}+{\frac {\lambda }{2\,Np}}\right)+\cdots }
=
−
λ
−
λ
2
2
N
p
−
λ
2
N
p
+
λ
3
6
N
2
p
2
+
⋯
{\displaystyle =-\,\lambda -{\frac {\lambda ^{2}}{2\,Np}}-{\frac {\lambda }{2\,Np}}+{\frac {\lambda ^{3}}{6\,N^{2}p^{2}}}+\cdots }
ed il secondo
−
(
N
q
−
λ
+
1
2
)
(
−
λ
N
q
−
λ
2
2
N
2
q
2
−
λ
3
3
N
3
q
3
−
⋯
)
=
{\displaystyle -\left(Nq-\lambda +{\frac {1}{2}}\right)\left(-{\frac {\lambda }{Nq}}-{\frac {\lambda ^{2}}{2\,N^{2}q^{2}}}-{\frac {\lambda ^{3}}{3\,N^{3}q^{3}}}-\cdots \right)=}
=
λ
−
λ
2
2
N
q
+
λ
2
N
q
−
λ
3
6
N
2
q
2
−
⋯
{\displaystyle =\lambda -{\frac {\lambda ^{2}}{2\,Nq}}+{\frac {\lambda }{2\,Nq}}-{\frac {\lambda ^{3}}{6\,N^{2}q^{2}}}-\cdots }
e sommando si ottiene
ln
κ
=
−
λ
2
2
N
p
q
−
λ
2
N
(
1
p
−
1
q
)
+
λ
3
6
N
2
(
1
p
2
−
1
q
2
)
+
⋯
{\displaystyle \ln \kappa =-{\frac {\lambda ^{2}}{2\,Npq}}-{\frac {\lambda }{2\,N}}\left({\frac {1}{p}}-{\frac {1}{q}}\right)+{\frac {\lambda ^{3}}{6\,N^{2}}}\left({\frac {1}{p^{2}}}-{\frac {1}{q^{2}}}\right)+\cdots }
Da questo sviluppo risulta che il solo termine che si mantiene finito al divergere di
N
{\displaystyle N}
λ
{\displaystyle \lambda }
N
p
q
{\displaystyle {\sqrt {Npq}}}
1
/
N
{\displaystyle 1/{\sqrt {N}}}
1
/
N
{\displaystyle 1/N}
|
λ
|
<
3
N
p
q
{\displaystyle |\lambda |<3{\sqrt {Npq}}}
N
{\displaystyle N}
κ
{\displaystyle \kappa }
ln
κ
≈
−
λ
2
2
N
p
q
{\displaystyle \ln \kappa \;\approx \;-\,{\frac {\lambda ^{2}}{2\,Npq}}}
e la probabilità dello scarto dalla media
λ
{\displaystyle \lambda }
P
(
λ
)
≈
1
2
π
N
p
q
e
−
1
2
λ
2
N
p
q
{\displaystyle P(\lambda )\;\approx \;{\frac {1}{\sqrt {2\pi Npq}}}\,e^{-{\frac {1}{2}}{\frac {\lambda ^{2}}{Npq}}}}
per la variabile
M
{\displaystyle M}
P
(
M
)
≈
1
2
π
N
p
q
e
−
1
2
(
M
−
N
p
)
2
N
p
q
.
{\displaystyle P(M)\;\approx \;{\frac {1}{\sqrt {2\pi Npq}}}\,e^{-{\frac {1}{2}}{\frac {(M-Np)^{2}}{Npq}}}.}
