| 112 |
|
Capitolo 8 - Esempi di distribuzioni teoriche |
|
![{\displaystyle =-{\frac {Np}{\sqrt {Npq}}}\,t+N\,\ln \left[1+p\left(e^{\frac {t}{\sqrt {Npq}}}-1\right)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dd58974697ee37b406c3468bc7dcd8c636f75e33)
|
|
|
|
![{\displaystyle \qquad \left.+{\frac {p^{3}}{3}}\left(e^{\frac {t}{\sqrt {Npq}}}-1\right)^{3}+\cdots \right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4bb741764596bc0dbe84619faee54e858279639b)
|
|
![{\displaystyle =-{\frac {Np}{\sqrt {Npq}}}\,t+N\left\{p\left[{\frac {t}{\sqrt {Npq}}}+{\frac {1}{2}}\,{\frac {t^{2}}{Npq}}+{\frac {1}{6}}\,{\frac {t^{3}}{(Npq)^{\frac {3}{2}}}}+\cdots \right]-\right.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/09eaab92f79f7dc6bc7aefb3a0b8d209fbe5e5f6)
|
|
![{\displaystyle \qquad -{\frac {p^{2}}{2}}\left[{\frac {t}{\sqrt {Npq}}}+{\frac {1}{2}}\,{\frac {t^{2}}{Npq}}+{\frac {1}{6}}\,{\frac {t^{3}}{(Npq)^{\frac {3}{2}}}}+\cdots \right]^{2}+}](https://wikimedia.org/api/rest_v1/media/math/render/svg/786ff7f5dcd4aeb5231de89a684d0fdaba9d22ff)
|
|
![{\displaystyle \qquad \left.+{\frac {p^{3}}{3}}\left[{\frac {t}{\sqrt {Npq}}}+{\frac {1}{2}}\,{\frac {t^{2}}{Npq}}+{\frac {1}{6}}\,{\frac {t^{3}}{(Npq)^{\frac {3}{2}}}}+\cdots \right]^{3}+\cdots \right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2646e6fd8e61fcaad821f7191ba3442c73694c87)
|
|
![{\displaystyle =N\left\{p\left[{\frac {1}{2}}\,{\frac {t^{2}}{Npq}}+{\frac {1}{6}}\,{\frac {t^{3}}{(Npq)^{\frac {3}{2}}}}+\cdots \right]-\right.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3193eedbbd1c62267f79fb4ca7070cb0c4a5e8d7)
|
|
![{\displaystyle \qquad \left.-{\frac {p^{2}}{2}}\left[{\frac {t^{2}}{Npq}}+{\frac {t^{3}}{(Npq)^{\frac {3}{2}}}}+\cdots \right]+{\frac {p^{3}}{3}}\left[{\frac {t^{3}}{(Npq)^{\frac {3}{2}}}}+\cdots \right]\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68f49c0bf2a9a945f31fea108bad704cf255da0c)
|
|
![{\displaystyle =N\left[{\frac {1}{2}}\,{\frac {t^{2}}{Npq}}\,p(1-p)+{\frac {t^{3}}{(Npq)^{\frac {3}{2}}}}\left({\frac {p}{6}}-{\frac {p^{2}}{2}}+{\frac {p^{3}}{3}}\right)+{\mathcal {O}}\left(t^{4}N^{-2}\right)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0da045372743c0866e126f251c3ad9da6481a7f7)
|
|
|
ove si è sviluppato in serie di McLaurin prima
e poi
e si sono svolti i prodotti tenendo solo i termini dei primi ordini.
Chiaramente quando 
viene fatto tendere all'infinito tutti i termini eccetto il primo tendono a zero, per cui
