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Gaussiana


$\displaystyle G(t\,\vert\,{\cal N}(\mu,\sigma))$ $\displaystyle =$ $\displaystyle \int_{-\infty}^{+\infty}
\frac{e^{tx}}{\sqrt{2\pi}\sigma}
e^{-\frac{(x-\mu)^2}{2\sigma^2}}$d$\displaystyle x$  
  $\displaystyle =$ $\displaystyle \int_{-\infty}^{+\infty}
\frac{1}{\sqrt{2\pi}\sigma}
e^{-(x^2+\mu^2-2\, x\, \mu-2\, \sigma^2\, t\, x)/2\sigma^2}$d$\displaystyle x\,.$  

Riscriviamo il numeratore del termine all'esponente nella seguente forma:

$\displaystyle x^2-2\, (\mu+\sigma^2\, t)x + (\mu+\sigma^2\, t)^2-
(\mu+\sigma^2\, t)^2+\mu^2\,. $

Introducendo il nuovo parametro $ \mu^\prime=\mu+\sigma^2\, t$ otteniamo:
$\displaystyle G(t\,\vert\,{\cal N}(\mu,\sigma))$ $\displaystyle =$ $\displaystyle e^{\frac{(\mu+\sigma^2\, t)^2-\mu^2}{2\sigma^2}}
\int_{-\infty}^{+\infty}
\frac{1}{\sqrt{2\pi}\sigma}
e^{-\frac{(x-\mu^\prime)^2}{2\sigma^2}}$d$\displaystyle x\, ,$  

da cui, finalmente:
$\displaystyle G(t\,\vert\,{\cal N}(\mu,\sigma))$ $\displaystyle =$ $\displaystyle e^{\mu\, t+\sigma^2\, t^2/2}$  
       
E$\displaystyle (X)$ $\displaystyle =$ $\displaystyle \left. (\mu+\sigma^2\, t)\,
e^{\mu\, t+\sigma^2\, t^2/2}\right\vert _{t=0} = \mu$  
E$\displaystyle (X^2)$ $\displaystyle =$ $\displaystyle \left. \left[(\mu+\sigma^2\, t)^2+\sigma^2\right]
\, e^{\mu\, t+\sigma^2\, t^2/2} \right\vert _{t=0}
= \mu^2+\sigma^2$  
Var$\displaystyle (X)$ $\displaystyle =$ $\displaystyle \sigma^2\,.$  



Giulio D'Agostini 2001-04-02